Quantum entanglement for systems of identical bosons: I. General features

Here the commonly applied physically based approach to mathematically defining entangled states will be described [18]. The definition involves vectors and density operators that represent states than can be prepared in real experiments, so the mathematical approach is to be physically based. The concept of quantum entanglement involves composite systems made up of component sub-systems each of which are distinguishable from the other sub-systems, and where each could constitute a stand-alone quantum system. This means the each sub-system will have its own set of physically realizable quantum states—mixed or pure—which could be prepared independently of the quantum states of the other sub-systems. As will be seen, the requirement that sub-systems be distinguishable and their states be physically preparable will have important consequences, especially in the context of identical particle systems. The formal definition of what is meant by an entangled state starts with the pure states, described via a vector in a Hilbert space. The formalism of quantum theory allows for pure states for composite systems made up of two or more distinct sub-systems via tensor products of sub-system states

$$\begin{eqnarray}&&| {\rm{\Phi }}\rangle =| {{\rm{\Phi }}}_{A}\rangle \otimes | {{\rm{\Phi }}}_{B}\rangle \otimes | {{\rm{\Phi }}}_{C}\rangle ....\end{eqnarray} \tag{ 1 }$$

Such products are called non-entangled or separable states. However, since these product states exist in a Hilbert space, it follows that linear combinations of such products of the form

$$\begin{eqnarray}&&| {\rm{\Phi }}\rangle =\sum _{\alpha \beta \gamma ..}{C}_{\alpha \beta \gamma ..}| {{\rm{\Phi }}}_{A}^{\alpha }\rangle \otimes | {{\rm{\Phi }}}_{B}^{\beta }\rangle \otimes | {{\rm{\Phi }}}_{C}^{\gamma }\rangle ...\end{eqnarray} \tag{ 2 }$$

could also represent possible pure quantum states for the system. Such quantum superpositions which cannot be expressed as a single product of sub-system states are known as entangled (or non-separable) states.

The concept of entanglement can be extended to mixed states, which are described via density operators in the Hilbert space. If A, B, ... are the sub-systems with ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, being density operators the sub-systems A, B, then a general non-entangled or separable state is one where the overall density operator $\widehat{\rho }$ can be written as the weighted sum of tensor products of these sub-system density operators in the form [7]

$$\begin{eqnarray}&&\widehat{\rho }=\displaystyle \sum _{R}{P}_{R}\,{\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}\otimes {\widehat{\rho }}_{R}^{C}\otimes ...\end{eqnarray} \tag{ 3 }$$

with ${\sum }_{R}{P}_{R}=1$ and ${P}_{R}\geqslant 0$ giving the probability that the specific product state ${\widehat{\rho }}_{R}={\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}\otimes {\widehat{\rho }}_{R}^{C}\otimes ...$ occurs. It is assumed that at least in principle such separable states can be prepared [7]. This implies the possibility of turning off the interactions between the different sub-systems, a task that may be difficult in practice except for well-separated sub-systems. Entangled states (or non-separable states) are those that cannot be written in this form, so in this approach knowing what the term entangled state refers to is based on first knowing what the general form is for a non-entangled state. The density operator $\widehat{\rho }=| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}|$ for the pure state in (2) is not of the form (3), as there are cross terms of the form ${C}_{\alpha \beta \gamma ..}{C}_{\theta \lambda \eta ..}^{* }(| {{\rm{\Phi }}}_{A}^{\alpha }\rangle \langle {{\rm{\Phi }}}_{A}^{\theta }| )\otimes (| {{\rm{\Phi }}}_{B}^{\beta }\rangle \langle {{\rm{\Phi }}}_{B}^{\lambda }| )\otimes ...$ involved.

The concepts of separability and entanglement based on the equations (1) and (3) for non-entangled states do not however just rest on the mathematical forms alone. Implicitly there is the assumption that separable quantum states described by the two expressions can actually be created in physical processes. The sub-systems involved must therefore be distinguishable quantum systems in their own right, and the sub-system states $| {{\rm{\Phi }}}_{A}\rangle ,| {{\rm{\Phi }}}_{B}\rangle ,\ldots$ or ${\widehat{\rho }}_{R}^{A},{\widehat{\rho }}_{R}^{B},\ldots$ must also be possible quantum states for the sub-systems. We will return to these requirements later. The issue of the physical preparation of non-entangled (separable) states starting from some uncorrelated fiducial state for the separate sub-systems was introduced by Werner [7], and discussed further by Bartlett et al (see [54], section IIB). This involves the ideas of LOCC dealt with in the next section.

The key requirement is that entangled states exhibit a novel quantum feature that is only found in composite systems. Separable states are such that the joint probability for measurements of all physical quantities associated with the sub-systems can be found from separate measurement probabilities obtained from the sub-system density operators ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, etc and the overall classical probability P_R (see section 2.3). This feature of separable probabilities is absent in certain entangled states, and because of this key non-separability feature Schrödinger called these states 'entangled'. The separability feature for the joint probabilities is essentially a classical feature and applies in HVTs (see section 2.5) applied to quantum systems—as well as to quantum separable states. The fact that entangled states are quantum states that can exhibit the failure of this separability feature for classical LHV theories highlights entanglement being a non-classical feature for composite systems.

An alternative operational approach to defining entangled states focuses on whether or not they exhibit certain non-classical features such as Bell Inequality violation or whether they satisfy certain mathematical tests such as having a non-negative partial transpose [36, 47], and a utilitarian approach focuses or whether entangled states have technological applications such as in various quantum information protocols. As will be seen in section 3.4, the particular definition of entangled states based on their non-creatability via LOCC essentially coincides with the approach used in the present paper. It has been realized for some time that different types of entangled states occur, for example states in which a Bell inequality is not violated or states demonstrating an EPR paradox [83]. Wiseman et al [27–29] and Reid et al [19, 20, 30, 84] discuss the concept of a hierarchy of entangled states, with states exhibiting Bell non-locality being a subset of states for which there is EPR steering, which in turn is a subset of all the entangled states, the latter being defined as states whose density operators cannot be written as in equation (3) though without further consideration if additional properties are required for the sub-system density operators. The operational approach could lead into a quagmire of differing interpretations of entanglement depending on which non-classical feature is highlighted, and the utilitarian approach implies that all entangled states have a technological use—which is by no means the case. For these reasons, the present physical approach based on the quantities involved representing allowed sub-system states is generally favored [18]. It is also compatible with later classifying entangled states in a hierarchy.

Finally, we should mention that in addition to quantum entanglement, there is a body of work dealing with so-called classical entanglement. This is essentially of mathematical rather than physical interest, but for completeness a brief summary is presented in appendix B.

As pointed out by Vedral [17], one reason for calling states such as in equations (1) and (3) separable is associated with the idea of performing operations on the separate sub-systems that do not affect the other sub-systems. Such operations on such local systems are referred to as local operations and include unitary operations ${\widehat{U}}_{A}$, ${\widehat{U}}_{B}$, that change the states via ${\widehat{\rho }}_{R}^{A}\to {\widehat{U}}_{A}{\widehat{\rho }}_{R}^{A}{\widehat{U}}_{A}^{-1}$, ${\widehat{\rho }}_{R}^{B}\to {\widehat{U}}_{B}{\widehat{\rho }}_{R}^{B}{\widehat{U}}_{B}^{-1}$, etc as in a time evolution, and could include processes by which the states ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, are separately prepared from suitable initial states.

We note that performing local operations on a separable state only produces another separable state, not an entangled state. Such local operations are obviously facilitated in experiments if the sub-systems are essentially non-interacting-such as when they are spatially well-separated, though this does not have to be the case. The local systems and operations could involve sub-systems whose quantum states and operators are just in different parts of Hilbert space, such as for cold atoms in different hyperfine states even when located in the same spatial region. Note the distinction between local and localized. As described by Werner [7], if one observer (Alice) is associated with preparing separate sub-system A in an allowed quantum state ${\widehat{\rho }}_{R}^{A}$ via local operations with a probability P_R, a second observer (Bob) could be then advised via a classical communication channel to prepare sub-system B in state ${\widehat{\rho }}_{R}^{B}$ via local operations. After repeating this process for different choices R of the correlated pairs of sub-system states, the overall quantum state prepared by both observers via this local operation and classical communication protocol (${\text{}}{\rm{LOCC}})$ would then be the bipartite non-entangled state $\widehat{\rho }={\sum }_{R}{P}_{R}{\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}$. Multipartite non-entangled states of the form (3) can also be prepared via LOCC protocols involving further observers. As will be seen, the separable or non-entangled states are just those that can be prepared by LOCC protocols.

A key issue however is whether density operators $\widehat{\rho }$ and ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, in equation (3) always represent possible quantum states, even if the operators $\widehat{\rho }$ and ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, etc satisfy all the standard mathematical requirements for density operators—Hermitiancy, positiveness, trace equal to unity, trace of density operator squared being not greater than unity. In this paper it will be argued that for systems of identical massive particles there are further requirements not only on the overall density operator, but also (for separable states) on those for the individual sub-systems that are imposed by symmetrization and SSRs.

For situations involving distinct sub-systems measurements can be carried out on all the sub-systems and the results expressed in terms of the joint probability for various outcomes. If ${\widehat{{\rm{\Omega }}}}_{A}$ is a physical quantity associated with sub-system A, with eigenvalues ${\lambda }_{i}^{A}$ and with ${\widehat{{\rm{\Pi }}}}_{i}^{A}$ the projector onto the subspace with eigenvalue ${\lambda }_{i}^{A}$, ${\widehat{{\rm{\Omega }}}}_{B}$ is a physical quantity associated with sub-system B, with eigenvalues ${\lambda }_{j}^{B}$ and with ${\widehat{{\rm{\Pi }}}}_{j}^{B}$ the projector onto the subspace with eigenvalue ${\lambda }_{j}^{B}$ etc, then the joint probability ${P}_{{AB}..}(i,j,\ldots )$ that measurement of ${\widehat{{\rm{\Omega }}}}_{A}$ leads to result ${\lambda }_{i}^{A}$, measurement of ${\widehat{{\rm{\Omega }}}}_{B}$ leads to result ${\lambda }_{j}^{B}$, etc is given by

$$\begin{eqnarray}&&{P}_{{AB}..}(i,j,\ldots )={\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{i}^{A}\,{\widehat{{\rm{\Pi }}}}_{j}^{B}\,\ldots \widehat{\rho }).\end{eqnarray} \tag{ 4 }$$

This joint probability depends on the full density operator $\widehat{\rho }$ representing the allowed quantum state as well as on the quantities being measured. Here the projectors (strictly ${\widehat{{\rm{\Pi }}}}_{i}^{A}\otimes {\widehat{1}}^{B}\otimes \,\ldots \,$, ${\widehat{1}}^{A}\otimes {\widehat{{\rm{\Pi }}}}_{j}^{B}\otimes ...$, etc) commute, so the order of measurements is immaterial. An alternative notation in which the physical quantities are also specified is ${P}_{{AB}..}({\widehat{{\rm{\Omega }}}}_{A},i;{\widehat{{\rm{\Omega }}}}_{B},j;\ldots )$.

The reduced density operator ${\widehat{\rho }}_{A}$ for sub-system A given by

$$\begin{eqnarray}&&{\widehat{\rho }}_{A}={{\rm{Tr}}}_{B,C,\ldots }(\widehat{\rho })\end{eqnarray} \tag{ 5 }$$

and enables the results for measurements on sub-system A to be determined for the situation where the results for all joint measurements involving the other sub-systems are discarded. The probability P_A(i) that measurement of ${\widehat{{\rm{\Omega }}}}_{A}$ leads to result ${\lambda }_{i}^{A}$ irrespective of the results for measurements on the other sub-systems is given by

$$\begin{eqnarray}{P}_{A}(i) & = & \displaystyle \sum _{j,k,\ldots }{P}_{{AB}..}(i,j,\ldots )\\ & = & {\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{i}^{A}\,\widehat{\rho })\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&=\,{{\rm{Tr}}}_{A}({\widehat{{\rm{\Pi }}}}_{i}^{A}\,{\widehat{\rho }}_{A})\end{eqnarray} \tag{ 7 }$$

using ${\sum }_{j}{\widehat{{\rm{\Pi }}}}_{j}^{B}=\widehat{1}$, etc. Hence the reduced density operator ${\widehat{\rho }}_{A}$ plays the role of specifying the quantum state for mode A considered as a separate sub-system, even if the original state $\widehat{\rho }$ is entangled. An alternative notation in which the physical quantity is also specified is ${P}_{A}({\widehat{{\rm{\Omega }}}}_{A},i)$.

The mean value for measuring a physical quantity ${\widehat{{\rm{\Omega }}}}_{A}$ will be given by

$$\begin{eqnarray}\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle & = & \displaystyle \sum _{{\lambda }_{i}^{A}}{\lambda }_{i}^{A}{P}_{A}(i)\\ & = & {{\rm{Tr}}}_{A}({\widehat{{\rm{\Omega }}}}^{A}\,{\widehat{\rho }}_{A}),\end{eqnarray} \tag{ 8 }$$

where we have used ${\widehat{{\rm{\Omega }}}}^{A}={\sum }_{{\lambda }_{i}^{A}}{\lambda }_{i}^{A}{\widehat{{\rm{\Pi }}}}_{i}^{A}$.

The variance of measurements of the physical quantity ${\widehat{{\rm{\Omega }}}}_{A}$ will be given by

$$\begin{eqnarray}\langle {({\rm{\Delta }}{\widehat{{\rm{\Omega }}}}^{A})}^{2}\rangle & = & \displaystyle \sum _{{\lambda }_{i}^{A}}{({\lambda }_{i}^{A}-\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle )}^{2}{P}_{A}(i)\\ & = & {{Tr}}_{A}({({\widehat{{\rm{\Omega }}}}^{A}-\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle )}^{2}\,{\widehat{\rho }}_{A})\end{eqnarray} \tag{ 9 }$$

so both the mean and variance only depend on the reduced density operator ${\widehat{\rho }}_{A}$.

On the other hand the mean value of a product of sub-system operators ${\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes ...$, where ${\widehat{{\rm{\Omega }}}}_{A}$, ${\widehat{{\rm{\Omega }}}}_{B}$, ${\widehat{{\rm{\Omega }}}}_{C}$, .. are Hermitian operators representing physical quantities for the separate sub-systems, is given by

$$\begin{eqnarray}\langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes .\rangle & = & \displaystyle \sum _{{\lambda }_{i}^{A}}\displaystyle \sum _{{\lambda }_{j}^{B}}...{\lambda }_{i}^{A}{\lambda }_{j}^{B}...{P}_{{AB}..}(i,j,\ldots )\\ & = & {Tr}({\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes .)\widehat{\rho }\end{eqnarray} \tag{ 10 }$$

which involves the overall system density operator, as expected.

Treating the case of two sub-systems for simplicity we can use Bayes theorem (see appendix C, equation (103)) to obtain expressions for conditional probabilities [18]. The conditional probability that if measurement of ${\widehat{{\rm{\Omega }}}}_{B}$ associated with sub-system B leads to eigenvalue ${\lambda }_{j}^{B}$ then measurement of ${\widehat{{\rm{\Omega }}}}_{A}$ associated with sub-system A leads to eigenvalue ${\lambda }_{i}^{A}$ is given by

$$\begin{eqnarray}&&{P}_{{AB}}(i| j)={\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{i}^{A}\,{\widehat{{\rm{\Pi }}}}_{j}^{B}\,\widehat{\rho })/{\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{j}^{B}\widehat{\rho }).\end{eqnarray} \tag{ 11 }$$

In general, the overall density operator is required to determine the conditional probability. An alternative notation in which the physical quantities are also specified is ${P}_{{AB}}({\widehat{{\rm{\Omega }}}}_{A},i| {\widehat{{\rm{\Omega }}}}_{B},j)$.

As shown in appendix C the conditional probability is given by

$$\begin{eqnarray}&&{P}_{{AB}}(i| j)={\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{i}^{A}{\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B},{\lambda }_{j}^{B})),\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B},{\lambda }_{j}^{B})={\widehat{{\rm{\Pi }}}}_{j}^{B}\,\widehat{\rho }\,{\widehat{{\rm{\Pi }}}}_{j}^{B}/{\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{j}^{B}\widehat{\rho })\end{eqnarray} \tag{ 13 }$$

is the so-called conditioned density operator, corresponding the quantum state produced following the measurement of ${\widehat{{\rm{\Omega }}}}_{B}$ that obtained the result ${\lambda }_{j}^{B}$. The conditional probability result is the same as

$$\begin{eqnarray}&&{P}_{{AB}}(i| j)={\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{i}^{A}{\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B},{\lambda }_{j}^{B}))\end{eqnarray} \tag{ 14 }$$

which is the same as the expression (6) with $\widehat{\rho }$ replaced by ${\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B},{\lambda }_{j}^{B})$. This is what would be expected for a conditioned measurement probability.

Also, if the measurement results for ${\widehat{{\rm{\Omega }}}}_{B}$ are not recorded the conditioned density operator now becomes

$$\begin{eqnarray}{\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B}) & = & \displaystyle \sum _{{\lambda }_{j}^{B}}{P}_{B}(j){\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B},{\lambda }_{j}^{B})\\ & = & \displaystyle \sum _{{\lambda }_{j}^{B}}{\widehat{{\rm{\Pi }}}}_{j}^{B}\,\widehat{\rho }\,{\widehat{{\rm{\Pi }}}}_{j}^{B}.\end{eqnarray} \tag{ 15 }$$

This is still different to the original density operator $\widehat{\rho }$ because a measurement of ${\widehat{{\rm{\Omega }}}}_{B}$ has occurred, even if we don't know the outcome. However, the measurement probability for ${\widehat{{\rm{\Omega }}}}_{A}$ is now

$$\begin{eqnarray}{P}_{{AB}}({i}| {Any}\,j) & = & {\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{{i}}^{A}{\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B}))\\ & = & {\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{{i}}^{A}\widehat{\rho })\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&=\,{P}_{A}(i),\end{eqnarray} \tag{ 17 }$$

where we have used the cyclic properties of the trace, ${({\widehat{{\rm{\Pi }}}}_{j}^{B})}^{2}={\widehat{{\rm{\Pi }}}}_{j}^{B}$ and ${\sum }_{{\lambda }_{j}^{B}}{\widehat{{\rm{\Pi }}}}_{j}^{B}=\widehat{1}$. The results in equations (16) and (17) are the same as the measurement probability for ${\widehat{{\rm{\Omega }}}}_{A}$ if no measurement for ${\widehat{{\rm{\Omega }}}}_{B}$ had taken place at all. This is perhaps not surprising, since the record of the latter measurements was discarded. Another way of showing this result is that Bayes theorem tells us that ${\sum }_{j}{P}_{{AB}}(i| j){P}_{B}(j)={\sum }_{j}{P}_{{AB}}(i,j)={P}_{A}(i)$, since ${\sum }_{j}{P}_{{AB}}(i,j)$ is the probability that measurement of ${\widehat{{\rm{\Omega }}}}_{A}$ will lead to ${\lambda }_{i}^{A}$ and measurement of ${\widehat{{\rm{\Omega }}}}_{B}$ will lead to any of the ${\lambda }_{j}^{B}$. This result is called the no-signalling theorem [18].

Also, as ${P}_{{AB}}(i| {Any}\,j)={\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{i}^{A}{\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B}))$ we see from (7) that

$$\begin{eqnarray}&&{\widehat{\rho }}_{A}={{\rm{Tr}}}_{B}({\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B}))\end{eqnarray} \tag{ 18 }$$

showing that the trace over B of the conditioned density operator for the state obtained by measuring any observable ${\widehat{{\rm{\Omega }}}}_{A}$ and then discarding the results just gives the reduced density operator for sub-system A.

As explained in appendix C, to determine the conditioned mean value of $\widehat{{\rm{\Lambda }}}$ after measurement of $\widehat{{\rm{\Omega }}}$ has led to the eigenvalue ${\lambda }_{i}$ we use ${\widehat{\rho }}_{{\rm{cond}}}(\widehat{{\rm{\Omega }}},i)$ rather than $\,\widehat{\rho }$ in the mean formula $\langle \widehat{{\rm{\Lambda }}}\rangle ={\rm{Tr}}(\widehat{{\rm{\Lambda }}}\widehat{\rho })$ and the result is given in terms of the conditional probability $P(\widehat{{\rm{\Lambda }}}j| \widehat{{\rm{\Omega }}}i)$. Here we refer to two commuting observables and include the operators in the notation to avoid any misinterpretation. Hence

$$\begin{eqnarray}{\langle \widehat{{\rm{\Lambda }}}\rangle }_{\widehat{{\rm{\Omega }}},i} & = & {\rm{Tr}}(\widehat{{\rm{\Lambda }}}{\widehat{\rho }}_{{\rm{cond}}}(\widehat{{\rm{\Omega }}},i))\\ & = & \displaystyle \sum _{j}{\mu }_{j}\,P(\widehat{{\rm{\Lambda }}},j| \widehat{{\rm{\Omega }}},i).\end{eqnarray} \tag{ 19 }$$

For the conditioned variance of $\widehat{{\rm{\Lambda }}}$ after measurement of $\widehat{{\rm{\Omega }}}$ has led to the eigenvalue ${\lambda }_{i}$ we use ${\widehat{\rho }}_{{\rm{cond}}}(\widehat{{\rm{\Omega }}},i)$ rather than $\,\widehat{\rho }$ and the conditioned mean ${\langle \widehat{{\rm{\Lambda }}}\rangle }_{\widehat{{\rm{\Omega }}},i}$ rather than $\langle \widehat{{\rm{\Lambda }}}\rangle$ in the variance formula $\langle {\rm{\Delta }}{\widehat{{\rm{\Lambda }}}}^{2}\rangle ={\rm{Tr}}({(\widehat{{\rm{\Lambda }}}-\langle \widehat{{\rm{\Lambda }}})}^{2}\widehat{\rho })$. Hence

$$\begin{eqnarray}{\langle {\rm{\Delta }}{\widehat{{\rm{\Lambda }}}}^{2}\rangle }_{\widehat{{\rm{\Omega }}},i} & = & {Tr}({(\widehat{{\rm{\Lambda }}}-{\langle \widehat{{\rm{\Lambda }}}\rangle }_{\widehat{{\rm{\Omega }}},i})}^{2}{\widehat{\rho }}_{{\rm{cond}}}(\widehat{{\rm{\Omega }}},i))\\ & = & \displaystyle \sum _{j}{({\mu }_{j}-{\langle \widehat{{\rm{\Lambda }}}\rangle }_{\widehat{{\rm{\Omega }}},i})}^{2}\,P(\widehat{{\rm{\Lambda }}},j| \widehat{{\rm{\Omega }}},i).\end{eqnarray} \tag{ 20 }$$

If we weighted the conditioned mean by the probability $P(\widehat{{\rm{\Omega }}},i)$ that measuring $\widehat{{\rm{\Omega }}}$ has led to the eigenvalue ${\lambda }_{i}$ and summed over the possible outcomes ${\lambda }_{i}$ for the $\widehat{{\rm{\Omega }}}$ measurement, then we obtain the mean for measurements of $\widehat{{\rm{\Lambda }}}$ after unrecorded measurements of $\widehat{{\rm{\Omega }}}$ have occurred. From Bayes theorem ${\sum }_{i}P(\widehat{{\rm{\Lambda }}},j| \widehat{{\rm{\Omega }}},i)P(\widehat{{\rm{\Omega }}},i)=P(\widehat{{\rm{\Lambda }}},j)$ so this gives the unrecorded mean ${\langle \widehat{{\rm{\Lambda }}}\rangle }_{\widehat{{\rm{\Omega }}}}$ as

$$\begin{eqnarray}{\langle \widehat{{\rm{\Lambda }}}\rangle }_{\widehat{{\rm{\Omega }}}} & = & \displaystyle \sum _{i}{\langle \widehat{{\rm{\Lambda }}}\rangle }_{\widehat{{\rm{\Omega }}},i}P(\widehat{{\rm{\Omega }}},i)\\ & = & \displaystyle \sum _{j}{\mu }_{j}\,P(\widehat{{\rm{\Lambda }}},j)\\ & = & \langle \widehat{{\rm{\Lambda }}}\rangle \end{eqnarray} \tag{ 21 }$$

which is the usual mean value for measurements of $\widehat{{\rm{\Lambda }}}$ when no measurements of $\widehat{{\rm{\Omega }}}$ have occurred. Note that no such similar result occurs for the unrecorded variance ${\langle {\rm{\Delta }}{\widehat{{\rm{\Lambda }}}}^{2}\rangle }_{\widehat{{\rm{\Omega }}}}$

$$\begin{eqnarray}{\langle {\rm{\Delta }}{\widehat{{\rm{\Lambda }}}}^{2}\rangle }_{\widehat{{\rm{\Omega }}}} & = & \displaystyle \sum _{i}{\langle {\rm{\Delta }}{\widehat{{\rm{\Lambda }}}}^{2}\rangle }_{\widehat{{\rm{\Omega }}},i}P(\widehat{{\rm{\Omega }}},i)\\ & \ne & \langle {\rm{\Delta }}{\widehat{{\rm{\Lambda }}}}^{2}\rangle .\end{eqnarray} \tag{ 22 }$$

In the case of the general non-entangled state we find that the joint probability is

$$\begin{eqnarray}&&{P}_{{AB}..}(i,j,\ldots )=\displaystyle \sum _{R}{P}_{R}\,{P}_{A}^{R}(i){P}_{B}^{R}(j)\ldots ,\end{eqnarray} \tag{ 23 }$$

where

$$\begin{eqnarray}&&{P}_{A}^{R}(i)={Tr}({\widehat{{\rm{\Pi }}}}_{i}^{A}\,{\widehat{\rho }}_{R}^{A})\qquad {P}_{B}^{R}(j)={Tr}({\widehat{{\rm{\Pi }}}}_{j}^{B}\,{\widehat{\rho }}_{R}^{B})..\end{eqnarray} \tag{ 24 }$$

are the probabilities for measurement results for ${\widehat{{\rm{\Omega }}}}_{A}$, ${\widehat{{\rm{\Omega }}}}_{B}$, ... on the separate sub-systems with density operators $\,{\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, etc and the overall joint probability is given by the products of the probabilities ${P}_{A}^{R}(i)$, ${P}_{B}^{R}(j)$, .. for the measurement results ${\lambda }_{i}^{A}$, ${\lambda }_{j}^{B}$, ... for physical quantities ${\widehat{{\rm{\Omega }}}}_{A}$, ${\widehat{{\rm{\Omega }}}}_{B}$, ... if the sub-systems are in the states ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, etc. Note that here ${P}_{A}^{R}(i)$, ${P}_{B}^{R}(j)$, are given by quantum theory formulae for the subsystem states. For simplicity only quantized measured values will be considered—the extension to continuous values is straightforward. Thus the results for the probabilities of joint measurements when the system is in a separable quantum state are determined by the measurement probabilities in possible quantum states for the sub-systems, combined with a classical probability for creating the particular set of sub-system quantum states. Note the emphasis on 'possible'—some of the separable states described in [56] are not possible.

Furthermore, if we consider measurements of the physical quantity ${\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}$ then for a separable state the mean value for measurement of this quantity is given by

$$\begin{eqnarray}&&\langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\rangle ={\rm{Tr}}({\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\,\widehat{\rho })=\displaystyle \sum _{R}{P}_{R}{\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle }_{R}^{A}{\langle {\widehat{{\rm{\Omega }}}}_{B}\rangle }_{R}^{B}\,,\end{eqnarray} \tag{ 25 }$$

where ${\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle }_{R}^{A}={{\rm{Tr}}}_{a}({\widehat{{\rm{\Omega }}}}_{A}\,{\widehat{\rho }}_{R}^{A})$ and ${\langle {\widehat{{\rm{\Omega }}}}_{B}\rangle }_{R}^{B}={{\rm{Tr}}}_{b}({\widehat{{\rm{\Omega }}}}_{B}\,{\widehat{\rho }}_{R}^{B})$ are the mean values of ${\widehat{{\rm{\Omega }}}}_{A}$ and ${\widehat{{\rm{\Omega }}}}_{B}$ for the sub-system states ${\widehat{\rho }}_{R}^{A}$ and ${\widehat{\rho }}_{R}^{B}$ respectively. If $\langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\rangle =\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle \langle {\widehat{{\rm{\Omega }}}}_{B}\rangle$ then the state is said to be uncorrelated. Separable states are correlated except for the case where ${\widehat{\rho }}_{{\rm{sep}}}={\widehat{\rho }}^{A}\otimes {\widehat{\rho }}^{B}$, but the correlation is essentially non-quantum and attributable to the classical probabilities P_R. However, for separable states the inequality $| \langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}^{\dagger }\rangle {| }^{2}\leqslant \langle {\widehat{{\rm{\Omega }}}}_{A}^{\dagger }{\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}^{\dagger }{\widehat{{\rm{\Omega }}}}_{B}\rangle$ applies, so that if $| \langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}^{\dagger }\rangle {| }^{2}\gt \langle {\widehat{{\rm{\Omega }}}}_{A}^{\dagger }{\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{A}^{\dagger }{\widehat{{\rm{\Omega }}}}_{A}\rangle$ then the state is entangled.

In the simple non-entangled pure state situation in equation (1) the joint probability only involves a single product of sub-system probabilities

$$\begin{eqnarray}&&{P}_{{AB}..}(i,j,\ ...)={P}_{A}(i){P}_{B}(j)...,\end{eqnarray} \tag{ 26 }$$

where

$$\begin{eqnarray}&&{P}_{A}(i)=\langle {{\rm{\Phi }}}_{A}| {\widehat{{\rm{\Pi }}}}_{i}^{A}| {{\rm{\Phi }}}_{A}\rangle \quad {P}_{B}(j)=\langle {{\rm{\Phi }}}_{B}| {\widehat{{\rm{\Pi }}}}_{j}^{B}| {{\rm{\Phi }}}_{B}\rangle ..\end{eqnarray} \tag{ 27 }$$

just give the probabilities for measurements in the separate sub-systems.

This key result (23) showing that the joint measurement probability for a separable state only depends on separate measurement probabilities for the sub-systems, together with the classical probability for preparing correlated product states of the sub-systems, does not necessarily apply for entangled states [7]. However the key quantum feature for composite systems of non-separability for joint measurement probabilities applies only to entangled states. This strange quantum feature of entangled states has been regarded as particularly unusual when the sub-systems are spatially well-separated (or non-local) because then measurement events can become space-like separated. This is relevant to quantum paradoxes such as Einstein–Poldolsky–Rosen and Bell's theorem which aim to show there could be no causal classical theory explaining quantum mechanics [1, 2]. Measurements on sub-system A of physical quantity ${\widehat{{\rm{\Omega }}}}_{A}$ affect the results of measurements of ${\widehat{{\rm{\Omega }}}}_{B}$ at the same time on a distant sub-system B, even if the choice of measured quantity ${\widehat{{\rm{\Omega }}}}_{B}$ is unknown to the experimenter measuring ${\widehat{{\rm{\Omega }}}}_{A}$. As will be shown below, a similar result to (23) also occurs in HVT—a classical theory—so non-separability for joint measurements resulting from entanglement is a truly non-classical feature of composite systems.

For the general non-entangled state, the reduced density operator for sub-system A is given by

$$\begin{eqnarray}&&{\widehat{\rho }}_{A}=\displaystyle \sum _{R}{P}_{R}\,{\widehat{\rho }}_{R}^{A}.\end{eqnarray} \tag{ 28 }$$

A key feature of a non-entangled state is that the results of a measurement on any one of the sub-systems is independent of the states for the other subsystems. From equations (7) and (28) the probability P_A(i) that measurement of ${\widehat{{\rm{\Omega }}}}_{A}$ leads to result ${\lambda }_{i}^{A}$ is given by

$$\begin{eqnarray}&&{P}_{A}(i)=\displaystyle \sum _{R}{P}_{R}\,{P}_{A}^{R}(i),\end{eqnarray} \tag{ 29 }$$

where the reduced density operator is given by equation (28) for the non-entangled state in equation (3). This result only depends on the reduced density operator ${\widehat{\rho }}_{A}$, which represents a state for sub-system A and which is a statistical mixture of the sub-system states $\,{\widehat{\rho }}_{R}^{A}$, with a probability P_R that is the same for all sub-systems. The result for the measurement probability P_A(i) is just the statistical average of the results that would apply if sub-system A were in possible states $\,{\widehat{\rho }}_{R}^{A}$. For all quantum states the final expression for the measurement probability P_A(i) only involves a trace of quantities ${\widehat{{\rm{\Pi }}}}_{i}^{A}$, ${\widehat{\rho }}_{A}$ that apply to sub-system A, but for a non-entangled state the reduced density operator ${\widehat{\rho }}_{A}$ is given by an expression (28) that does not involve density operators for the other sub-systems. Thus for a non-entangled state, the probability P_A(i) is independent of the states ${\widehat{\rho }}_{R}^{B}$, ${\widehat{\rho }}_{R}^{C}$, associated with the other sub-systems. Analogous results apply for measurements on the other sub-systems.

For a general non-entangled bipartite mixed state the conditional probability is given by

$$\begin{eqnarray}&&{P}_{{AB}}(i| j)=\displaystyle \sum _{R}{P}_{R}\,{P}_{A}^{R}(i){P}_{B}^{R}(j)/\displaystyle \sum _{R}{P}_{R}{P}_{B}^{R}(j)\end{eqnarray} \tag{ 30 }$$

which in general depends on ${\widehat{{\rm{\Omega }}}}_{B}$ associated with sub-system B and the eigenvalue ${\lambda }_{j}^{B}$. This may seem surprising for the case where A and B are localized sub-systems which are well separated. Even for separable states a measurement result for sub-system B will give immediate information about a totally separated measurement on sub-system A—which is space-like separated. However it should be remembered that the general separable system can still be a correlated state, since each sub-system density operator ${\widehat{\rho }}_{R}^{B}$ for sub-system B is matched with a corresponding density operator ${\widehat{\rho }}_{R}^{A}$ for sub-system A. Results at A can be correlated with those at B, so the observer at A can potentially infer from a local measurement on the sub-system A the result of a local measurement on sub-system B. It is therefore not necessarily the case that measurement results for A are independent of those for B. However, as we will see below, such correlations (usually) have a classical interpretation. Result (30) is not a case of the 'spooky action at a distance' that Einstein [1] referred to.

However, for a non-entangled pure state where $\widehat{\rho }={\widehat{\rho }}^{A}\otimes {\widehat{\rho }}^{B}$ we do find that

$$\begin{eqnarray}&&{P}_{{AB}}(i| j)={P}_{A}(i),\end{eqnarray} \tag{ 31 }$$

where ${P}_{A}(i)={\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{i}^{A}{\widehat{\rho }}^{A})$. For separable pure states the conditional probability is independent of ${\widehat{{\rm{\Omega }}}}_{B}$ associated with sub-system B and the eigenvalue ${\lambda }_{j}^{B}$.

Also of course ${\sum }_{j}{P}_{{AB}}(i| j){P}_{B}(j)={P}_{A}(i)$ is true for separable states since it applies to general bipartite states. Hence if the measurement results for ${\widehat{{\rm{\Omega }}}}_{B}$ are discarded then the probability distribution for measurements on ${\widehat{{\rm{\Omega }}}}_{A}$ will be determined from the conditioned density operator ${\widehat{\rho }}_{{\rm{cond}}}({\widehat{{\rm{\Omega }}}}_{B})$ and just result in P_A(i)—as in shown in equation (17) for any quantum state.

For non-entangled states as in equation (3) the mean value for measuring a physical quantity ${\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes ...$, where ${\widehat{{\rm{\Omega }}}}_{A}$, ${\widehat{{\rm{\Omega }}}}_{B}$, ${\widehat{{\rm{\Omega }}}}_{C}$, .. are Hermitian operators representing physical quantities for the separate sub-systems can be obtained from equations (3) and (10), and is given by

$$\begin{eqnarray}&&\langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes .\rangle =\displaystyle \sum _{R}{P}_{R}{\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle }_{R}^{A}{\langle {\widehat{{\rm{\Omega }}}}_{B}\rangle }_{R}^{B}{\langle {\widehat{{\rm{\Omega }}}}_{C}\rangle }_{R}^{C}...,\end{eqnarray} \tag{ 32 }$$

where

$$\begin{eqnarray}&&{\langle {\widehat{{\rm{\Omega }}}}_{K}\rangle }_{R}^{K}={Tr}({\widehat{{\rm{\Omega }}}}_{K}\,{\widehat{\rho }}_{R}^{K}),\quad (K=A,B,\ldots )\end{eqnarray} \tag{ 33 }$$

is the mean value for measuring ${\widehat{{\rm{\Omega }}}}_{K}$ in the K sub-system when its density operator is ${\widehat{\rho }}_{R}^{K}$. Since the overall mean value is not equal to the product of the separate mean values, the measurements on the sub-systems are said to be correlated. However, for the general non-entangled state as the mean value is just the products of mean values weighted by the probability of preparing the particular product state—which involves a LOCC protocol, as we have seen—the correlation is classical rather than quantum [18]. In the case of a single product state where $\widehat{\rho }={\widehat{\rho }}^{A}\otimes {\widehat{\rho }}^{B}\otimes {\widehat{\rho }}^{C}\otimes ...$ we have $\langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes \mathrm{...}\rangle ={\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle }^{A}{\langle {\widehat{{\rm{\Omega }}}}_{B}\rangle }^{B}{\langle {\widehat{{\rm{\Omega }}}}_{C}\rangle }^{C}...$ which is just the product of mean values for the separate sub-systems, and in this case the measurements on the sub-systems are said to be uncorrelated. For entangled states however the last result for $\langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes \mathrm{...}\rangle$ does not apply, and the correlation is strictly quantum.

The actual values that would be assigned to the physical quantities ${{\rm{\Omega }}}_{A}$, ${{\rm{\Omega }}}_{B}$ etc will depend on the hidden variables but can be taken as the mean values of the possible values ${\lambda }_{i}^{A}$, ${\lambda }_{i}^{A}$ etc. We denote these mean values as $\langle {{\rm{\Omega }}}_{A}(\xi )\rangle$, $\langle {{\rm{\Omega }}}_{B}(\xi )\rangle$ etc where

$$\begin{eqnarray}&&\langle {{\rm{\Omega }}}_{K}(\xi )\rangle =\sum _{{\lambda }_{k}^{K}}{\lambda }_{k}^{K}\,{P}_{K}(k,\xi )\quad (K=A,B,\ldots ).\end{eqnarray} \tag{ 35 }$$

These expressions my be compared to equation (33) for the mean values of physical quantities ${\widehat{{\rm{\Omega }}}}_{A}$, ${\widehat{{\rm{\Omega }}}}_{B}$ etc in quantum separable states.

We can then obtain an expression for the mean value in HVT of the physical quantity ${{\rm{\Omega }}}_{A}\times {{\rm{\Omega }}}_{B}\times {{\rm{\Omega }}}_{C}\times ...$, where ${{\rm{\Omega }}}_{A}$, ${{\rm{\Omega }}}_{B}$, etc are physical quantities for the separate sub-systems. This is obtained from equations (34) and (35) and is given by

$$\begin{eqnarray}&&{\langle {{\rm{\Omega }}}_{A}\times {{\rm{\Omega }}}_{B}\times {{\rm{\Omega }}}_{C}\times .\rangle }_{{\rm{LHV}}}\\ &&\quad =\displaystyle \int {\rm{d}}\xi \,P(\xi )\langle {{\rm{\Omega }}}_{A}(\xi )\rangle \langle {{\rm{\Omega }}}_{B}(\xi )\rangle \langle {{\rm{\Omega }}}_{C}(\xi )\rangle ...\end{eqnarray} \tag{ 36 }$$

This may be compared to equation (32) for the mean value of the physical quantity ${\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}\otimes {\widehat{{\rm{\Omega }}}}_{C}\otimes ..$ in quantum separable states.

In the original version of the EPR paradox, Einstein et al [1] considered a two-particle system A, B in which the particles were associated with positions ${\widehat{x}}_{A}$, ${\widehat{x}}_{B}$ and momenta ${\widehat{p}}_{A}$, ${\widehat{p}}_{B}$. They envisaged a quantum state in which the pairs of physical quantities ${\widehat{x}}_{A}$, ${\widehat{x}}_{B}$ or ${\widehat{p}}_{A}$, ${\widehat{p}}_{B}$ had highly correlated values—measured or otherwise. To be specific, one may consider a simultaneous eigenstate $| {\rm{\Phi }}\rangle$ of the two commuting operators ${\widehat{x}}_{A}-{\widehat{x}}_{B}$ and ${\widehat{p}}_{A}+{\widehat{p}}_{B}$, where $({\widehat{x}}_{A}-{\widehat{x}}_{B})| {\rm{\Phi }}\rangle =2x| {\rm{\Phi }}\rangle$ and $({\widehat{p}}_{A}+{\widehat{p}}_{B})| {\rm{\Phi }}\rangle =0| {\rm{\Phi }}\rangle$. This state is an example of an entangled state, as may be seen if it is expanded in terms of position eigenstates $| {x}_{A}{x}_{B}\rangle$. If the system is in state $| {\rm{\Phi }}\rangle$ then from standard quantum theory if A had a mean momentum p then B would have a mean momentum $-p$. Alternatively, if A had a mean position x then B would have a mean position $-x$. Then if the eigenvalue $2x$ is very large so that the two particles will be well-separated (in quantum theory their spatial wave functions would be localized in separate spatial regions) it follows that if the position of B was measured then the position of A would be immediately known, even if the particles were light years apart. On the other hand, if the momentum of B was measured instead, then the momentum of A would immediately be known. From the realist point of view both A and B always have definite positions and momenta, even if these are not known, so all these measurements do is reveal these (hidden) values. It would seem then that measurements of position and momentum on particle B could lead to a knowledge of the position and momentum at a far distant particle A, perhaps with an accuracy that would violate the HUP. As we will see, this is not the case when quantum theory is applied correctly. However, what Einstein et al pointed out as being particularly strange was that the choice of whether the momentum or position of B was measured (and found to have a definite value) would instantly determine which of the position or momentum of A would then have a definite value—even if A and B were separated by such a large distance that no signal could have been passed from B to A regarding which quantity was measured. Einstein referred to this as 'spooky action at a distance' to highlight the strangeness of what came to be referred to as entangled states. Thus a somewhat paradoxical situation would seem to arise. Einstein stated that this did not demonstrate that quantum theory was wrong, only that it was incomplete.

The EPR argument assumes local realism, to justify that the possibility of an exact prediction of the position of the far-away particle A (based on the measurement of the position for the particle B) implies the realist viewpoint that the position of particle A was predetermined. The same argument applies to the momentum of particle A, and hence EPR conclude that both the position and momentum of particle A are precisely predetermined—in conflict with the HUP derived from quantum mechanics. Since the argument is based on the assumption of local realism, the modern interpretation of the EPR analysis is that it reveals (for the appropriate entangled state) the inconsistency of local realism with the completeness of quantum mechanics.

Discussions of the EPR paradox [1] in terms of HVTs has been given by numerous authors (see [17–19, 89] for example). The papers and reviews by Reid et al [19, 83, 89], give a full account taking into consideration the 'fuzzy' version of local HVT (LHV) and determining the predictions for the conditional variances for x_A and p_A based both on separable quantum states and states described via local HVT. This treatment successfully quantifies the somewhat qualitative considerations described in the previous paragraph. If the position for particle B is measured and the result is x, then the original density operator $\widehat{\rho }$ for the two particle system is changed into the conditional density operator ${\widehat{\rho }}_{{\rm{cond}}}({\widehat{x}}_{B},x)={\widehat{{\rm{\Pi }}}}_{x}^{B}\widehat{\rho }\,{\widehat{{\rm{\Pi }}}}_{x}^{B}/{\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{x}^{B}\widehat{\rho })$, where ${\widehat{{\rm{\Pi }}}}_{x}^{B}={(| x\rangle \langle x| )}_{B}$ is the projector onto the eigenvector ${| x\rangle }_{B}$ (the eigenvalues x are assumed for simplicity to form a quasi-continuum). Similarly, if the momentum for particle B is measured and the result is p, then the original density operator $\widehat{\rho }$ for the two particle system is changed into the conditional density operator ${\widehat{\rho }}_{{\rm{cond}}}({\widehat{p}}_{B},p)\,={\widehat{{\rm{\Pi }}}}_{p}^{B}\,\widehat{\rho }\,{\widehat{{\rm{\Pi }}}}_{p}^{B}/{\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{p}^{B}\widehat{\rho })$, where ${\widehat{{\rm{\Pi }}}}_{p}^{B}={(| p\rangle \langle p| )}_{B}$ is the projector onto the eigenvector ${| p\rangle }_{B}$ (the eigenvalues p are assumed for simplicity to form a quasi-continuum). Here we outline the discussion based on quantum separable states. Conditional variances for position and momentum for sub-system A are considered based on measurements for sub-system B of position. It can be shown that for these conditional variances the HUP still applies. The same conclusion is obtained if the measurements on sub-system B had been the momentum. As the experimenter on sub-system A could not know whether the measurement on sub-system B was on position or momentum, the action at a distance feature of quantum entanglement is confirmed.

The question is whether the conditional variances ${\langle {\rm{\Delta }}{\widehat{x}}_{A}^{2}\rangle }_{{\widehat{x}}_{B}}$ for measuring ${\widehat{x}}_{A}$ for sub-system A having measured ${\widehat{x}}_{B}$ for sub-system B, and ${\langle {\rm{\Delta }}{\widehat{p}}_{A}^{2}\rangle }_{{\widehat{p}}_{B}}$ for measuring ${\widehat{p}}_{A}$ for sub-system A having measured ${\widehat{p}}_{B}$ for sub-system B violate the HUP [83]

$$\begin{eqnarray}&&{\langle {\rm{\Delta }}{\widehat{x}}_{A}^{2}\rangle }_{{\widehat{x}}_{B}}{\langle {\rm{\Delta }}{\widehat{p}}_{A}^{2}\rangle }_{{\widehat{p}}_{B}}\lt \displaystyle \frac{1}{4}{{\hslash }}^{2},\end{eqnarray} \tag{ 37 }$$

where the measurements on sub-system B are left unrecorded. If this inequality holds we have an EPR violation. However for separable states it can be shown that ${\langle {\rm{\Delta }}{\widehat{x}}_{A}^{2}\rangle }_{{\widehat{x}}_{B}}{\langle {\rm{\Delta }}{\widehat{p}}_{A}^{2}\rangle }_{{\widehat{p}}_{B}}\geqslant \,\tfrac{1}{4}{{\hslash }}^{2}$. The proof of this result is set out in appendix F. Thus if the EPR violations as defined in equation (37) are to occur then the state must be entangled. Progress towards experimental confirmation of EPR violations is reviewed in [9, 19].

In [89] an analogous treatment based on LHV theory also shows that the HUP is satisfied for the conditioned variances. The details of this treatment will not be given here, but the formal similarity of expressions for conditional probabilities in LHV theories and for separable states indicates the steps involved.

The EPR paradox is not confined to position and momentum measurements on two sub-systems. A related paradox [93] occurs in the case of measurements on spin components ${\widehat{S}}_{\alpha 1}$ and ${\widehat{S}}_{\alpha 2}$—with $\alpha =x,y,z$—associated with two sub-systems 1 and 2. The spin operators also satisfy non-zero commutation rules (see paper II for details)

$$\begin{eqnarray}&&[{\widehat{S}}_{\alpha 1},{\widehat{S}}_{\beta 1}]={\rm{i}}{\widehat{S}}_{\gamma 1}\qquad [{\widehat{S}}_{\alpha 2},{\widehat{S}}_{\beta 2}]={\rm{i}}{\widehat{S}}_{\gamma 2},\end{eqnarray} \tag{ 38 }$$

where $\alpha ,\beta ,\gamma$ are $x,y,z$ in cyclic order. The question is whether the conditional variances ${\langle {\rm{\Delta }}{\widehat{S}}_{x1}^{2}\rangle }_{{\widehat{S}}_{x2}}{\langle {\rm{\Delta }}{\widehat{x}}_{A}^{2}\rangle }_{{\widehat{x}}_{B}}$ for measuring ${\widehat{S}}_{x1}$ for sub-system 1 having measured ${\widehat{S}}_{x2}$ for sub-system 2, and ${\langle {\rm{\Delta }}{\widehat{S}}_{y1}^{2}\rangle }_{{\widehat{S}}_{y2}}$ for measuring ${\widehat{S}}_{y1}$ for sub-system 1 having measured ${\widehat{S}}_{y2}$ for sub-system 2 violate the HUP

$$\begin{eqnarray}&&{\langle {\rm{\Delta }}{\widehat{S}}_{x1}^{2}\rangle }_{{\widehat{S}}_{x2}}{\langle {\rm{\Delta }}{\widehat{S}}_{y1}^{2}\rangle }_{{\widehat{S}}_{y2}}\lt \displaystyle \frac{1}{4}| \langle {\widehat{S}}_{z1}\rangle {| }^{2}.\end{eqnarray} \tag{ 39 }$$

Again we find that for separable states that the product of conditional variances ${\langle {\rm{\Delta }}{\widehat{S}}_{x1}^{2}\rangle }_{{\widehat{S}}_{x2}}{\langle {\rm{\Delta }}{\widehat{S}}_{y1}^{2}\rangle }_{{\widehat{S}}_{y2}}$ $\geqslant \tfrac{1}{4}| \langle {\widehat{S}}_{z1}\rangle {| }^{2}$ showing that if the EPR violations as defined in (39) occur, then the state must be entangled. For completeness, this spin version of the EPR paradox is set out in appendix F.

An effect related to the EPR paradox is EPR steering. As we have seen, the measurement of the position for particle B changes the density operator and consequently the probability distributions for measurements on particle A will now be determined from the conditional probabilities, such as ${P}_{{AB}}({\widehat{x}}_{A},{x}_{A}| {\widehat{x}}_{B},{x}_{B})$ or ${P}_{{AB}}({\widehat{p}}_{A},{p}_{A}| {\widehat{x}}_{B},{x}_{B}).$ Thus measurements on B are said to steer the results for measurements on A. Steering will of course only apply if the measurement results for ${\widehat{x}}_{B}$ are recorded, and not discarded. A discussion of EPR steering (see [19, 30]) is beyond the scope of this article.

The Schrödinger cat paradox [2, 94] relates to composite systems where one sub-system (the cat) is macroscopic and the other sub-system is microscopic (the radioactive atom). The paradox is a clear consequence of quantum theory allowing the existence of entangled states. Schrödinger envisaged a state in which an alive cat and an undecayed atom existed at an initial time, and because the decayed atom would be associated with a dead cat, the system after a time of one hour corresponding to the half-life for radioactive decay would be described in quantum theory via the entangled state

$$\begin{eqnarray}&&| {\rm{\Psi }}\rangle =\displaystyle \frac{1}{\sqrt{2}}({| e\rangle }_{{\rm{Atom}}}{| {\rm{Alive}}\rangle }_{{\rm{Cat}}}+{| g\rangle }_{{\rm{Atom}}}{| {\rm{Dead}}\rangle }_{{\rm{Cat}}})\end{eqnarray} \tag{ 40 }$$

in an obvious notation. The quantum state defined by (40) represents the knowledge that an observer outside the box would have about the combined atom-cat system one hour after the live cat was placed in the box along with an undecayed atom. The combined system is in an enclosed box, and opening the box and observing what is inside constitutes a measurement on the system. According to quantum theory if the box was opened at this time there would be a probability of 1/2 of finding the atom undecayed and the cat alive, with the same probability for finding a decayed atom and a dead cat. From the realist viewpoint the cat should be either dead or it should be alive irrespective of whether the box is opened or not, and it was regarded as a paradox that in the quantum theory description of the state prior to measurement the cat is in some sense both dead and alive. This paradox is made worse because the cat is a macroscopic system—how could a cat be either dead or alive at the same time, it must be one or the other? From the quantum point of view in which the actual values of physical quantities only appear when measurement occurs, the Schrödinger cat presents no paradox. The two possible values signifying the health of the cat are 'alive' and 'dead', and these values are found with a probability of 1/2 when measurement takes place on opening the box, and this would entirely explain the results if such an experiment were to be performed. Thus the cat is neither dead nor alive until measurement has taken place when the box is opened. There is of course no paradox if the quantum state is only considered to represent the observer's information about what is inside the box. If the box is closed then at one half life after the cat was put into the box, the state vector (40) enables the outside observer to correctly assess the probability that the cat will be alive is 1/2. If the box is then opened and the cat is found to be dead, then the observer's information changes and the state vector for the cat-atom system is now

$$\begin{eqnarray}&&| {\rm{\Psi }}^{\prime} \rangle ={| g\rangle }_{{\rm{Atom}}}{| {\rm{Dead}}\rangle }_{{\rm{Cat}}}.\end{eqnarray} \tag{ 41 }$$

In this interpretation of quantum states, the notion of there being some sort of underlying reality that exists prior to measurement is rejected. It is only this notion that such a reality must exist—perhaps described via hidden variables—that leads to the paradox. EPR paradoxes can also be constructed from the entangled state (40), as outlined in [95, 96].

In recent times, experiments based on a Rydberg atom in a microwave cavity [97] involving states such as (40) have been performed showing that entanglement can occur between macroscopic and microscopic systems, and it is even possible to prepare states analogous to $\tfrac{1}{\sqrt{2}}({| {\rm{Alive}}\rangle }_{{\rm{Cat}}}+{| {\rm{Dead}}\rangle }_{{\rm{Cat}}})$ in the macroscopic system itself. In such experiments the different macroscopic states are large amplitude coherent states of the cavity mode. Coherent states are possible for microwave photons as they are created from classical currents with well-defined phases. A coherent superposition of an alive and dead cat within the cat sub-system itself can be created by measurement. The entangled state in (40) can also be written as

$$\begin{eqnarray}| {\rm{\Psi }}\rangle & = & \displaystyle \frac{1}{\sqrt{2}}\left\{\displaystyle \frac{1}{\sqrt{2}}({| e\rangle }_{{\rm{Atom}}}+{| g\rangle }_{{\rm{Atom}}})\displaystyle \frac{1}{\sqrt{2}}\right.\\ & & \times ({| {\rm{Alive}}\rangle }_{{\rm{Cat}}}+{| {\rm{Dead}}\rangle }_{{\rm{Cat}}})\\ & & +\displaystyle \frac{1}{\sqrt{2}}({| e\rangle }_{{\rm{Atom}}}-{| g\rangle }_{{\rm{Atom}}})\displaystyle \frac{1}{\sqrt{2}}\\ & & \times ({| {\rm{Alive}}\rangle }_{{\rm{Cat}}}-{| {\rm{Dead}}\rangle }_{{\rm{Cat}}})\phantom{\displaystyle \frac{1}{\sqrt{2}}}\,\}\end{eqnarray} \tag{ 42 }$$

so that measurement on the atom for an observable in which the superposition states $\tfrac{1}{\sqrt{2}}({| e\rangle }_{{\rm{Atom}}}\pm {| g\rangle }_{{\rm{Atom}}})$ are the eigenstates for this observable would result in the cat then being in the corresponding macroscopic superposition states $\tfrac{1}{\sqrt{2}}({| {\rm{Alive}}\rangle }_{{\rm{Cat}}}\pm {| {\rm{Dead}}\rangle }_{{\rm{Cat}}})$ of an alive and dead cat.

A key feature of entangled states is that they are associated with violations of Bell inequalities [4] and hence can exhibit this particular non-classical feature. The Bell inequalities arise in attempts to restore a classical interpretation of quantum theory via hidden variable treatments, where actual values are assigned to all measurable quantities—including those which in quantum theory are associated with non-commuting Hermitian operators. In this case we consider two different physical quantities ${{\rm{\Omega }}}_{A}$ for sub-system A, which are listed A₁, A₂, etc, and two ${{\rm{\Omega }}}_{B}$ for sub-system B, which are listed B₁, B₂, etc. The corresponding quantum Hermitian operators ${\widehat{{\rm{\Omega }}}}_{A}$, ${\widehat{{\rm{\Omega }}}}_{B}$, etc are ${\widehat{A}}_{1}$, ${\widehat{A}}_{2}$ and, ${\widehat{B}}_{1}$, ${\widehat{B}}_{2}$. The Bell inequalities involve the mean value ${\langle {A}_{i}\times {B}_{j}\rangle }_{{\rm{HVT}}}$ of the product of observables A_i and B_j for subsystems A, B respectively, for which there are two possible measured values, +1 and −1. For simplicity we consider a local HVT. In a LHV theory we see using (34) for Bell local states that the mean values ${\langle {A}_{i}\times {B}_{j}\rangle }_{{\rm{LHV}}}$ are given by

$$\begin{eqnarray}&&{\langle {A}_{i}\times {B}_{j}\rangle }_{{\rm{LHV}}}=\displaystyle \int {\rm{d}}\xi \,P(\xi )\langle {A}_{i}(\xi )\rangle \langle {B}_{j}(\xi )\rangle ,\end{eqnarray} \tag{ 43 }$$

where $\langle {A}_{i}(\xi )\rangle$ and $\langle {B}_{j}(\xi )\rangle$ (as in equation (35)) are the values are assigned to A_i and B_j when the hidden variables are ξ, and $P(\xi )$ is the hidden variable probability distribution function. If the corresponding quantum Hermitian operators are such that their eigenvalues are +1 and −1—as in the case of Pauli spin operators—then the only possible values for $\langle {A}_{i}(\xi )\rangle$ and $\langle {B}_{j}(\xi )\rangle$ are between +1 and −1, since HVT does not conflict with quantum theory regarding allowed values for physical quantities. However, LHV theory predicts certain inequalities for the mean values of products of physical quantities for the two sub-systems.

The form given by Clauser et al [86] for Bell's inequalityis

$$\begin{eqnarray}&&| S| \leqslant 2,\end{eqnarray} \tag{ 44 }$$

where

$$\begin{eqnarray}S & = & {\langle {A}_{1}\times {B}_{1}\rangle }_{{\rm{LHV}}}+{\langle {A}_{1}\times {B}_{2}\rangle }_{{\rm{LHV}}}\\ & & +{\langle {A}_{2}\times {B}_{1}\rangle }_{{\rm{LHV}}}-{\langle {A}_{2}\times {B}_{2}\rangle }_{{\rm{LHV}}}.\end{eqnarray} \tag{ 45 }$$

The minus sign can actually be attached to any one of the four terms The proof of this important result that all Bell local states must satisfy is given in [18] but for completeness is set out in appendix G.

We also confirm in appendix G that for a general two mode non-entangled state, $| S|$ cannot violate the Bell inequality upper bound of 2. This of course also follows directly from separable states being seen as examples of Bell local states. Thus, the violation of Bell inequalities proves that the quantum state must be entangled for the sub-systems involved, so Bell inequality violations are a test of entanglement. An example of an entangled state that violates the Bell inequality is the Bell state $| {{\rm{\Psi }}}_{-}\rangle$ (see [18], section 2.5) written in terms of eigenstates of ${\widehat{\sigma }}_{z}^{A}$ and ${\widehat{\sigma }}_{z}^{B}$

$$\begin{eqnarray}&&| {{\rm{\Psi }}}_{-}\rangle =\displaystyle \frac{1}{\sqrt{2}}({| +1\rangle }_{A}\otimes {| -1\rangle }_{B}-{| -1\rangle }_{A}\otimes {| +1\rangle }_{B})\end{eqnarray} \tag{ 46 }$$

we find that

$$\begin{eqnarray}&&E(\mathop{a}\limits_{}\cdot \mathop{{\widehat{\sigma }}^{A}}\limits_{}\otimes \mathop{b}\limits_{}\cdot \mathop{{\widehat{\sigma }}^{B}}\limits_{})=-\mathop{a}\limits_{}\cdot \mathop{b}\limits_{}.\end{eqnarray} \tag{ 47 }$$

The Bell inequality given in the appendix equation (179) can be violated for the choice where b₁and b₂ are orthogonal and a₁, a₂ are parallel to ${b}_{1}+{b}_{2}$, ${b}_{1}-{b}_{2}$ respectively (see [18], section 5.1). Furthermore, such a quantum state cannot be described via a HVT, since Bell inequalities are always satisfied using a HVT. Experiments have been carried out in optical systems providing strong evidence for the existence of quantum states that violate Bell inequalities with only a few loopholes remaining (see [9, 19, 36] for references to experiments). Such violation of Bell inequalities is clearly a non-classical feature, since the experiments rule out all LHV theories. As Bell inequalities do not occur for separable states, the experimental observation of a Bell inequality indicates the presence of an entangled state. These violations are not without applications, since such Bell entangled states can be useful in device-independent quantum key distribution [17, 18, 36].

Consider two operators ${\widehat{{\rm{\Omega }}}}_{A}$ and ${\widehat{{\rm{\Omega }}}}_{B}$ associated with sub-systems A and B. These would be Hermitian if observables are involved, but for generality this is not required. In a LHV theory these would be associated with functions ${{\rm{\Omega }}}_{C}(\xi )$ $(C=A,B)$ of the LHVs ξ, with the Hermitian adjoints ${\widehat{{\rm{\Omega }}}}_{C}^{\dagger }$ being associated with the complex conjugates ${{\rm{\Omega }}}_{C}^{* }(\xi )$. In LHV theory correlation functions are given by the following mean values

$$\begin{eqnarray}&&{\langle {{\rm{\Omega }}}_{A}^{* }\times {{\rm{\Omega }}}_{B}\rangle }_{{\rm{LHV}}}=\displaystyle \int {\rm{d}}\xi \,P(\xi ){{\rm{\Omega }}}_{A}^{* }(\xi ){{\rm{\Omega }}}_{B}(\xi )\\ &&{\langle {{\rm{\Omega }}}_{A}^{* }{{\rm{\Omega }}}_{A}\times {{\rm{\Omega }}}_{B}^{* }{{\rm{\Omega }}}_{B}\rangle }_{{\rm{LHV}}}=\displaystyle \int {\rm{d}}\xi \,P(\xi )\\ &&\quad \times {{\rm{\Omega }}}_{A}^{* }(\xi ){{\rm{\Omega }}}_{A}(\xi ){{\rm{\Omega }}}_{B}^{* }(\xi ){{\rm{\Omega }}}_{B}(\xi )\end{eqnarray} \tag{ 48 }$$

which then can be shown to satisfy the following correlation inequality

$$\begin{eqnarray}&&| {\langle {{\rm{\Omega }}}_{A}^{* }\times {{\rm{\Omega }}}_{B}\rangle }_{{\rm{LHV}}}\,{| }^{2}\leqslant {\langle {{\rm{\Omega }}}_{A}^{* }{{\rm{\Omega }}}_{A}\times {{\rm{\Omega }}}_{B}^{* }{{\rm{\Omega }}}_{B}\rangle }_{{\rm{LHV}}}.\end{eqnarray} \tag{ 49 }$$

This result is based on the inequality

$$\begin{eqnarray}&&\int {\rm{d}}\xi \,P(\xi )C(\xi )\geqslant {\left(\int {\rm{d}}\xi P(\xi )\sqrt{C(\xi )}\right)}^{2}\end{eqnarray} \tag{ 50 }$$

for real, positive functions $C(\xi ),P(\xi )$ and where $\int {\rm{d}}\xi \,P(\xi )=1$, and which is proved in appendix E. In the present case we have $C(\xi )={{\rm{\Omega }}}_{A}^{* }(\xi ){{\rm{\Omega }}}_{A}(\xi ){{\rm{\Omega }}}_{B}^{* }(\xi ){{\rm{\Omega }}}_{B}(\xi )$, which is real, positive. A violation of the inequality in equation (49) is an indication of strong correlation between sub-systems A and B. It would also demonstrate Bell non-locality.

As separable states are particular cases of Bell local states it follows that the corresponding result to (49), namely

$$\begin{eqnarray}&&| \langle {\widehat{{\rm{\Omega }}}}_{A}^{\dagger }\otimes {\widehat{{\rm{\Omega }}}}_{B}\rangle {| }^{2}=| \langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}^{\dagger }\rangle {| }^{2}\leqslant \langle {\widehat{{\rm{\Omega }}}}_{A}^{\dagger }{\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}^{\dagger }{\widehat{{\rm{\Omega }}}}_{B}\rangle \end{eqnarray} \tag{ 51 }$$

applies for separable states. The direct proof of this result is included for completeness in appendix H.

Hence if it is found that the correlation inequality is violated $| \langle {\widehat{{\rm{\Omega }}}}_{A}^{\dagger }\otimes {\widehat{{\rm{\Omega }}}}_{B}\rangle {| }^{2}=| \langle {\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}^{\dagger }\rangle {| }^{2}\gt \langle {\widehat{{\rm{\Omega }}}}_{A}^{\dagger }{\widehat{{\rm{\Omega }}}}_{A}\otimes {\widehat{{\rm{\Omega }}}}_{B}^{\dagger }{\widehat{{\rm{\Omega }}}}_{B}\rangle$ then the state must be entangled, so the correlation inequality violation is also a sufficiency test for entanglement.

It is useful to clarify some of the issues involved by considering a simple example. Since density operators can always be expressed in a diagonal form involving their orthonormal eigenstates $| {\rm{\Phi }}\rangle$ with real, positive eigenvalues $P({\rm{\Phi }})$ as $\widehat{\rho }={\sum }_{{\rm{\Phi }}}$ $P({\rm{\Phi }})| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}|$ and each $| {\rm{\Phi }}\rangle$ can always be written as a linear combination of basis vectors $| {\rm{\Psi }}\rangle$, we will focus on these basis vectors and their forms in both first and second quantization. We consider a system with N = 2 particles, which may be identical and are labeled 1 and 2, or they may be distinguishable and labeled α and β. In each case a particle has a choice of two modes which it may occupy. Thus there are two distinct single particle states (modes) designated as $| A\rangle$ and $| B\rangle$ in the identical particle case, and four distinct single particle states (modes) designated as $| {A}_{\alpha }\rangle$, $| {B}_{\alpha }\rangle$ and $| {A}_{\beta }\rangle$, $| {B}_{\beta }\rangle$ in the distinguishable particle case for particles α and β respectively. The notation in first quantization is that $| C(i)\rangle$ refers to a vector in which particle i is in mode $| C\rangle$. The notation in second quantization is that ${| n\rangle }_{C}$ refers to a vector where there are n particles in mode $| C\rangle$.

For the case of the identical particles we consider basis states for two bosons or for two fermions, which are written in terms of first quantization as

$$\begin{eqnarray}&&{| {\rm{\Psi }}\rangle }_{{\rm{boson}}}=\displaystyle \frac{1}{\sqrt{2}}(| A(1)\rangle \otimes | B(2)\rangle +| B(1)\rangle \otimes | A(2)\rangle ),\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{| {\rm{\Psi }}\rangle }_{{\rm{fermion}}}=\displaystyle \frac{1}{\sqrt{2}}(| A(1)\rangle \otimes | B(2)\rangle -| B(1)\rangle \otimes | A(2)\rangle )\end{eqnarray} \tag{ 53 }$$

and clearly satisfy the symmetrization principle. In second quantization the basis state in both the fermion and boson cases is

$$\begin{eqnarray}&&{| {\rm{\Psi }}\rangle }_{{\rm{boson}},{\rm{fermion}}}={| 1\rangle }_{A}\otimes {| 1\rangle }_{B}.\end{eqnarray} \tag{ 54 }$$

In both first and second quantization this basis state involves one identical particle in mode $| A\rangle$ and the other in mode $| B\rangle$.

These examples highlight two possibilities for specifying sub-systems for systems of identical particles. The two possibilities have differing consequences in terms of whether specific pure states are regarded as separable or entangled in terms of the general form in equation (1) for separable pure states, depending on whether the first or second quantization approach is used. The first option is to regard the labeled identical particles as sub-systems—in which case using first quantization the boson or fermion basis states in equations (52) and (53) would be regarded as entangled states of the two sub-systems consisting of particle 1 and particle 2 [15, 74, 77]. This is a more mathematical approach, and suffers from the feature that the sub-systems are not distinguishable and measurements cannot be made on specifically labeled identical particles. In the case of identical particles the option of regarding labeled identical particles as the sub-systems leads to the concept of entanglement due to symmetrization. In the textbook by Peres ([15], see pp 126–8) it is stated that 'two particles of the same type are always entangled'. Peres obviously considers such entanglement is a result of symmetrization. The second option would be to regard the modes or single particle states as sub-systems [35]—in which case using second quantization the basis state for both fermions or bosons in equation (54) would be regarded as a separable state of two sub-systems consisting of modes $| A\rangle$ and $| B\rangle$. This is a more physically based approach, and has the advantage that the sub-systems are distinguishable and measurements can be made on specific modes. Noting that in the example the same quantum state is involved with one identical particle in mode $| A\rangle$ and the other in mode $| B\rangle$, the different categorization is disconcerting. It indicates that a choice must be made in regard to defining sub-systems when identical particles are involved (see section 3.1.2).

Now consider the case where the particles are distinguishable. Each distinguishable particle α, β has its own unique set of modes ${A}_{\alpha }$, ${B}_{\alpha }$, ${A}_{\beta }$, ${B}_{\beta }$. There are two cases in which one particle α occupies mode $| {A}_{\alpha }\rangle$ or $| {B}_{\alpha }\rangle$ and the other particle β occupies mode $| {A}_{\beta }\rangle$ or $| {B}_{\beta }\rangle$. Basis states analogous to the previous ones are given in first quantization as

$$\begin{eqnarray}{| {\rm{\Psi }}\rangle }_{{\rm{dist}}} & = & | {A}_{\alpha }(\alpha )\rangle \otimes | {B}_{\beta }(\beta )\rangle \qquad {\rm{or}}\\ {| {\rm{\Psi }}\rangle }_{{\rm{dist}}} & = & | {B}_{\alpha }(\alpha )\rangle \otimes | {A}_{\beta }(\beta )\rangle .\end{eqnarray} \tag{ 55 }$$

The somewhat surplus particle labels $(\alpha )$ and $(\beta )$ have been added for comparison with (52) and (53). The states (57) are not required to satisfy the symmetrization principle since the particles are not identical. Each may be either a boson or a fermion. In second quantization the basis states are

$$\begin{eqnarray}{| {\rm{\Psi }}\rangle }_{{\rm{dist}}} & = & ({| 1\rangle }_{{A}_{\alpha }}\otimes {| 0\rangle }_{{B}_{\alpha }})\otimes ({| 0\rangle }_{{A}_{\beta }}\otimes {| 1\rangle }_{{B}_{\beta }})\\ & & \qquad {\rm{or}}\\ {| {\rm{\Psi }}\rangle }_{{\rm{dist}}} & = & ({| 0\rangle }_{{A}_{\alpha }}\otimes {| 1\rangle }_{{B}_{\alpha }})\otimes ({| 1\rangle }_{{A}_{\beta }}\otimes {| 0\rangle }_{{B}_{\beta }}).\end{eqnarray} \tag{ 56 }$$

In both first and second quantization, the first case corresponds to particle α being in mode $| {A}_{\alpha }\rangle$ and particle β being in mode $| {B}_{\beta }\rangle$ with the other two modes empty, and the second case corresponds to particle α being in mode $| {B}_{\alpha }\rangle$ and particle β being in mode $| {A}_{\beta }\rangle$ with the other two modes empty.

These examples also highlight two possibilities for specifying sub-systems for systems of distinguishable particles. In this case the two possibilities have similar consequences in terms of whether specific pure states are regarded as separable or entangled, based on the general form in equation (1) for separable pure states, irrespective of whether the first or second quantization approach is used. Here the first option is to regard the labeled distinguishable particles as sub-systems—in which case using first quantization the boson or fermion basis states in equation (55) would be regarded as separable states of the two sub-systems consisting of particle α and particle β. The second option would be to regard the modes or single particle states as sub-systems—in which case using second quantization the basis state for both fermions or bosons in equation (56) would be regarded as a separable state of four sub-systems consisting of modes $| {A}_{\alpha }\rangle$, $| {B}_{\alpha }\rangle$ and $| {A}_{\beta }\rangle$, $| {B}_{\beta }\rangle$. Both expressions refer to the same quantum state, and the same result regarding separability is obtained in both first and second quantization, even though the number of sub-systems differ. It indicates that either option may be chosen in regard to defining sub-systems when distinguishable particles are involved. However, it is simpler if the same option—particles or modes as sub-systems—is made for treating either identical or distinguishable particle systems and we will adopt this approach.

To highlight the distinction between the identical and distinguishable particles situation, we note that for the two distinguishable particle case treated previously we can also form entangled states from the basis states (55) or (56)

$$\begin{eqnarray}&&| {\rm{\Psi }}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| {A}_{\alpha }(\alpha )\rangle \otimes | {B}_{\beta }(\beta )\rangle \pm | {B}_{\alpha }(\alpha )\rangle \otimes | {A}_{\beta }(\beta )\rangle )\end{eqnarray} \tag{ 57 }$$

which are similar in mathematical form to (52) and (53) when written in first quantization, and which are given by

$$\begin{eqnarray}| {\rm{\Psi }}\rangle & = & \displaystyle \frac{1}{\sqrt{2}}(({| 1\rangle }_{{A}_{\alpha }}\otimes {| 0\rangle }_{{B}_{\alpha }})\otimes ({| 0\rangle }_{{A}_{\beta }}\otimes {| 1\rangle }_{{B}_{\beta }})\\ & \pm & ({| 0\rangle }_{{A}_{\alpha }}\otimes {| 1\rangle }_{{B}_{\alpha }})\otimes ({| 1\rangle }_{{A}_{\beta }}\otimes {| 0\rangle }_{{B}_{\beta }}))\end{eqnarray} \tag{ 58 }$$

when written in second quantization. However, in this case both the first and second quantization forms are clearly cases of entangled states. Whether they are regarded as entangled states of two sub-systems consisting of particle α and particle β (first option) or entangled states of the four sub-systems consisting of modes $| {A}_{\alpha }\rangle$, $| {B}_{\alpha }\rangle$ and $| {A}_{\beta }\rangle$, $| {B}_{\beta }\rangle$ (second option) depends on whether particle or modes are chosen as sub-systems.

Note however that not all basis states result in separable/entangled distinctions even in the case of identical particles. For the same two mode, two particle case as considered previously for bosons the basis vectors $| A(1)\rangle \otimes | A(2)\rangle$ or $| B(1)\rangle \otimes | B(2)\rangle$ (first quantization) or equivalently ${| 2\rangle }_{A}\otimes {| 0\rangle }_{B}$ or ${| 0\rangle }_{A}\otimes {| 2\rangle }_{B}$ (second quantization) would be regarded as separable states irrespective of whether particle or modes were chosen as the sub-systems. Entangled states such as $(| A(1)\rangle \otimes | A(2)\rangle \pm | B(1)\rangle \otimes | B(2))/\sqrt{2}$ (first quantization) or equivalently $({| 2\rangle }_{A}\otimes {| 0\rangle }_{B}\pm {| 0\rangle }_{A}\otimes {| 2\rangle }_{B})/\sqrt{2}$ (second quantization) can also be formed from the two doubly occupied basis states. There are no analogous states for fermions due to the Pauli principle.

It is worth noting that these examples illustrate the general point that just the mathematical form of the state vector or the density operator alone is not enough to determine whether a separable or an entangled state is involved. The meaning of the factors involved also has to be taken into account. Failure to realize this may lead to states being regarded as separable when they are not (see section 3.4 for further examples).

In the above discussion the symmetrization principle was complied with both in the first and second quantization treatments. It should be noted however that some authors disregard the symmetrization principle. In describing BECs [74, 98] consider states of the form

$$\begin{eqnarray}&&\widehat{\rho }=\displaystyle \sum _{R}{P}_{R}\,{\widehat{\rho }}_{R}^{1}\otimes {\widehat{\rho }}_{R}^{2}\otimes {\widehat{\rho }}_{R}^{3}\otimes ...\end{eqnarray} \tag{ 59 }$$

as defining non-entangled states, where ${\widehat{\rho }}_{R}^{i}$ is a density operator for particle i. However such a state would not in general be allowed, since the symmetrization principle would be violated unless the ${\widehat{\rho }}_{R}^{i}$ were related. For example, consider the state for two identical bosonic atoms given by

$$\begin{eqnarray}&&\widehat{\rho }={P}_{\sigma \xi }\,{\widehat{\sigma }}^{1}\otimes {\widehat{\xi }}^{2}+{P}_{\theta \eta }\,{\widehat{\theta }}^{1}\otimes {\widehat{\eta }}^{2}\end{eqnarray} \tag{ 60 }$$

and apply the permutation $\widehat{P}=\widehat{P}(1\leftrightarrow 2)$. The invariance of $\widehat{\rho }$ in general requires $\,\widehat{\sigma }=\widehat{\xi }$ and $\widehat{\theta }=\widehat{\eta }$, giving $\widehat{\rho }={P}_{\sigma }\,{\widehat{\sigma }}^{1}\otimes {\widehat{\sigma }}^{2}+{P}_{\theta }\,{\widehat{\theta }}^{1}\otimes {\widehat{\theta }}^{2}$. This is a statistical mixture of two states, one with both atoms in state $\widehat{\sigma }$, the other with both atoms in state $\widehat{\theta }$. Thus only special cases of (60) are compatible with the symmetrization principle. Of course if the atoms were all different (atom 1 a Rb⁸⁷ atom, atom 2 a Na²³ atom, ...) then the expression (60) would be a valid non-entangled state, but there the atomic sub-systems are distinguishable and symmetrization is not required. Such authors are really ignoring the symmetrization principle, and in addition are treating the individual identical particles in the BEC as separate sub-systems—a viewpoint we have described previously and will discuss further in the next section. For the present we just point out that valid quantum states must comply with the symmetrization principle.

As highlighted in the previous section 3.1, when the quantum system involves identical particles the very definition of entanglement itself requires special care in regard to identifying legitimate sub-systems. There is a long-standing debate on the issue, with at present two schools of thought—see reviews such as [36] or [40]. As explained in the previous section, the first approach is to identify mathematically labeled individual identical particles as the sub-systems [15, 74, 77, 98]. Sub-systems may of course also be sets of such individually labeled particles. This approach leads to the conclusion that symmetrization creates entanglement of identical particles. The second approach is to identify single particle states or modes that the identical particles may occupy as the sub-systems [35]. The sub-systems may of course also be sets of distinguishable modes. This approach leads to the conclusion that it is interaction processes between modes that creates entanglement of distinguishable modes.

The approach based on particle entanglement is still being used [77]. As explained in section 3.1 this is not the same as mode entanglement so tests and measures for particle entanglement will differ from those for mode entanglement. A further discussion about the distinction is given in [41]. In a recent paper Killoran et al [99] considered original states such as $(| a0(1)\rangle \otimes | a1(2)\rangle \pm | a0(2)\rangle \otimes | a1(1))/\sqrt{2}$ involving two modes a0 and a1—which were considered (based on first quantization) as an entangled state for two sub-systems consisting of particles 1 and 2, but would be considered (in second quantization) as a separable state ${| 1\rangle }_{a0}\otimes {| 1\rangle }_{a1}$ for two sub-systems consisting of modes a0 and a1. In addition there were two modes b0 and b1 which are initially unoccupied. The particles may be bosons or fermions. They envisaged converting such an input state using interferometer processes which couple A modes a0 and a1 to previously unoccupied B modes b0 and b1, into an output state—which is different. Projective measurements would then be made on the output state, based on having known numbers of particles in each of the A mode pairs a0 and a1 and in the B mode pairs b0 and b1. The projected state with one particle in the A modes and one particle in the B modes would be of the form (in second quantization) $({| 1\rangle }_{a0}\otimes {| 0\rangle }_{a1}\,\otimes$ ${| 0\rangle }_{b0}\otimes {| 1\rangle }_{b1}\,\pm$ ${| 0\rangle }_{a0}\otimes {| 1\rangle }_{a1}\otimes {| 1\rangle }_{b0}\otimes {| 0\rangle }_{b1})/\sqrt{2}$, which is a bipartite entangled state for the two pairs of modes A and B and is mathematically of the same form as the first quantization form for the original A modes state considered as an example of particle entanglement if the correspondences $| a0(1)\rangle \ \to {| 1\rangle }_{a0}\otimes {| 0\rangle }_{a1}$, $| a1(2)\rangle \to {| 0\rangle }_{b0}\otimes {| 1\rangle }_{b1}$, $| a0(2)\rangle \to {| 1\rangle }_{b0}{| 0\rangle }_{b1}$ and $| a1(1)\rangle \to$ ${| 0\rangle }_{a0}{| 1\rangle }_{a1}$ are made. Even the minus sign is obtained in the fermion case. Details are given in appendix I. Killoran et al stated that this represented a way of extracting the original symmetrization generated entanglement. However, another point of view is that the two mode interferometer process created an entangled state from a non-entangled state, and as the final measurements are still based on entanglement of modes it is hard to justify the claim that entanglement due to symmetrization exists as a directly observable basic feature in composite quantum systems—though the mapping identified in [99] is mathematically correct. Furthermore, all quantum states for identical particles are required to be symmetrized, so if symmetrization causes entanglement it differs from the numerous other controllable processes that produce entanglement by coupling the sub-systems. Since the idea of extracting entanglement due to symmetrization is of current interest, a fuller discussion of the approach by Killoran et al [99] is set out in appendix I.

However, it is generally recognized that sub-systems consisting of individually labeled identical particles are not amenable to measurements. What is distinguishable for systems of identical bosons or fermions is not the individual particles themselves—which do not carry labels, boson 1, boson 2, etc—but the single particle states or modes that the bosons may occupy. For bosonic or fermionic atoms with several hyperfine components, each component will have its own set of modes. For photons the modes may be specified via wave vectors and polarizations. Although the quantum pure states can be specified via symmetrized products of single particle states occupied by specific particles using a first quantization approach, it is more suitable to use second quantization. Here, a basis set for the quantum states of such sub-systems are the Fock states ${| n\rangle }_{A}$ ($n=0,1,2,\ldots$) etc, which specify the number of identical particles occupying the mode A, etc, so in this approach the mode is the sub-system and the Fock states give different quantum states for this sub-system. Symmetrization is built into the definition of the Fock states, so the symmetrization principle is automatically adhered to. If the atoms were fermions rather than bosons the Pauli exclusion principle would of course restrict n = 0, 1 only. In this second quantization approach situations with differing numbers of identical particles are recognized as being different states of a system consisting of a set of modes, not different systems as would be the case in first quantization. The overall system will be associated with quantum states represented in the theory by density operators and state vectors in Fock space, which includes states with total numbers of identical particles ranging from zero in the vacuum state right up to infinity. Finally, the artificial concept of entanglement due to symmetrization is replaced by the physically realistic concept of entanglement due to mode coupling.

The point of view in which the possible sub-systems A, B, etc are modes (or sets of modes) rather than particles has been adopted by several authors [17, 33–35, 51, 52] and will be the approach used here—as in [5]. To emphasize—what are or are not entangled in the present treatment involving systems of identical particles are distinguishable modes not labeled—indistinguishable—particles. Overall, the system is a collection of modes, not particles. Particles are associated with mode occupancies, and therefore related to specifying the quantum states of the system, rather than the system itself.

As pointed out in section 3.1, in the case of systems consisting entirely of distinguishable particles the sub-systems may still be regarded as sets of modes, namely those single particle states associated with the particular distinguishable particle. In this case the particle descriptor (He atom, Na atom, ...) is synonymous with its collection of modes. Here all the sub-system states are one particle states.

In terms of this approach, for non-interacting identical particles at zero temperature, the ground states for BECs and Fermi gases trapped in a harmonic potential provide examples of non-entangled states for bosonic and fermionic atoms respectively, when the sub-systems are chosen as the HO modes. In the bosonic case all the bosons occupy the lowest energy HO state, in the fermionic case one fermion occupies each HO state from the lowest up to a high energy state (the Fermi energy) until all the fermions are accommodated. On the other hand, if one particle position states spatially localized in two different regions are chosen as two sub-systems, then the same zero temperature state for the identical particle system is spatially entangled, as pointed out by Goold et al [39]. Note that in this approach states where there is only a single atom may still be entangled states—for example with two spatial modes $A,B$ the states which are a quantum superposition of the atom in each of these modes, such as the Bell state $({| 1\rangle }_{A}{| 0\rangle }_{B}+{| 0\rangle }_{A}{| 1\rangle }_{B})/\sqrt{2}$ are entangled states. For entangled states associated with the EPR paradox or for quantum teleportation, the mode functions may be localized in well-separated spatial regions—spooky action at a distance—but spatially overlapping mode functions apply in other situations. This distinction is important in discussions of quantum non-locality. Atoms in states with overall spin zero only have one internal state, but two mode systems can be created for their spatial motion using double-well trap potentials. If the wells are separated then two spatially separated modes can be created for studies of quantum non-locality. On the other hand atoms with spin 1/2 have two internal states, which constitute a two mode system. However these two modes may be associated with the same or overlapping spatial wave functions, in which case studies of quantum non-locality are precluded. These latter situation can however still lead to what is referred to as intrasystem entanglement [100]. Furthermore, as well as being distinguishable the modes can act as separate systems, with other modes being ignored. For interacting bosonic atoms this is much harder to accomplish experimentally than for the case of photons, where the relatively slow processes in which photons are destroyed in one EM field mode and created in another may require the presence of atoms as intermediaries. Two bosonic atoms in one mode may collide and rapidly disappear into other modes. However, atomic boson interactions can be made very small via Feshbach resonance methods. Near absolute zero the basic physics of a BEC in a single trap potential is describable via a one mode theory. Hence with A, B, ... signifying distinct modes, the general non-entangled state is given in equation (3) though the present paper mainly involves only two modes.

As well as the simple case where the sub-systems are all individual modes, the concept of entanglement may be extended to situations where the sub-systems are sets of modes, rather than individual modes, In this case entanglement or non-entanglement will be of these distinct sets of modes. Such a case in considered in section 4.3 of paper II, where pairs of modes associated with distinct lattice sites are considered as the sub-systems. Another example is treated in He et al [31], which involves a double well potential with each well associated with two bosonic modes, these pairs of modes being the two sub-systems. Entanglement criteria for the mode pairs based on local spin operators associated with each potential well are considered (see sections 4.2 and 5.3 of paper II). A further example is treated by Heaney et al [101], again involving four modes associated with a double well potential. As in the previous example, each mode pair is associated with the same well in the potential, but here a Bell entanglement test was obtained for pairs of modes in the different wells. The concept of entanglement of sets of modes is a straightforward extension of the basic concept of entanglement of individual modes.

The question of what quantum states—entangled or not—are possible in the non-relativistic quantum physics of a system of identical bosonic particles—such as bosonic atoms or photons—has been the subject of much discussion. Whether entangled or not it is generally accepted that there is a SSR that prohibits quantum superposition states of the form

$$\begin{eqnarray}| {\rm{\Phi }}\rangle & = & \displaystyle \sum _{N=0}^{\infty }{C}_{N}| N\rangle ,\qquad \widehat{\rho }=\displaystyle \sum _{N=0}^{\infty }{| {C}_{N}| }^{2}| N\rangle \langle N| \\ & & +\displaystyle \sum _{N=0}^{\infty }\ \displaystyle \,\text{}\sum _{M=0}^{\infty }(1-{\delta }_{N,M}){C}_{N}\,{C}_{M}^{* }| N\rangle \langle M| \end{eqnarray} \tag{ 61 }$$

being quantum states when they involve Fock states $| N\rangle$ with differing total numbers N of particles. The density operator for such a state would involve coherences between states with differing N. Although such superpositions—such as the Glauber coherent state $| \alpha \rangle$, where ${C}_{N}=\exp (-| \alpha {| }^{2}/2){\alpha }^{N}/\sqrt{N!}$—do have a useful mathematical role, they do not represent actual quantum states according to the SSR. The papers by Sanders et al [68] and Cable et al [102] are examples of applying the SSR for optical fields, but also using the mathematical features of coherent states to treat phenomena such as interference between independent lasers. The SSR indicates that the most general quantum state for a system of identical bosonic particles can only be of the form

$$\begin{eqnarray}\widehat{\rho } & = & \displaystyle \sum _{N=0}^{\infty }\displaystyle \sum _{{\rm{\Phi }}}{P}_{{\rm{\Phi }}N}(| {{\rm{\Phi }}}_{N}\rangle \langle {{\rm{\Phi }}}_{N}| )\\ | {{\rm{\Phi }}}_{N}\rangle & = & \displaystyle \sum _{i}{C}_{i}^{N}| N\,i\rangle ,\end{eqnarray} \tag{ 62 }$$

where $| {{\rm{\Phi }}}_{N}\rangle$ is a quantum superposition of states $| N\,i\rangle$ each of which involves exactly N particles, and where different states with the same N are designated as $| N\,i\rangle$. This state $\widehat{\rho }$ is a statistical mixture of states, each of which contains a specific number of particles. Such a SSR is referred to as a global SSR, as it applies to the system as a whole. Mathematically, the global particle number SSR can be expressed as

$$\begin{eqnarray}&&[\widehat{N},\widehat{\rho }]=0,\end{eqnarray} \tag{ 63 }$$

where $\widehat{N}$ is the total number operator.

Examples of a state vector $| {{\rm{\Phi }}}_{N}\rangle$ for an entangled pure state [34] and a density operator $\widehat{\rho }$ for a non-entangled mixed [39] state for a two mode bosonic system, both of which are possible quantum states are

$$\begin{eqnarray}&&| {{\rm{\Phi }}}_{N}\rangle =\sum _{k=0}^{N}C(N,k){| k\rangle }_{A}\otimes {| N-k\rangle }_{B},\end{eqnarray} \tag{ 64 }$$

$$\begin{eqnarray}&&\widehat{\rho }=\sum _{k=0}^{N}P(k){| k\rangle }_{A}{\langle k| }_{A}\,\otimes {| N-k\rangle }_{B}{\langle N-k| }_{B}.\end{eqnarray} \tag{ 65 }$$

The entangled pure state is a superposition of product states with k bosons in mode A and the remaining N − k bosons in mode B. Every term in the superposition is associated with the same total boson number N. The non-entangled mixed state is a statistical mixture of product states also with k bosons in mode A and the remaining N − k bosons in mode B. Every term in the statistical mixture is associated with the same total boson number N. For the case of a two mode fermionic system the Pauli exclusion principle restricts the number of possible fermions to two, with at most one fermion in each mode. Expressions for a state with exactly N = 2 fermions are

$$\begin{eqnarray}&&| {{\rm{\Phi }}}_{2}\rangle ={| 1\rangle }_{A}\otimes {| 1\rangle }_{B},\end{eqnarray} \tag{ 66 }$$

$$\begin{eqnarray}&&\widehat{\rho }={| 1\rangle }_{A}{\langle 1| }_{A}\,\otimes {| 1\rangle }_{B}{\langle 1| }_{B}.\end{eqnarray} \tag{ 67 }$$

Neither state is entangled and both are the same pure state since $\widehat{\rho }=| {{\rm{\Phi }}}_{2}\rangle \langle {{\rm{\Phi }}}_{2}|$. Although the SSRs and symmetrization principle also applies to fermions, as indicated in the Introduction this paper is focused on bosonic systems, and it will be assumed that the modes are bosonic unless indicated otherwise.

Bell states [17, 18] for N = 1 bosons provide important examples of entangled two mode pure quantum states that are compliant with the global particle number SSR. The modes are designated $A,B$ and the Fock states are in general $| {n}_{A},{n}_{B}\rangle$. These Bell states may be written

$$\begin{eqnarray}| {{\rm{\Psi }}}_{{AB}}^{-}\rangle & = & \displaystyle \frac{1}{\sqrt{2}}(| {0}_{A},{1}_{B}\rangle -| {1}_{A},{0}_{B})\\ | {{\rm{\Psi }}}_{{AB}}^{+}\rangle & = & \displaystyle \frac{1}{\sqrt{2}}(| {0}_{A},{1}_{B}\rangle +| {1}_{A},{0}_{B})\end{eqnarray} \tag{ 68 }$$

Neither of these states is separable. There are also two other two mode Bell states given by

$$\begin{eqnarray}| {{\rm{\Phi }}}_{{AB}}^{-}\rangle & = & \displaystyle \frac{1}{\sqrt{2}}(| {0}_{A},{0}_{B}\rangle -| {1}_{A},{1}_{B}\rangle )\\ | {{\rm{\Phi }}}_{{AB}}^{+}\rangle & = & \displaystyle \frac{1}{\sqrt{2}}(| {0}_{A},{0}_{B}\rangle +| {1}_{A},{1}_{B}\rangle ).\end{eqnarray} \tag{ 69 }$$

These however are not compliant with the global particle number SSR. Linear combinations $(| {{\rm{\Phi }}}_{{AB}}^{-}\rangle +| {{\rm{\Phi }}}_{{AB}}^{+}\rangle )/\sqrt{2}=| {0}_{A},{0}_{B}\rangle$ and $(-| {{\rm{\Phi }}}_{{AB}}^{-}\rangle +| {{\rm{\Phi }}}_{{AB}}^{+}\rangle )/\sqrt{2}=| {1}_{A},{1}_{B}\rangle$ are global particle number SSR compliant and also separable, corresponding to states with N = 0 and N = 2 bosons respectively.

It is important to realize that such SSRs [59] are different constraints to those imposed by conservation laws, as emphasized by Bartlett et al [67]. For example, the conservation law on total particle number only leads to the requirement on the superposition state $| {\rm{\Phi }}\rangle$ that the $| {C}_{N}{| }^{2}$ are time independent, it does not require only one C_N being non-zero. They are however related, as is discussed in section 3.3.1 and appendix K where the SSRs based on particle number are related to invariances of the density operator under changes of phase reference frames. This involves considering groups of phase changing operators $\widehat{T}({\theta }_{a})=\exp ({\rm{i}}{\widehat{N}}_{a}{\theta }_{a})$ when considering local particle number SSR for single modes in the context of separable states, or $\widehat{T}(\theta )=\exp ({\rm{i}}\widehat{N}\theta )$ when considering global particle number SSR in the context of multimode entangled states. SSRs are broad in their scope, forbidding quantum superpositions of states of systems with differing charge, differing baryon number and differing statistics. Thus a combined system of a hydrogen atom and a helium ion does not exist in quantum states that are linear combinations of hydrogen atom states and helium ion states—the SSRs on both charge and baryon number preclude such states. The basis quantum states for such a combined system would involve symmetrized tensor products of hydrogen atom and helium ion states, not linear combinations—symmetrization being required because the system contains two identical electrons. On the other hand, SSRs do not prohibit quantum superpositions of states of systems with differing energy, angular or linear momenta—other physical quantities that may also be conserved. Thus in a hydrogen atom quantum superpositions of states with differing energy and angular momentum quantum numbers are allowed quantum states.

However, conservation laws on total particle number (such as apply in the case of massive bosons) are relevant to showing that multi-mode states generated via total particle number conserving processes from an initial separable state will be global SSR compliant if the sub-systems in the initial state are local particle number SSR compliant, and will not be if the initial state involves a sub-system state that is not local particle number SSR compliant. For simplicity we consider two sub-systems A and B with the initial state

$$\begin{eqnarray}&&\widehat{\rho }(0)=\displaystyle \sum _{R}{P}_{R}\,{\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}.\end{eqnarray} \tag{ 70 }$$

If $\widehat{U}(t)$ is the evolution operator where $\widehat{\rho }(t)\,={\widehat{U}(t)\widehat{\rho }(0)\widehat{U}(t)}^{\dagger }$ and the processes are number conserving then $[\widehat{N},\widehat{U}(t)]=0$. We then have

$$\begin{eqnarray}[\widehat{N},\widehat{\rho }(t)] & = & \widehat{U}(t)[\widehat{N},\widehat{\rho }(0)]\,\widehat{U}{(t)}^{\dagger }\\ & = & \widehat{U}(t)\displaystyle \sum _{R}{P}_{R}([{\widehat{N}}_{A},{\widehat{\rho }}_{R}^{A}]\otimes {\widehat{\rho }}_{R}^{B}\\ & & +{\widehat{\rho }}_{R}^{A}\otimes [{\widehat{N}}_{B},{\widehat{\rho }}_{R}^{B}])\widehat{U}{(t)}^{\dagger }.\end{eqnarray} \tag{ 71 }$$

Hence if ${\widehat{\rho }}_{R}^{A}$ and ${\widehat{\rho }}_{R}^{B}$ are local particle number SSR compliant, then $[{\widehat{N}}_{A},{\widehat{\rho }}_{R}^{A}]$ and $[{\widehat{N}}_{B},{\widehat{\rho }}_{R}^{B}]$ are zero, showing that $[\widehat{N},\widehat{\rho }(t)]=0$ so the state is global particle number SSR compliant. On the other hand if $[\widehat{N},\widehat{\rho }(t)]=0$ we see that $[\widehat{N},\widehat{\rho }(0)]={\sum }_{R}{P}_{R}([{\widehat{N}}_{A},{\widehat{\rho }}_{R}^{A}]\otimes {\widehat{\rho }}_{R}^{B}+{\widehat{\rho }}_{R}^{A}\otimes [{\widehat{N}}_{B},{\widehat{\rho }}_{R}^{B}])=0$. By taking Tr_A and Tr_B of this result gives ${\sum }_{R}{P}_{R}[{\widehat{N}}_{A},{\widehat{\rho }}_{R}^{A}]={\sum }_{R}{P}_{R}\,[{\widehat{N}}_{B},{\widehat{\rho }}_{R}^{B}]=0$. This shows that both of the reduced density operators ${\sum }_{R}{P}_{R}{\widehat{\rho }}_{R}^{A}$ and ${\sum }_{R}{P}_{R}{\widehat{\rho }}_{R}^{B}$ must be local particle number SSR compliant, which amounts to requiring the sub-system density operators to be local particle number SSR compliant. This situation applies even when there is coupling between modes, provided the interaction is number conserving—such as a coupling given by $\widehat{V}=\lambda \widehat{a}{\widehat{b}}^{\dagger }+{HC}$. Analogous results apply for systems of massive bosons if there are more than two modes involved, where again global SSR compliance involves the total particle number since even with interactions there is total number conservation. For example with three modes in coupled BECs, interactions of the form $\widehat{V}=\lambda {(\widehat{c})}^{2}{\widehat{a}}^{\dagger }{\widehat{b}}^{\dagger }+{HC}$ in which two bosons are annihilated in mode C and one boson is created in each of modes A and B are consistent with total particle number conservation and lead to global SSR involving the total particle number.

Although outside the focus of this paper, it is worth pointing out that somewhat different considerations apply to photons. Single non-interacting modes, such as are discussed in the context of separable states do have a conservation law for the photon number in that mode. The applicability (or otherwise) of the local particle number SSR for the sub-system density operators in a separable state is discussed in appendix L. In the case of interacting photonic modes there may be no conservation law associated with total photon number and it may be thought that no global SSR would apply. However, other global SSR involving combinations of the mode photon numbers may still apply. As an example, we consider a three mode situation in a non-degenerate parametric amplifier, where the basic generation process involves one pump photon of frequency ${\omega }_{C}={\omega }_{A}+{\omega }_{B}$ being destroyed and one photon created in each of modes A and B. The interaction term is $\widehat{V}=\lambda \widehat{c}{\widehat{a}}^{\dagger }{\widehat{b}}^{\dagger }+{HC}$. It is straight-forward to show that a total quanta number operator ${\widehat{N}}_{{\rm{tot}}}={\widehat{N}}_{A}+{\widehat{N}}_{B}+2{\widehat{N}}_{C}$ commutes with the Hamiltonian. The situation is analogous to the atom–molecule system treated in appendix M. Thus ${\widehat{N}}_{{\rm{tot}}}$ is conserved and we can then consider a group of phase changing operators $\widehat{T}(\theta )=\exp ({\rm{i}}{\widehat{N}}_{{\rm{tot}}}\theta )$ and show that there could be a global SSR for the three mode system, but now involving the total quanta number ${N}_{A}+{N}_{B}+2{N}_{C}$. The pure state which is often used in a quantum treatment of the non-degenerate parametric amplifier $| {\rm{\Psi }}\rangle =$ ${\sum }_{n}{C}_{n}{| n\rangle }_{A}\otimes {| n\rangle }_{B}\otimes {| N-n\rangle }_{C}$ is global SSR compliant in terms of the modified ${\widehat{N}}_{{\rm{tot}}}$, since in every term ${N}_{A}+{N}_{B}+2{N}_{C}=2N$ and there are no coherences between terms with different N_tot. For the non-degenerate parametric amplifier case an analogous treatment to that for number conserving processes shows that if

$$\begin{eqnarray}&&\widehat{\rho }(0)=\displaystyle \sum _{R}{P}_{R}\,{\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}\otimes {\widehat{\rho }}_{R}^{C}\end{eqnarray} \tag{ 72 }$$

then using $[{\widehat{N}}_{{\rm{tot}}},\widehat{U}(t)]=0$ we have

$$\begin{eqnarray}&&[{\widehat{N}}_{{\rm{tot}}},\widehat{\rho }(t)]=\widehat{U}(t)[{\widehat{N}}_{{\rm{tot}}},\widehat{\rho }(0)]\,\widehat{U}{(t)}^{\dagger }=\widehat{U}(t)\displaystyle \sum _{R}{P}_{R}\\ &&\quad \times \,\left(\begin{array}{c}[{\widehat{N}}_{A},{\widehat{\rho }}_{R}^{A}]\otimes {\widehat{\rho }}_{R}^{B}\otimes {\widehat{\rho }}_{R}^{C}+{\widehat{\rho }}_{R}^{A}\otimes [{\widehat{N}}_{B},{\widehat{\rho }}_{R}^{B}]\otimes {\widehat{\rho }}_{R}^{C}\\ +2{\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}\otimes [{\widehat{N}}_{C},{\widehat{\rho }}_{R}^{C}]\end{array}\right)\widehat{U}{(t)}^{\dagger }.\end{eqnarray} \tag{ 73 }$$

Hence if ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$ and ${\widehat{\rho }}_{R}^{C}$ are local particle number SSR compliant, then  $[{\widehat{N}}_{{\rm{tot}}},\widehat{\rho }(t)]=0$ so the state is SSR compliant, but with global total quanta number ${\widehat{N}}_{{\rm{tot}}}$. On the other hand if $[{\widehat{N}}_{{\rm{tot}}},\widehat{\rho }(t)]=0$ we find that ${\sum }_{R}{P}_{R}\,[{\widehat{N}}_{A},{\widehat{\rho }}_{R}^{A}]\,={\sum }_{R}{P}_{R}\,[{\widehat{N}}_{B},{\widehat{\rho }}_{R}^{B}]={\sum }_{R}{P}_{R}\,[{\widehat{N}}_{C},{\widehat{\rho }}_{R}^{C}]=0$. This shows that each of the reduced density operators ${\sum }_{R}{P}_{R}{\widehat{\rho }}_{R}^{A}$, ${\sum }_{R}{P}_{R}{\widehat{\rho }}_{R}^{B}$ and ${\sum }_{R}{P}_{R}{\widehat{\rho }}_{R}^{C}$ must be local particle number SSR compliant, which amounts to requiring the sub-system density operators to be local particle number SSR compliant.

There are two types of justification for applying the SSRs for systems of identical particles. The first approach is based on simple considerations and will be outlined below in this subsection. The second approach [40, 60, 63, 64, 66–71] is more sophisticated and involves linking the absence or presence of SSR to whether or not there is a suitable reference frame in terms of which the quantum state is described, and is outlined in the next subsection and appendix K. The key idea is that SSR are a consequence of considering the description of a quantum state by an external observer (Charlie) whose phase reference frame has an unknown phase difference from that of an observer ((Alice) more closely linked to the system being studied. Thus, while Alice's description of the quantum state may violate the SSR, the description of the same quantum state by Charlie will not. In the main part of this paper the density operator $\widehat{\rho }$ used to describe the various quantum states will be that of the external observer (Charlie).

There is a further reference frame based justification for the SSRs proposed by Stenholm [103] involving Galilean transformations—corresponding to describing the system from the point of view of an observer moving with a constant velocity with respect to the original observer, and where the two observers have identical clocks. A consideration of where the relative velocity is unknown might lead to the conclusion that SSR non-compliant coherences do not occur. For completeness, a brief discussion is included in appendix K.

A number of straightforward reasons have been given in the Introduction for why it is appropriate to apply the superselection rule to exclude quantum superposition states of the form (61) as quantum states for systems of identical particles, and these will now be considered in more detail.

Firstly, no way is known for creating such states. The Hamiltonian for such a system commutes with the total boson number operator, resulting in the ${| {C}_{N}| }^{2}$ remaining constant, so the quantum superposition state would need to have existed initially. In the simplest case of non-interacting bosonic atoms, the Fock states are also energy eigenstates, such Fock states involve total energies that differ by energies of order the rest mass energy mc², so a coherent superposition of states with such widely differing energies would at least seem unlikely in a non-relativistic theory, though for massless photons this would not be an issue as the energy differences are of order the photon energy ${\hslash }\omega$. The more important question is: Is there a non-relativistic quantum process could lead to the creation of such a state? Processes such as the dissociation of M diatomic molecules into up to $2M$ bosonic atoms under Hamiltonian evolution involve entangled atom–molecule states of the form

$$\begin{eqnarray}&&| {\rm{\Phi }}\rangle =\sum _{m=0}^{M}{C}_{m}{| M-m\rangle }_{{\rm{mol}}}\otimes {| 2m\rangle }_{{\rm{atom}}}\end{eqnarray} \tag{ 74 }$$

but the reduced density operator for the bosonic atoms is

$$\begin{eqnarray}&&{\widehat{\rho }}_{{\rm{atoms}}}=\sum _{m=0}^{M}{| {C}_{m}| }^{2}{(| 2m\rangle \langle 2m| )}_{{\rm{atom}}}\end{eqnarray} \tag{ 75 }$$

which is a statistical mixture of states with differing atom numbers with no coherence terms between such states. Such statistical mixtures are valid quantum states, corresponding to a lack of a priori knowledge of how many atoms have been produced. To obtain a quantum superposition state for the atoms alone, the atom–molecule state vector would need to evolve at some time into the form

$$\begin{eqnarray}&&| {\rm{\Phi }}\rangle =\sum _{m=0}^{M}{B}_{m}{| M-m\rangle }_{{\rm{mol}}}\otimes \sum _{n=0}^{M}{A}_{2n}{| 2n\rangle }_{{\rm{atom}}},\end{eqnarray} \tag{ 76 }$$

where the separate atomic system is in the required quantum superposition state. However if such a state existed there would be terms with at least one non-zero product of coefficient ${B}_{m}{A}_{2n}$ involving product states ${| M-m\rangle }_{{\rm{mol}}}\,\otimes {| 2n\rangle }_{{\rm{atom}}}$ with $n\ne m$ if the state $| {\rm{\Phi }}\rangle$ is not just in the entangled form (74). However, the presence of such a term would mean that the conservation law involving the number of molecules plus two times the number of atoms was violated. This is impossible, so such an evolution is not allowed.

Secondly, no way is known for measuring all the properties of such states, even if they existed. If a state such as (61) did exist then the amplitudes C_N would oscillate with frequencies that differ by frequencies of order ${\omega }_{C}={{mc}}^{2}/{\hslash }$ (the Compton frequency, which is $\gtrsim$10²⁵ Hz for massive bosons) even if boson–boson interactions were included Suppose a SSR violating state such as (61) could be created. To distinguish the phases of the ${C}_{N}\,$ in order to verify the existence of the non-SSR-complying state, we need to measure the mean value $\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle$ of observables—such as quadrature operators in the case of a single mode $\ {\widehat{x}}_{A}=(\widehat{a}+{\widehat{a}}^{\dagger })/\sqrt{2}$ and ${\widehat{p}}_{A}=(\widehat{a}-{\widehat{a}}^{\dagger })/{\rm{i}}\sqrt{2}$,—which have non-zero matrix elements between states with differing N. To observe the oscillations in $\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle$ that would demonstrate the existence of a non-SSR-complying state, the mean value $\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle$ of such an observable would need to change by a detectable amount in a time scale ${\rm{\Delta }}t$ short compared to the period of the oscillations—that is, short compared to the Compton period $2\pi /{\omega }_{C}$. A reasonable measure for the minimal detectable change would be the original standard deviation ${\rm{\Delta }}{{\rm{\Omega }}}_{A}=\sqrt{\langle {\rm{\Delta }}{\widehat{{\rm{\Omega }}}}_{A}^{2}\rangle }$ for measurements on ${\widehat{{\rm{\Omega }}}}_{A}$ (where the fluctuation is ${\rm{\Delta }}{\widehat{{\rm{\Omega }}}}_{A}={\widehat{{\rm{\Omega }}}}_{A}-\langle {\widehat{{\rm{\Omega }}}}_{A}\rangle$). However there is an uncertainty principle relationship between the time scale ${\rm{\Delta }}t$ during which the mean value changes by a standard deviation and the standard deviation in the energy ${\rm{\Delta }}{H}_{A}=\sqrt{\langle {\rm{\Delta }}{\widehat{H}}_{A}^{2}\rangle }$ that applies for the quantum state being studied, namely ${\rm{\Delta }}{H}_{A}\,{\rm{\Delta }}t\geqslant {\hslash }/2$. As we have seen, we require ${\rm{\Delta }}t\ll 2\pi /{\omega }_{C}$ so that from the energy–time uncertainty principle we see that ${\rm{\Delta }}{H}_{A}\gg {\hslash }{\omega }_{C}={{mc}}^{2}$. The consequence is that the uncertainty in energy for the proposed SSR violating quantum state must be large compared to the rest mass energy in order to observe the effects of SSR-non-compliance. This shows that the proposed observation is not possible for the non-relativistic quantum systems we are considering, since the energy uncertainty is large enough to allow the existence of boson–anti-boson pairs. This argument about the non-observability of non-SSR-complying states in the case of massive particles is an in-principle demonstration that the SSR non-complying oscillations are too large to be followed, and is much stronger than one that is merely based on the current lack of an appropriate technology. We note in passing that the same discussion does not apply to photons, whose rest mass energy is zero.

Thirdly, there is no need to invoke the existence of such states in order to understand coherence and interference effects. It is sometimes thought that states involving quantum superpositions of number states are needed for discussing coherence and interference properties of BECs, and some papers describe the state via the Glauber coherent states. However, as Leggett [78] has pointed out (see also Bach et al [104], Dalton and Ghanbari [38]), a highly occupied number state for a single mode with N bosons has coherence properties of high order n, as long as $n\ll N$. The introduction of a Glauber coherent state is not required to account for coherence effects. Even the well-known presence of spatial interference patterns produced when two independent BECs are overlapped can be accounted for via treating the BECs as Fock states. The interference pattern is built up as a result of successive boson position measurements [68, 102, 105].

We now prove a theorem concerning quantum correlation functions for bosonic systems with two modes A and B. This theorem is relevant for possible tests on whether the SSR apply, as we will see in the next section.

Theorem. If a state is global particle number SSR compliant then all quantum correlation functions $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ for which $n+l\ne m+k$ must be zero. The proof of this theorem is given in appendix J.

The last result for the general two mode quantum correlation function $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ is relevant to the various experimental measurements that are discussed in the accompanying paper II. For example, as we will see ${\langle {\widehat{S}}_{x}\rangle }_{\rho }$ is a combination of $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ with $n=1,m=0,l=0,k=1$ and $n=0,m=1,l=1,k=0$, and ${\langle {\rm{\Delta }}{\widehat{S}}_{x}^{2}\rangle }_{\rho }$ $=\,{\langle {\widehat{S}}_{x}^{2}\rangle }_{\rho }-{\langle {\widehat{S}}_{x}\rangle }_{\rho }^{2}$ would involve terms such as $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ with $n=2,m=0,l=0,k=2$ and $n=0,m=2,l=2,k=0$, and $n=1,m=1,l=1,k=1$ from ${\langle {\widehat{S}}_{x}^{2}\rangle }_{\rho }$. All of these have $n+l=m+k$, so they can be non-zero for globally SSR compliant states. The question then arises—what sort of quantity of the form $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ could be used to see if the quantum state was not globally SSR compliant? The answer is seen in terms of two corollaries to the last theorem. The proof of these corollaries is given in appendix J.

Corollary 1. If we find that any of the quantum correlation functions $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ are non-zero when $n+l\,\ne m+k$ then the state is not global particle number SSR compliant.

Corollary 2. Measurements of the QCF $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ when $n+l=m+k$ cannot determine whether or not the state includes a contribution that is global particle number SSR non-compliant.

The first corollary indicates what type of measurement is needed to see if SSR non compliant states exist. Quantities of the type $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ are measured for which $n+l\ne m+k$. If we find any that are non-zero we can then conclude that we have found a state which is not global particle number SSR compliant. The second corollary shows that measurements of this type with $n+l=m+k$ would not respond to the presence of contribution to the density operator that is not globally SSR compliant.

Hence the conclusion is that a quantum correlation function of the form $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}{({\widehat{b}}^{\dagger })}^{l}{(\widehat{b})}^{k}\rangle$ must be measured for cases where $n+l\ne m+k$ and a non-zero measurement result must be found. If it is, then we would have demonstrated that the state is not globally SSR compliant. The simplest case would be to find a non-zero result for ${\langle \widehat{a}\rangle }_{\rho }$ or ${\langle \widehat{b}\rangle }_{\rho }$.

Similar considerations apply to local SSR compliance in the sub-system states. For sub-system a a QCF of the form $\langle {({\widehat{a}}^{\dagger })}^{n}{(\widehat{a})}^{m}\rangle$ must be measured for cases where $n\ne m$ and a non-zero measurement result must be found. If it is, then we would have demonstrated that the state is not locally SSR compliant. The simplest case would be to find a non-zero result for ${\langle \widehat{a}\rangle }_{\rho }$.

Various measures of entanglement have been referred to in the Introduction. The so-called particle entanglement measure is one that takes into account both the SSR and particle identity. Wiseman et al have also treated entanglement for pure states [45] and mixed states [106] in identical particle systems, applying both the symmetrization principle and SSRs, invoking the argument that without a phase reference the quantum state must be comply with the local (and global) particle number SSR. This is essentially the same approach as in [5, 60, 71] and in the present paper. For two mode systems the observable system density operator $\widetilde{\widehat{\rho }}$ is obtained from the density operator $\widehat{\rho }$ that would apply if such a phase reference existed via the expression

$$\begin{eqnarray}&&\widetilde{\widehat{\rho }}=\sum _{{n}_{A}{n}_{B}}\,{\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}\,\widehat{\rho }\,{\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}=\sum _{{n}_{A}{n}_{B}}\,{\widehat{\rho }}^{({n}_{A}{n}_{B})},\end{eqnarray} \tag{ 77 }$$

where ${\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}\,=$ $\,{\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}^{2}$ is a projector onto sub-system states with n_A, n_B particles in modes A, B respectively. Note that ${\widehat{\rho }}^{({n}_{A}{n}_{B})}={\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}$ $\widehat{\rho }\,{\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}$ is not normalized to unity. In fact the probability that there are n_A, n_B particles in modes A, B respectively is given by ${P}_{{n}_{A}{n}_{B}}={\rm{Tr}}({\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}$ $\widehat{\rho }\,{\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}})\,={\rm{Tr}}({\widehat{\rho }}^{({n}_{A}{n}_{B})})$, so ${\rm{Tr}}(\widetilde{\widehat{\rho }})={\sum }_{{n}_{A}{n}_{B}}{P}_{{n}_{A}{n}_{B}}=1$. For separable states defined here as in equation (3), the expression in (77) for the density operator is the same as that used here, since with $\widehat{\rho }$ given by equation (3) and with ${\sum }_{{n}_{A}{n}_{B}}\,{\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}({\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}){\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}\,={\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}$ it is easy to show that $\widetilde{\widehat{\rho }}={\widehat{\rho }}_{{\rm{sep}}}$. For general mixed states Wiseman et al introduce in [45] the idea of particle entanglement by defining its measure ${E}_{P}(\widehat{\rho }\,)$ by

$$\begin{eqnarray}&&{E}_{P}(\widehat{\rho }\,)=\sum _{{n}_{A}{n}_{B}}\,{P}_{{n}_{A}{n}_{B}}{E}_{M}({\widehat{\rho }}^{({n}_{A}{n}_{B})})={E}_{P}(\widetilde{\widehat{\rho }}\,),\end{eqnarray} \tag{ 78 }$$

where ${E}_{M}({\widehat{\rho }}^{({n}_{A}{n}_{B})})$ is a measure of the mode entanglement associated with the (unnormalized) state ${\widehat{\rho }}^{({n}_{A}{n}_{B})}$. This might be taken as the entropy of mode entanglement ${E}_{M}(\widehat{\sigma })=-{\rm{Tr}}({\widehat{\sigma }}_{A}\,\mathrm{ln}{\widehat{\sigma }}_{A})$ for normalized density operators $\widehat{\sigma }$, where the reduced density operator for mode A is ${\widehat{\sigma }}_{A}={{\rm{Tr}}}_{B}(\widehat{\sigma })$. Note that from ${\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}{\widehat{{\rm{\Pi }}}}_{{m}_{A}{m}_{B}}={\delta }_{{n}_{A}{m}_{A}}{\delta }_{{n}_{B}{m}_{B}}$ ${\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}$ the particle entanglement measure ${E}_{P}(\widehat{\rho })$ is the same for $\widetilde{\widehat{\rho }}$, the observable density operator for the system. The operational definition of E_P is the maximal amount of entanglement which can be produced between the two (non-SSR-constrained sub-systems by local operations. In the case of the separable state for modes A, B given in (3) it is straightforward to show that

$$\begin{eqnarray}&&{\widehat{\rho }}_{{\rm{sep}}}^{({n}_{A}{n}_{B})}=\displaystyle \sum _{R}{P}_{R}({\widehat{{\rm{\Pi }}}}_{{n}_{A}}{\widehat{\rho }}_{R}^{A}\,{\widehat{{\rm{\Pi }}}}_{{n}_{A}})\otimes ({\widehat{{\rm{\Pi }}}}_{{n}_{B}}{\widehat{\rho }}_{R}^{B}\,{\widehat{{\rm{\Pi }}}}_{{n}_{B}}),\end{eqnarray} \tag{ 79 }$$

$$\begin{eqnarray}&&{P}_{{n}_{A}{n}_{B}}^{{\rm{sep}}}=\displaystyle \sum _{R}{P}_{R}\,{P}_{{n}_{A}}({\widehat{\rho }}_{R}^{A}){P}_{{n}_{B}}({\widehat{\rho }}_{R}^{B}),\end{eqnarray} \tag{ 80 }$$

where ${\widehat{{\rm{\Pi }}}}_{{n}_{A}}$ and ${\widehat{{\rm{\Pi }}}}_{{n}_{B}}$ are projectors onto sub-system states in modes A, B respectively with n_A and n_B particles in the respective modes (${\widehat{{\rm{\Pi }}}}_{{n}_{A}{n}_{B}}={\widehat{{\rm{\Pi }}}}_{{n}_{A}}\otimes {\widehat{{\rm{\Pi }}}}_{{n}_{B}}$), with ${P}_{{n}_{A}}({\widehat{\rho }}_{R}^{A})\,={{\rm{Tr}}}_{A}({\widehat{{\rm{\Pi }}}}_{{n}_{A}}{\widehat{\rho }}_{R}^{A})$ and ${P}_{{n}_{B}}({\widehat{\rho }}_{R}^{B})={{\rm{Tr}}}_{B}({\widehat{{\rm{\Pi }}}}_{{n}_{B}}{\widehat{\rho }}_{R}^{B})$ being the probabilities of finding n_A and n_B particles in the respective modes when the corresponding sub-system states are ${\widehat{\rho }}_{R}^{A}$ and ${\widehat{\rho }}_{R}^{B}$. Since the state ${\widehat{\rho }}_{{\rm{sep}}}^{({n}_{A}{n}_{B})}$ is clearly a separable state of the form (3) for the modes A, B, the corresponding measure of mode entanglement must be zero. It then follows from the general expression (78) that the particle entanglement measure is also zero for the separable state. This is as expected.

$$\begin{eqnarray}&&{E}_{P}({\widehat{\rho }}_{{\rm{sep}}}\,)=0.\end{eqnarray} \tag{ 81 }$$

For the pure states considered in [45] we note that among them is the two boson state ${| 1\rangle }_{A}\otimes {| 1\rangle }_{B}$ which has one boson in each of the two modes A, B. The particle entanglement measure ${E}_{P}(\widehat{\rho })$ is zero for this state (where $\widehat{\rho }={(| 1\rangle \langle 1)}_{A}\otimes {(| 1\rangle \langle 1)}_{B}$), consistent with it being a separable rather than an entangled state. This indicates that Wiseman et al [45] do not consider that entanglement occurs due to symmetrization, as the first quantization form for the state might indicate. However, finding ${E}_{P}(\widehat{\rho })$ to be zero does not always shows that the state is separable, as the case of the relative phase state (defined in appendix O of paper II, see also [38]) shows. As is shown there, ${E}_{P}(\widehat{\rho })=0$ for the relative phase state, yet the state is clearly an entangled one. Just as some entangled states have zero spin squeezing, some entangled states may be associated with a zero particle entanglement measure. Nevertheless a non-zero result for the particle entanglement measure ${E}_{P}(\widehat{\rho }\,)$ shows that the state must be entangled—again we have a sufficiency test. However, as in the case of other entanglement measures the problem with using the particle entanglement measure to detect entangled states is that there is no obvious way to measure it experimentally.

The discussion of the SSR issue in terms of reference systems is quite complex and too lengthy to be covered in the body of this paper. However, in view of the wide use of the reference frame approach a full outline is presented in appendix K. The key idea is that there are two observers—Alice and Charlie—who are describing the same prepared system in terms of their own reference frames and hence their descriptions involve two different quantum states. The reference systems are macroscopic systems in states where the behavior is essentially classical, such as large magnets that can be used to define Cartesian axes or BEC in Glauber coherent states that are introduced to define a phase reference. The relationship between the two reference systems is represented by a group of unitary transformation operators listed as $\widehat{T}(g)$, where the particular transformation (translation or rotation of Cartesian axes, phase change of phase references, ...) that changes Alice's reference system into Charlie's is denoted by g. Alice describes the quantum state via her density operator, whereas Charlie is the external observer whose specification of the same quantum state via his density operator is of most interest. There are two cases of importance, Situation A—where the relationship between Alice's and Charlie's reference frame is is known and specified by a single parameter g, and Situation B—where on the other hand the relationship between frames is completely unknown, all possible transformations g must be given equal weight. Situation A is not associated with SSR, whereas Situation B leads to SSR. The relationship between Alice's and Charlie's density operators is given in terms of the transformation operators (see appendix equation (221) for Situation A and appendix equation (222) for Situation B). In Situation B there is often a qualitative change between Alice's and Charlie's description of the same quantum state, with pure states as described by Alice becoming mixed states when described by Charlie. It is Situation B with the U(1) transformation group—for which number operators are the generators—that is of interest for the single or multi-mode systems involving identical bosons on which the present paper focuses. An example of the qualitative change of behavior for the single mode case is that if it is assumed that Alice could prepare the system in a Glauber coherent pure state—which involves SSR breaking coherences between differing number states—then Charlie would describe the same state as a Poisson statistical mixture of number states—which is consistent with the operation of the SSR. Thus the SSR applies in terms of external observer Charlie's description of the state. This is how the dispute on whether the state for single mode laser is a coherent state or a statistical mixture is resolved—the two descriptions apply to different observers—Alice and Charlie. On the other hand there are quantum states such as Fock states and Bell states which are described the same way by both Alice and Charlie, even in Situation B. The general justification of the SSR for Charlie's density operator description of the quantum state in Situation B is derived in terms of the irreducible representations of the transformation group, there being no coherences between states associated with differing irreducible representations (see appendix equation (246)). For the particular case of the U(1) transformation group the irreducible representations are associated with the total boson number for the system or sub-system, hence the SSR that prohibits coherences between states where this number differs. Finally, it is seen that if Alice describes a general non-entangled state of sub-systems—which being separable have their own reference frames—then Charlie will also describe the state as a non-entangled state and with the same probability for each product state (see appendix equations (251) and (252)). For systems involving identical bosons Charlie's description of the sub-system density operators will only involve density operators that conform to the SSR. This is in accord with the key idea of the present paper.

Based around the reference frame approach Dowling et al [106] and Terra Cunha et al [35] propose processes using a BEC as a reference system that would create a coherent superposition of an atom and a molecule, or a boson and a fermion [106]. Dunningham et al [107] consider a scheme for observing a superposition of a one boson state and the vacuum state. Obviously if SSRs can be overcome in these instances, it might be possible to produce coherent superpositions of Fock states with differing particle numbers such as Glauber coherent states, though states with $\overline{N}$ ∼ 10⁸ would presumably be difficult to produce. However, detailed considerations of such papers indicate that the states actually produced in terms of Charlie's description are statistical mixtures consistent with the SSRs rather than coherent superpositions, which are only present in Alice's description of the state (see appendix K). Also, although coherence and interference effects are demonstrated, these can also be accounted for without invoking the presence of coherent superpositions that violate the SSRe. As the paper by Dowling et al [106] entitled 'Observing a coherent superposition of an atom and a molecule' is a good example of where the SSRs are challenged, the key points are described in appendix M. Essentially the process involves one atom A interacting with a BEC of different atoms B leading to the creation of one molecule AB, with the BEC being depleted by one B atom. There are three stages in the process, the first being with the interaction that turns separate atoms A and B into the molecule AB turned on at Feshbach resonance for a time t related to the interaction strength and the mean number of bosons in the BEC reference system, the second being free evolution at large Feshbach detuning Δ for a time τ leading to a phase factor $\phi ={\rm{\Delta }}\tau$, the third being again with the interaction turned on at Feshbach resonance for a further time t. However, it is pointed out in appendix M that Charlie's description of the state produced for the atom plus molecule system is merely a statistical mixture of a state with one atom and no molecules and a state with no atom and one molecule, the mixture coefficients depending on the phase ϕ imparted during the process. However a coherent superposition is seen in Alice's description of the final state, though this is not surprising since a SSR violating initial state was assumed. The feature that in Charlie's description of the final state no coherent superposition of an atom and a molecule is produced in the process is not really surprising, because of the averaging over phase differences in going from Alice's reference frame to Charlie's. It is the dependence on the phase ϕ imparted during the process that demonstrates coherence (Ramsey interferometry) effects, but it is shown in appendix M that exactly the same results can be obtained via a treatment in which states which are coherent superpositions of an atom and a molecules are never present, the initial BEC state being chosen as a Fock state. In terms of the description by an external observer (Charlie) the claim of violating the SSR has not been demonstrated via this particular process.

Whether such SSR violating states can be detected has also not been justified. For example, consider the state given by a superposition of a one boson state and the vacuum state (as discussed in [107]). We consider an interferometric process in which one mode A for a two mode BEC interferometer is initially in the state $\alpha | 0\rangle +\beta | 1\rangle$, and the other mode B is initially in the state $| 0\rangle$—thus $| {\rm{\Psi }}(i)\rangle ={(\alpha | 0\rangle +\beta | 1\rangle )}_{A}\otimes | 0{\rangle }_{B}$ in the usual occupancy number notation, where $| \alpha {| }^{2}+| \beta {| }^{2}=1$. The modes are first coupled by a beam splitter, then a free evolution stage occurs for time τ associated with a phase difference $\phi ={\rm{\Delta }}\tau$ (where ${\rm{\Delta }}={\omega }_{B}-{\omega }_{A}$ is the mode frequency difference), the modes are then coupled again by the beam splitter and the probability of an atom being found in modes A, B finally being measured. The probabilities of finding one atom in modes A, B respectively are found to only depend on $| \beta {| }^{2}$ and ϕ. Details are given in appendix M. There is no dependence on the relative phase between α and β, as would be required if the superposition state $\alpha | 0\rangle +\beta | 1\rangle$ is to be specified. Exactly the same detection probabilities are obtained if the initial state is the mixed state $\widehat{\rho }(i)\,=| \alpha {| }^{2}({| 0\rangle }_{A}{\langle 0| }_{A}\,\otimes$ ${| 0\rangle }_{B}{\langle 0| }_{B})+| \beta {| }^{2}({| 1\rangle }_{A}{\langle 1| }_{A}\otimes {| 0\rangle }_{B}{\langle 0| }_{B})$, in which the vacuum state for mode A occurs with a probability $| \alpha {| }^{2}$ and the one boson state for mode A occurs with a probability $| \beta {| }^{2}$. In this example the proposed coherent superposition associated with the SSR violating state would not be detected in this interferometric process, nor in the more elaborate scheme discussed in [107].

Of course, the claim that in isolated systems of massive particles it is not possible in non-relativistic quantum physics to create states that violate the particle number SSR—either for the sub-system states in a separable state or for any quantum state of the overall system—can be questioned. Ideally the claim should be tested by experiment, in particular when the number of particles is large in view of the interest in macroscopic entanglement since the Schrödinger cat was first described. The simplest situation would be to test whether states that violate the (local) particle number SSR could be created for a single mode system. Clearly, a specific proposal for an experiment in which the SSR could be violated is required, but to our knowledge no such proposal has been presented. BECs, in which all the bosons can occupy a single mode would seem an ideal candidate as a suitable bosonic system, and the Glauber coherent state is an example of a non-SSR compliant state. For fermions, the Pauli exclusion principle would limit the number of fermions in a one mode system to be zero or one, but coherent superpositions of a zero and one fermion state are examples of non-SSR compliant pure states. As pointed out above, some authors such as [56, 57, 61, 62], base their definition of entanglement by allowing for the possible presence of non-SSR compliant sub-system states when defining separable states. The approach in [5] is based on the physical assumption that states that are non-compliant with particle number SSR—both local and global—do not come into the realm of non-relativistic quantum physics, in which the concept of entanglement is useful. Until clear evidence is presented that non-SSR-compliant states can be prepared, and in view of the theoretical reasons why they cannot, it seems preferable to base the theory of entanglement on their absence when defining separable and entangled quantum states.

We now consider the role of the SSR for the case of non-entangled states. The global SSR on total particle number has restricted the physical quantum state for a system of identical bosons to be of the form (62). Such states may or may not be entangled states of the modes involved. The question is—do similar restrictions involving the sub-system particle number apply to the modes, considered as separate sub-systems in the definition of non-entangled states ? The viewpoint in this paper is that this is so. Note that applying the SSR on the separate sub-system density operators ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, ... is only in the context of non-entangled states. Such a SSR is referred to as a local SSR, as it applies to each of the separate sub-systems. Mathematically, the local particle number SSR can be expressed as

$$\begin{eqnarray}&&[{\widehat{N}}_{X},{\widehat{\rho }}_{R}^{X}]=0,\end{eqnarray} \tag{ 82 }$$

where ${\widehat{N}}_{X}$ is the number operator for sub-system $X=A,B,\ldots \,$.The SSR restriction is based on the proposition that the density operators ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, ... for the separate sub-systems A, B, ... should themselves represent possible quantum states for each of the sub-systems, considered as a separate system and thus be required to satisfy the SSR that forbids quantum superpositions of Fock states with differing boson numbers. Note that if the local particle number SSR applies in each sub-system the global particle number SSR applies to any separable state. The proof is trivial and just requires showing that $[\widehat{N},{\widehat{\rho }}_{{\rm{sep}}}]=0$

The justification of applying the local particle number SSR to the density operators ${\widehat{\rho }}_{R}^{a}$, ${\widehat{\rho }}_{R}^{b}$, ... for the sub-system quantum states that occur in any separable state is simply that these are possible quantum states of the sub-systems when the latter are considered as separate quantum systems before being combined as in the Werner protocol [7] to form the separable state. Hence all the justifications based either on simple physical considerations or phase reference systems that were previously invoked for the density operator $\widehat{\rho }$ of any general quantum states of the combined sub-systems to establish the global particle number SSR apply equally well here. No more need be said. It is contended that expressions for the non-entangled quantum state $\widehat{\rho }$ in which ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, ${\widehat{\rho }}_{R}^{C}$... were not allowed quantum states for the sub-systems would only be of mathematical interest.

Applying the local particle number SSR to the sub-system density operators for non-entangled states is discussed in papers by Bartlett et al [54, 60] as one of several operational approaches for defining entangled states. As pointed out above other authors [56, 57] define separable (and hence entangled) states differently by specifically allowing sub-system density operators that are not consistent with the local particle number SSR, though the overall density operator is globally SSR consistent. The corresponding overall states are termed separable but non-local, and states that they would regard as separable would here be regarded as entangled. Examples of such states are given in equations (83) and (85). There are also other authors [61, 62] who define separable (and hence entangled) states via (3) but leave unspecified whether the sub-system density operators are consistent or inconsistent with the local particle number SSR. Note that any inequalities involving measured quantities that are found for separable states in which local SSR compliance is neglected must also apply to separable states where it is required. The consequent implications for entanglement tests where local particle number SSR compliance is required is discussed in the accompanying paper II. Hence, in this paper we are advocating a revision to a widely held notion of entanglement in identical particle systems, the consequence being that the set of entangled states is now much larger. This is a key idea in this paper—not only should SSRs on particle numbers be applied to the the overall quantum state, entangled or not, but it also should be applied to the density operators that describe states of the modal sub-systems involved in the general definition of non-entangled states. The reasons for adopting this viewpoint have been discussed above—basically it is because in separable states the sub-system density operators must represent possible quantum states for the sub-systems considered as isolate quantum systems, so the general reasons for applying the SSR will apply to these density operators also. Apart from the papers by Bartlett et al [54, 60] we are not aware that this definition of non-entangled states has been invoked previously, indeed the opposite approach has been proposed [56, 57]. However, the idea of considering whether sub-system states should satisfy the local particle number SSR has been presented in several papers—[54, 56, 57, 60, 63–65], mainly in the context of pure states for bosonic systems, though in these papers the focus is on issues other than the definition of entanglement, such as quantum communication protocols [56], multicopy distillation [54], mechanical work and accessible entanglement [63, 64] and Bell inequality violation [65]. However, there are a number of papers that do not apply the SSR to the sub-system density operators, and those that do have not studied the consequences for various entanglement tests. These tests are also discussed in the accompanying paper II.

There is a connection between global SSR compliance for separable states in general and local SSR compliance for the component sub-system states. This may be stated in the form of a theorem:

Theorem. A necessary and sufficient condition for all separable states for a given set of sub-system density operators ${\widehat{\rho }}_{R}^{a},{\widehat{\rho }}_{R}^{b}$ to be global particle number SSR compliant is that all such sub-system states are local particle number SSR compliant.

The proof of this theorem is set out in appendix N. It deals with the case where the probabilities P_R for preparing a particular product ${\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}$ of sub-system states in the Werner process can be arbitrarily varied. The theorem shows that in general global and local SSR compliance always occur together in separable states.

However, it should be noted that some authors [56, 57] consider sub-system density operators in the context of two mode systems which comply with the global particle number SSR but not the local particle number SSR. These involve special choices of both the sub-system states and the product preparation probabilities. Such a case involving four zero and one boson superpositions is presented by Verstraete et al [56, 57]. The overall density operator is a statistical mixture

$$\begin{eqnarray}\widehat{\rho } & = & \displaystyle \frac{1}{4}{{(| {\psi }_{1}\rangle \langle {\psi }_{1}| )}_{A}\otimes | {\psi }_{1}\rangle \langle {\psi }_{1}| )}_{B}\\ & & +\displaystyle \frac{1}{4}{{(| {\psi }_{i}\rangle \langle {\psi }_{i}| )}_{A}\otimes | {\psi }_{i}\rangle \langle {\psi }_{i}| )}_{B}\\ & & +\displaystyle \frac{1}{4}{(| {\psi }_{-1}\rangle \langle {\psi }_{-1}| )}_{A}\otimes | {\psi }_{-1}\rangle {\langle {\psi }_{-1}| )}_{B}\\ & & +\displaystyle \frac{1}{4}{(| {\psi }_{-i}\rangle \langle {\psi }_{-i}| )}_{A}\otimes | {\psi }_{-i}\rangle {\langle {\psi }_{-i}| )}_{B},\end{eqnarray} \tag{ 83 }$$

where $| {\psi }_{\omega }\rangle =(| 0\rangle +\omega | 1)/\sqrt{2}$, with $\omega =1,i,-1,-i$. The $| {\psi }_{\omega }\rangle$ are superpositions of zero and one boson states and consequently the local particle number SSR is violated by each of the sub-system density operators $| {\psi }_{\omega }\rangle \langle {\psi }_{\omega }| {)}_{A}$ and $| {\psi }_{\omega }\rangle \langle {\psi }_{\omega }| {)}_{B}$. Although the expression in equation (83) is of the form in equation (3), the subsystem density operators $| {\psi }_{\omega }\rangle \langle {\psi }_{\omega }| {)}_{A}$ and $| {\psi }_{\omega }\rangle \langle {\psi }_{\omega }| {)}_{B}$ do not comply with the local particle number SSR, so in the present paper and in [5] the state would be regarded as entangled. However, Verstraete et al [56, 57] regard it as separable. They refer to such a state as separable but non-local.

On the other hand, the global particle number SSR is obeyed since the density operator can also be written as

$$\begin{eqnarray}\widehat{\rho } & = & \displaystyle \frac{1}{4}{(| 0\rangle \langle 0| )}_{A}\otimes | 0\rangle {\langle 0| )}_{B}+\displaystyle \frac{1}{4}{(| 1\rangle \langle 1| )}_{A}\otimes | 1\rangle {\langle 1| )}_{B}\\ & & +\displaystyle \frac{1}{2}{(| {{\rm{\Psi }}}_{+}\rangle \langle {{\rm{\Psi }}}_{+}| )}_{{AB}},\end{eqnarray} \tag{ 84 }$$

where ${| {{\rm{\Psi }}}_{+}\rangle }_{{AB}}=({| 0\rangle }_{A}{| 1\rangle }_{B}+{| 1\rangle }_{A}{| 0\rangle }_{B})/\sqrt{2}$. This is a statistical mixture of $N=0,1,2$ boson states. Note that equation (84) indicates that the state could be prepared as a mixed state containing two terms that comply with the local particle number SSR in each of the sub-systems plus a term which is an entangled state of the two sub-systems. The presence of an entangled state in such an obvious preparation process challenges the description of the state as being separable.

To further illustrate some of the points made about SSRs—local and global—it is useful to consider a second specific case also presented by Verstraete et al [56, 57]. This mixture of two mode coherent states is represented by the two mode density operator

$$\begin{eqnarray}\widehat{\rho } & = & \displaystyle \int \displaystyle \frac{{\rm{d}}\theta }{2\pi }| \alpha ,\alpha \rangle \langle \alpha ,\alpha | \\ & = & \displaystyle \int \displaystyle \frac{{\rm{d}}\theta }{2\pi }{(| \alpha \rangle \langle \alpha | )}_{A}\otimes {(| \alpha \rangle \langle \alpha | )}_{B},\end{eqnarray} \tag{ 85 }$$

where ${| \alpha \rangle }_{C}$ is a one mode coherent state for mode $C=A,B$ with $\alpha =| \alpha | \,\exp (-{\rm{i}}\theta )$, and modes $A,B$ are associated with bosonic annihilation operators $\widehat{a}$, $\widehat{b}$. The magnitude $| \alpha |$ is fixed.

This density operator appears to be that for a non-entangled state of modes $A,B$ in the form

$$\begin{eqnarray}&&\widehat{\rho }=\sum _{R}{P}_{R}\,{\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}\end{eqnarray} \tag{ 86 }$$

with ${\sum }_{R}{P}_{R}\to \int \tfrac{{\rm{d}}\theta }{2\pi }$ and ${\widehat{\rho }}_{R}^{A}\to {(| \alpha \rangle \langle \alpha | )}_{A}$ and ${\widehat{\rho }}_{R}^{B}\,\to {(| \alpha \rangle \langle \alpha | )}_{B}$. However although this choice of ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$ satisfy the Hermitiancy, unit trace, positivity features they do not conform to the requirement of satisfying the (local) sub-system boson number SSR. From equation (85) we have

$$\begin{eqnarray}\langle n| (| \alpha \rangle \langle \alpha | ){| m\rangle }_{A} & = & \exp (-| \alpha {| }^{2})\displaystyle \frac{{\alpha }^{n}}{\sqrt{n!}}\displaystyle \frac{{(\alpha )}^{* m}}{\sqrt{m!}}\\ \langle p| (| \alpha \rangle \langle \alpha | ){| q\rangle }_{B} & = & \exp (-| \alpha {| }^{2})\displaystyle \frac{{\alpha }^{p}}{\sqrt{p!}}\displaystyle \frac{{(\alpha )}^{* q}}{\sqrt{q!}}\end{eqnarray} \tag{ 87 }$$

so clearly for each of the separate modes there are coherences between Fock states with differing boson occupation numbers. In the approach in the present paper the density operator in equation (85) does not represent a non-entangled state. However, in the papers of Verstraete et al [56, 57], Hillery et al [61, 62] and others it would represent an allowable non-entangled (separable) state. Indeed, Verstraete et al [56] specifically state '..., this state is obviously separable, though the states $| \alpha \rangle$ are incompatible with the (local) super-selection rule'. Verstraete et al [56] introduce the state defined in equation (85) as an example of a state that is separable (in their terms) but which cannot be prepared locally, because it is incompatible with the local particle number SSR.

The mixture of two mode coherent states does of course satisfy the total or global boson number SSR. The matrix elements between two mode Fock states are

$$\begin{eqnarray}&&({\langle n| }_{A}\otimes {\langle p| }_{B})\widehat{\rho }({| m\rangle }_{A}\otimes {| q\rangle }_{B})\\ &&\quad =\exp (-2| \alpha {| }^{2})\displaystyle \frac{| \alpha {| }^{n+m}}{\sqrt{n!}\sqrt{m!}}\displaystyle \frac{| \alpha {| }^{p+q}}{\sqrt{p!}\sqrt{q!}}\displaystyle \int \displaystyle \frac{{\rm{d}}\theta }{2\pi }\\ &&\qquad \times \exp (-{\rm{i}}(n-m+p-q)\theta )\\ &&\quad =\exp (-2| \alpha {| }^{2})\displaystyle \frac{| \alpha {| }^{n+m}}{\sqrt{n!}\sqrt{m!}}\displaystyle \frac{| \alpha {| }^{p+q}}{\sqrt{p!}\sqrt{q!}}\,{\delta }_{n+p,m+q}.\end{eqnarray} \tag{ 88 }$$

These overall matrix elements are zero unless $n+p=m+q$, showing that there are no coherences between two mode Fock states where the total boson number differs. The mixture of two mode coherent states has the interesting feature of providing an example of a two mode state which satisfies the global but not the local SSR.

The reduced density operators for modes $A,B$ are

$$\begin{eqnarray*}&&{\widehat{\rho }}_{A}=\int \displaystyle \frac{{\rm{d}}\theta }{2\pi }{(| \alpha \rangle \langle \alpha | )}_{A}\qquad {\widehat{\rho }}_{B}=\int \displaystyle \frac{{\rm{d}}\theta }{2\pi }{(| \alpha \rangle \langle \alpha | )}_{B}\end{eqnarray*}$$

and a straightforward calculation gives

$$\begin{eqnarray*}{\widehat{\rho }}_{A} & = & \exp (-| \alpha {| }^{2})\displaystyle \sum _{n}\displaystyle \frac{| \alpha {| }^{2n}}{n!}{(| n\rangle \langle n| )}_{A}\\ {\widehat{\rho }}_{B} & = & \exp (-| \alpha {| }^{2})\displaystyle \sum _{p}\displaystyle \frac{| \alpha {| }^{2p}}{p!}{(| p\rangle \langle p| )}_{B}\end{eqnarray*}$$

which are statistical mixtures of Fock states with the expected Poisson distribution associated with coherent states. This shows that the reduced density operators are consistent with the separate mode local SSR, whereas the density operators ${\widehat{\rho }}_{R}^{A}={(| \alpha \rangle \langle \alpha | )}_{A}$, ${\widehat{\rho }}_{R}^{B}$ $={(| \alpha \rangle \langle \alpha | )}_{B}$ are not. Later we will revisit this example in the context of entanglement tests.

Note that if a twirling operation (see appendix equation (263)) were to be applied to mode A, the result would be equivalent to applying two independent twirling operations to each mode. In this case the density operator for each mode is a Poisson statistical mixture of number states, so each mode has a density operator that complies with the local particle number SSR.

To summarize: basically the sub-systems are single modes that the identical bosons can occupy, the SSR for identical bosons, massive or otherwise, prohibits states which are coherent superpositions of states with different numbers of bosons, and the only physically allowable ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, ... for the separate mode sub-systems that are themselves compatible with the local particle number SSR are allowed. For single mode sub-systems these can be written as statistical mixtures of states with definite numbers of bosons in the form

$$\begin{eqnarray}&&{\widehat{\rho }}_{R}^{A}=\displaystyle \sum _{{n}_{A}}{P}_{{n}_{A}}^{A}| {n}_{A}\rangle \langle {n}_{A}| \quad {\widehat{\rho }}_{R}^{B}=\displaystyle \sum _{{n}_{B}}{P}_{{n}_{B}}^{B}| {n}_{B}\rangle \langle {n}_{B}| ...\end{eqnarray} \tag{ 89 }$$

However, in cases where the sub-systems are pairs of modes the density operators ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, ... for the separate sub-systems are still required to conform to the symmetrization principle and the SSR. The forms for ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$, ... are now of course more complex, as entanglement within the pairs of modes A₁, A₂ associated with sub-system A, the pairs of modes B₁, B₂ associated with sub-system B, etc is now possible within the definition for the general non-entangled state equation (3) for these pairs of modes. Within each pair of modes A₁, A₂ statistical mixtures of states with differing total numbers N_A bosons in the two modes are possible and the sub-system density operators are based on states of the form given in equation (64). We have

$$\begin{eqnarray}{| {{\rm{\Phi }}}_{{N}_{A}}\rangle }_{A} & = & \displaystyle \sum _{k=0}^{{N}_{A}}{C}_{A{\rm{\Phi }}}({N}_{A},k){| k\rangle }_{{A}_{1}}\otimes {| {N}_{A}-k\rangle }_{{A}_{2}}\\ {\widehat{\rho }}_{R}^{A} & = & \displaystyle \sum _{{N}_{A}=0}^{\infty }\displaystyle \sum _{{\rm{\Phi }}}{P}_{{\rm{\Phi }}{N}_{A}}{| {{\rm{\Phi }}}_{{N}_{A}}\rangle }_{A}{\langle {{\rm{\Phi }}}_{{N}_{A}}| }_{A}\end{eqnarray} \tag{ 90 }$$

with analogous expressions for the density operators ${\widehat{\rho }}_{R}^{B}$ etc for the other pairs of modes. Note that ${| {{\rm{\Phi }}}_{{N}_{A}}\rangle }_{A}$ only involves quantum superpositions of states with the same total number of bosons ${N}_{A.}$ The expression (248) in appendix D of paper II is of this form.

The general non-entangled state for modes $\widehat{a}$ and $\widehat{b}$ is given by

$$\begin{eqnarray}&&\widehat{\rho }=\displaystyle \sum _{R}{P}_{R}\,{\widehat{\rho }}_{R}^{A}\otimes {\widehat{\rho }}_{R}^{B}\end{eqnarray} \tag{ 91 }$$

and as a consequence of the requirement that ${\widehat{\rho }}_{R}^{A}$ and ${\widehat{\rho }}_{R}^{B}$ are allowed quantum states for modes $\widehat{a}$ and $\widehat{b}$ satisfying the SSR, it follows that

$$\begin{eqnarray}{\langle {(\widehat{a})}^{n}\rangle }_{a} & = & {\rm{Tr}}({\widehat{\rho }}_{R}^{A}{(\widehat{a})}^{n})=0\\ {\langle {({\widehat{a}}^{\dagger })}^{n}\rangle }_{a} & = & {\rm{Tr}}({\widehat{\rho }}_{R}^{A}{({\widehat{a}}^{\dagger })}^{n})=0\\ {\langle {(\widehat{b})}^{m}\rangle }_{b} & = & {\rm{Tr}}({\widehat{\rho }}_{R}^{B}{(\widehat{b})}^{m})=0\\ {\langle {({\widehat{b}}^{\dagger })}^{m}\rangle }_{b} & = & {\rm{Tr}}({\widehat{\rho }}_{R}^{B}{({\widehat{b}}^{\dagger })}^{m})=0.\end{eqnarray} \tag{ 92 }$$

Thus coherence terms are zero. As we will see these results will limit spin squeezing to entangled states of modes $\widehat{a}$ and $\widehat{b}$. Note that similar results also apply when non-entangled states for the original modes $\widehat{c}$ and $\widehat{d}$ are considered-${\langle {(\widehat{c})}^{n}\rangle }_{c}=0$, etc.

In this case the general non-entangled state where A and B are pairs of modes—${\widehat{a}}_{1}$, ${\widehat{a}}_{2}$ associated with sub-system A, and modes ${\widehat{b}}_{1}$, ${\widehat{b}}_{2}$ associated with sub-system B, the overall density operator is of the form (91). Consistent with the requirement that the sub-system density operators ${\widehat{\rho }}_{R}^{A}$, ${\widehat{\rho }}_{R}^{B}$ conform to the symmetrization principle and the SSR, these density operators will not in general represent separable states for their single mode sub-systems ${\widehat{a}}_{1}$, ${\widehat{a}}_{2}$ or ${\widehat{b}}_{1}$, ${\widehat{b}}_{2}$—and may even be entangled states. As a result when considering non-entangled states for the sub-systems A and B we now have

$$\begin{eqnarray}{\langle {({\widehat{a}}_{i}^{\dagger }{\widehat{a}}_{j})}^{n}\rangle }_{A} & = & {\rm{Tr}}({\widehat{\rho }}_{R}^{A}{({\widehat{a}}_{i}^{\dagger }{\widehat{a}}_{j})}^{n})\ne 0\quad i,j=1,2\\ {\langle {({\widehat{b}}_{i}^{\dagger }{\widehat{b}}_{j})}^{n}\rangle }_{A} & = & {\rm{Tr}}({\widehat{\rho }}_{R}^{B}{({\widehat{b}}_{i}^{\dagger }{\widehat{b}}_{j})}^{n})\ne 0\quad i,j=1,2\end{eqnarray} \tag{ 93 }$$

in general. In this case where the sub-systems are pairs of modes the spin squeezing entanglement tests as in equations (37)–(39) in paper II for sub-systems consisting of single modes cannot be applied, as we will see. Nevertheless, there are still tests of bipartite entanglement involving spin operators.