Distinguishability of countable quantum states and von Neumann lattice

Different states of a system are assumed to be distinguishable in classical mechanics. This fundamental assumption, however, is abandoned in quantum mechanics. States cannot be distinguished without error unless they are orthogonal. One would then like to consider the problem of distinguishing states in a given set, which is sometimes called the state discrimination problem.

Strategies for state discrimination can be classified into two types. In the first type, one makes a measurement with n outcome in order to distinguish n input states. If a certain outcome is detected, we presume that the system is in the input state corresponding to that outcome. This strategy is widely employed. For example, Helstrom [1] considered measurements of this type to derive the minimum average error probability associated with the discrimination of arbitrary two states. In the second type, a measurement has $n+1$ outcomes. The extra outcome corresponds to the answer 'do not know'. At the expense of this inconclusive outcome, it is possible, under rather weak conditions on the input states, to distinguish the inputs with certainty when the other n outcomes is obtained. A measurement with this property is called an unambiguous measurement, which is the central subject of this paper. Note that all measurements of the second type should be regarded as belonging to the first type, if we obliged to estimate the input even when we obtain the inconclusive outcome. Consequently, unambiguous measurements cannot make the average error probability smaller than that in measurements of the first type. In other words, unambiguous measurements are less suitable for quantitative studies of state discrimination. On the contrary, they are appropriate for qualitative studies, which we shall carry out in this paper.

Unambiguous measurements for two pure states were discussed by Ivanovic [2] for the first time, and later its theory was developed by Dieks [3], Peres [4], Jaeger and Shimony [5]. Chefles [6] obtains a necessary and sufficient condition in order for a family which consists of finitely many pure states to be unambiguously measured. The condition is the linear independence of the given family of states. Sun et al [7] and Eldar [8] discuss optimal unambiguous measurements in relation to semidefinite programming. A necessary and sufficient condition for mixed states to be unambiguously measured was presented by Feng et al [9], which is slightly complicated. Note that all the studies above concern a family consisting of a finite number of states.

Another subject of this paper is a von Neumann lattice. This is a family of states which corresponds to the lattice on the phase space in the classical mechanics. This family is investigated in several contexts. Originally von Neumann [10] examines it for simultaneous measurement of position and momentum. Gabor [11] discussed these families in the context of communication theory and electrical engineering; this is a pioneering work in time-frequency analysis. An interpolation problem for entire functions also has a relation to von Neumann lattices [12]. The properties of a von Neumann lattice depend on the area of its fundamental region in the phase space; von Neumann stated without proof that this family is complete when the area is roughly smaller than the Planck constant h. However, it was about 40 years later that Perelomov [13] and Bargmann et al [14] gave the proof for this fact. Today, many of the properties have been revealed, which potentially offer measurement theoretical interpretations.

In this paper, we investigate unambiguous measurements on countably many states. First, we develop a general theory of the distinguishability of countably many states. We define the distinguishability of states as the possibility of unambiguous measurements on them. Then, we provide a necessary and sufficient condition for countable pure states to be distinguishable. We also consider uniform distinguishability, and give the maximum value of uniform success probability. We point out that there is a difference between distinguishability and uniform distinguishability in the case of infinitely many states. Second, we apply the criterion of distinguishability developed in the first part of this paper to von Neumann lattices. We find the measurement theoretical meaning of the Planck constant h, the smallest unit of area in the classical phase space. Depending on whether the spaces between the states are larger or smaller than the Planck constant, the distinguishability of states changes drastically.

This paper is organized as follows. In section 2, we define distinguishability of the states using unambiguous measurements. Then we briefly review the properties of vectors in a Hilbert space, which can be considered as generalizations of liner independence, in section 3. In section 4, We show that the distinguishability of countable pure states is equivalent to the properties of vectors which we see in the previous section. Section 5 is devoted to investigations of von Neumann lattices. Conclusions and discussions are given in sections 6 and 7. We discuss the case of uncountable states in appendix A.

We shall discuss the problem of distinguishing a quantum state in a given family by a single measurement. We allow an answer 'do not know' or 'unknown', but do not allow taking one state for another in the family. The problem is referred to as unambiguous measurement. We shall consider an arbitrary quantum system described by a Hilbert space H. Let ${({\rho }_{i})}_{i\in I}$ to be a given family of countable states, where ${\rho }_{i}$'s are density operators on H. In our terminology, countable includes finite. We will often denote a family ${(| {\psi }_{i}\rangle \langle {\psi }_{i}| )}_{i\in I}$ of pure states simply by ${({\psi }_{i})}_{i\in I}$ in the following.

A quantum measurement and the resulting probability density of the outcome is described by a POVM (e.g. [15 section 3.1]). A POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in J}$ on J, where J is a countable set, is a list of bounded operators ${{\rm{\Pi }}}_{j}$ on H that satisfies the positivity, ${{\rm{\Pi }}}_{j}\geqslant 0$ for all $j\in J$, and the normalization, ${\sum }_{j\in J}{{\rm{\Pi }}}_{j}=1$. The sum should be understood in the sense of the weak operator topology. The conditional probability of obtaining an outcome $j\in J$ when the input was ${\rho }_{i}$ is given by

$$\begin{eqnarray}&&{q}_{{ji}}({\rm{\Pi }}):= \mathrm{tr}[{{\rm{\Pi }}}_{j}{\rho }_{i}].\end{eqnarray} \tag{ 1 }$$

When $I\subset J$, the success probability of obtaining the outcome $i\in I\subset J$ for the input $i\in I$ is given by

$$\begin{eqnarray}&&{q}_{i}({\rm{\Pi }}):= {q}_{{ii}}({\rm{\Pi }})=\mathrm{tr}[{{\rm{\Pi }}}_{i}{\rho }_{i}].\end{eqnarray} \tag{ 2 }$$

Though these quantities depend on ${({\rho }_{i})}_{i\in I}$, we omit them in the notation since we usually fix a family ${({\rho }_{i})}_{i\in I}$ in our discussion.

We shall define the distinguishability of each state in a given family of states.

Definition 1 Let ${({\rho }_{i})}_{i\in I}$ be a countable family of states.

• i.  
  A POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in J}$ distinguishes (the states in) ${({\rho }_{i})}_{i\in I}$ if $J=I\bigsqcup \{?\}$, a disjoint union of I and a set containing one element which we denote '?', and the following hold:

  $$\begin{eqnarray}&&{q}_{{ji}}({\rm{\Pi }})=0\ \ \mathrm{for}\ \mathrm{all}\ i,j\in I,\quad i\ne j,\end{eqnarray} \tag{ 3 }$$

  $$\begin{eqnarray}&&{q}_{i}({\rm{\Pi }})\gt 0\ \mathrm{for}\ \mathrm{all}\ i\in I.\end{eqnarray} \tag{ 4 }$$
• ii.  
  A POVM Π uniformly distinguishes the states in ${({\rho }_{i})}_{i\in I}$ if Π distinguishes ${({\rho }_{i})}_{i\in I}$ and has constant ${q}_{i}({\rm{\Pi }})$. The constant is called the uniform success probability.
• iii.  
  A POVM Π perfectly distinguishes the states in ${({\rho }_{i})}_{i\in I}$ if Π distinguishes ${({\rho }_{i})}_{i\in I}$ and ${q}_{i}({\rm{\Pi }})=1$ for all $i\in I$.

The family ${({\rho }_{i})}_{i\in I}$ of states is said distinguishable, uniformly distinguishable, or perfectly distinguishable if there exists a POVM that distinguishes, uniformly distinguishes, or perfectly distinguishes, respectively, the states in ${({\rho }_{i})}_{i\in I}$.

It is obvious by definition that perfect distinguishability implies uniform distinguishability, and that uniform distinguishability implies distinguishability.

Uniform distinguishability allows another characterization: (ii)' There exists a POVM Π such that it distinguishes the states ${({\rho }_{i})}_{i\in I}$ and ${\mathrm{inf}}_{i\in I}{q}_{i}({\rm{\Pi }})\gt 0$ holds. Necessity is trivial. For sufficiency, assume that a POVM ${{\rm{\Pi }}}^{\prime }$ satisfies the condition (ii)'. Let ${q}_{i}^{\prime }:= {q}_{i}({{\rm{\Pi }}}^{\prime })$ and ${q}^{\prime }:= \mathrm{inf}{q}_{i}^{\prime }\gt 0$. Then the POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in I\bigsqcup \{?\}},$ where

$$\begin{eqnarray*}{{\rm{\Pi }}}_{j} & := & \displaystyle \frac{{q}^{\prime }}{{q}_{i}^{\prime }}{{\rm{\Pi }}}_{j}^{\prime },\quad j\in I,\\ {{\rm{\Pi }}}_{?} & := & {{\rm{\Pi }}}_{?}^{\prime }+\displaystyle \sum _{i\in I}\left(1-\displaystyle \frac{{q}^{\prime }}{{q}_{i}^{\prime }}\right){{\rm{\Pi }}}_{i}^{\prime },\end{eqnarray*}$$

satisfies the condition (ii), because its success probabilities ${q}_{i}({\rm{\Pi }})={q}^{\prime }\gt 0$ do not depend on $i\in I$.

Distinguishability and uniform distinguishability are equivalent if the family ${({\rho }_{i})}_{i\in I}$ consists of only finite number of states, because the infimum of finitely many positive numbers is positive.

The conditions equivalent to distinguishability and perfect distinguishability were discussed for finite number of states in [6]. Uniform distinguishability, which is different from the condition only when the family contains infinite number of states, is newly defined in this paper. In the case that the number of states is countable, we will derive the necessary and sufficient condition for each type of distinguishability, which is the theme of this paper. The assumption of countability is not a restriction if the Hilbert space is separable. See the appendix A for details.

It is worth noting that all conditions defined here are invariant under any unitary transformation, in particular, under any unitary time evolution. In fact, if a POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in I\bigsqcup \{?\}}$ distinguishes the states ${({\rho }_{i})}_{i\in I}$, then the POVM ${{\rm{\Pi }}}^{\prime }={(U{{\rm{\Pi }}}_{j}{U}^{\ast })}_{j\in I\bigsqcup \{?\}}$ distinguishes the states ${(U{\rho }_{i}{U}^{\ast })}_{i\in I}$, for any unitary operator U.

In this section, we shall review some properties of a family of vectors in a Hilbert space, which are related to the notion of linear independence. Careful discussion is necessary if the number of vectors is infinite. They will turn out to have substantial relation to distinguishability of quantum states in the next section. Let $\mathrm{span}\ S$ be the minimal linear subspace containing S, where S is a subset of H. In other words, $\mathrm{span}\ S$ is the set of all linear combinations of finitely many elements of S. Let $\bar{{span}}\ S$ be the norm closure of $\mathrm{span}\ S$.

Definition 2 Let ${({\psi }_{i})}_{i\in I}$ be a family of vectors in H.

• The family ${({\psi }_{i})}_{i\in I}$ is linearly independent if ${\psi }_{i}\notin \mathrm{span}\{{\psi }_{j}\in H| j\ne i,j\in I\}$ for each $i\in I$.
• The family ${({\psi }_{i})}_{i\in I}$ is minimal if ${\psi }_{i}\notin \bar{{span}}\{{\psi }_{j}\in H| j\ne i,j\in I\}$ for each $i\in I$.
• The family ${({\psi }_{i})}_{i\in I}$ is Riesz–Fischer if there exists $A\gt 0$ such that
  $$\begin{eqnarray}&&A\displaystyle \sum _{i\in I}\;{| {\alpha }_{i}| }^{2}\leqslant {\| \displaystyle \sum _{i\in I}{\alpha }_{i}{\psi }_{i}\| }^{2}\end{eqnarray} \tag{ 5 }$$
  holds for scalars ${({\alpha }_{i})}_{i\in I}$ with all but finitely many being zero. We call the positive number A a Riesz-Fischer bound.
• The family ${({\psi }_{i})}_{i\in I}$ is orthonormal if $\langle {\psi }_{j},{\psi }_{i}\rangle ={\delta }_{{ij}}$ holds for all $i,j\in I$.

Orthonormality obviously implies the Riesz–Fischer property. The Riesz–Fischer property implies minimality, because otherwise there would be $i\in I$ such that ${\psi }_{i}\in \bar{{span}}\{{\psi }_{j}| j\ne i\}$, and one could make the norm of ${\psi }_{i}-{\sum }_{j\ne i}{\alpha }_{j}{\psi }_{j}$ smaller than any given positive number by choosing ${\alpha }_{j}\in {\mathbb{C}}$ accordingly, thus violating (5) for any $A\gt 0$. Minimality implies linear independence by definition.

In the particular case that I is a finite set, linear independence, minimality and the Riesz–Fischer property are all equivalent. Indeed, if I is finite, then $\left\{({\alpha }_{i})\in {{\mathbb{C}}}^{I}| {\sum }_{i\in I}{| {\alpha }_{i}| }^{2}=1\right\}$ is compact, and we deduce the Riesz–Fischer property from linear independence.

In our discussion on distinguishability, the dual of a family ${({\psi }_{i})}_{i\in I}$ of vectors in H play a crucial role. Two lists ${({\psi }_{i})}_{i\in I}$ and ${({\phi }_{i})}_{i\in I}$ of vectors in H are said to be dual or biorthogonal to each other, if $\langle {\phi }_{j},{\psi }_{i}\rangle ={\delta }_{{ij}}$ for all $i,j\in I$. The condition for the existence of a dual is given by the following proposition. For the proof, see [16 p. 28], [17 lemma 3.3.1]. We also attach a proof in appendix B for the convenience of the reader.

Proposition 1. A family ${({\psi }_{i})}_{i\in I}$ of vectors in H admits a biorthogonal family if and only if it is minimal. In that case, the biorthogonal family is unique if ${({\psi }_{i})}_{i\in I}$ is complete i.e. $\bar{{span}}\{{\psi }_{i}| i\in I\}=H$.

The Riesz–Fischer property has a dual notion.

Definition 3. The list ${({\phi }_{i})}_{i\in I}$ of vectors in H is Bessel if there exists $B\lt \infty$, called a Bessel bound, such that the following equivalent conditions are fulfilled.

• 1.  
  For scalars ${({\alpha }_{i})}_{i\in I}$ with all but finitely many being zero,

  $$\begin{eqnarray}&&{\| \displaystyle \sum _{i\in I}{\alpha }_{i}{\phi }_{i}\| }^{2}\leqslant B\displaystyle \sum _{i\in I}\;{| {\alpha }_{i}| }^{2}\end{eqnarray} \tag{ 6 }$$

  holds.
• 2.  
  For each vector $\psi \in H$,

  $$\begin{eqnarray}&&\sum {| \langle \psi ,{\phi }_{i}\rangle | }^{2}\leqslant B{\parallel \psi \parallel }^{2}\end{eqnarray} \tag{ 7 }$$

  holds.

For the proof of equivalence of two conditions, see [16 chapter 4, section 2, theorem 3].

The Riesz–Fischer property allows characterizations by the dual ([16 chapter 4], see also [18 proposition 2.3 (ii)]).

Proposition 2. For a family ${({\psi }_{i})}_{i\in I}$ of vectors and $A\gt 0$, the following conditions are equivalent.

• 1.  
  ${({\psi }_{i})}_{i\in I}$ is Riesz–Fischer with bound A.
• 2.  
  ${({\psi }_{i})}_{i\in I}$ admits a biorthogonal family which is Bessel with bound ${A}^{-1}$.
• 3.  
  For each ${({\gamma }_{i})}_{i\in I}\in {{\mathbb{C}}}^{I}$ with ${\sum }_{i\in I}{| {\gamma }_{i}| }^{2}\lt \infty$, there exists $\phi \in H$ that satisfies $\langle \phi ,{\psi }_{i}\rangle ={\gamma }_{i}$ and ${\parallel \phi \parallel }^{2}\leqslant {A}^{-1}{\sum }_{i\in I}{| {\gamma }_{i}| }^{2}$.

The moment problem is to find a vector $\phi \in H$ that satisfies $\langle \phi ,{\psi }_{i}\rangle ={\gamma }_{i}$, $i\in I$, for a given family ${({\psi }_{i})}_{i\in I}$ and a numerical sequence ${({\gamma }_{i})}_{i\in I}$, In that context, the equivalence of (i) and (iii) states that the Riesz–Fischer property guarantees existence of a solution of the moment problem for each square summable ${({\gamma }_{i})}_{i\in I}$.

Now we shall state the condition for each type of distinguishability for a family ${({\psi }_{i})}_{i\in I}$ of pure states.

Theorem 1. Let ${({\psi }_{i})}_{i\in I}$ be a family of countably many pure states.

• i.  
  The states in ${({\psi }_{i})}_{i\in I}$ are distinguishable if and only if ${({\psi }_{i})}_{i\in I}$ is minimal.
• ii.  
  The states in ${({\psi }_{i})}_{i\in I}$ are uniformly distinguishable with uniform success probability q if and only if ${({\psi }_{i})}_{i\in I}$ is Riesz–Fischer with bound q.
• iii.  
  The states in ${({\psi }_{i})}_{i\in I}$ are perfectly distinguishable if and only if ${({\psi }_{i})}_{i\in I}$ is an orthonormal family.

Only the if part of (i) needs the assumption of countability.

It should be noted that the statement (ii) of theorem 1 reveals the significance of the Riesz–Fischer bound in quantum measurement theory: success probability of uniform distinction.

We remark that in the particular case of finite family ${({\psi }_{i})}_{i\in I}$, the statements (i) and (ii) of theorem 1 are identical, which reproduces the results obtained by Chefles [6]. As was discussed in the previous sections, linear independence, minimality and the Riesz–Fischer property are equivalent, for finitely many vectors.

We begin the proof of theorem 1 with

Lemma 1. If the POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in I\bigsqcup \{?\}}$ distinguishes the pure states in ${({\psi }_{i})}_{i\in I}$, then the following holds for $i,j,k\in I$.

• 1.  
  ${{\rm{\Pi }}}_{j}{\psi }_{i}=0$ for $i\ne j$.
• 2.  
  ${q}_{i}({\rm{\Pi }})=1$ if and only if ${{\rm{\Pi }}}_{?}{\psi }_{i}=0$.
• 3.  
  $\langle {\psi }_{i},{{\rm{\Pi }}}_{k}{\psi }_{j}\rangle ={q}_{i}({\rm{\Pi }}){\delta }_{{ik}}{\delta }_{{jk}}$.

Proof. The condition $\mathrm{tr}[\rho E]=0$ for $\rho =| \psi \rangle \langle \psi |$ and $E\geqslant 0$ implies $E\rho =0$. This fact together with the assumption of the lemma imply the first and second claims. The third follows directly from the first. □

Proof of theorem 1. We must prove the six claims below.

(1) (Distinguishability implies minimality). Suppose Π distinguishes ${({\psi }_{i})}_{i\in I}$. Then it follows from lemma 1 that ${({\phi }_{i})}_{i\in I}$ defined by

$$\begin{eqnarray*}&&{\phi }_{i}:= \displaystyle \frac{{{\rm{\Pi }}}_{i}{\psi }_{i}}{\langle {\psi }_{i},{{\rm{\Pi }}}_{i}{\psi }_{i}\rangle }\end{eqnarray*}$$

is biorthogonal to ${({\psi }_{i})}_{i\in I}$. Thus ${({\psi }_{i})}_{i\in I}$ is minimal by proposition 1.

(2) (Minimality implies distinguishability). Suppose ${({\psi }_{i})}_{i\in I}$ is minimal. Then by proposition 1 there exits a family ${({\phi }_{i})}_{i\in I}$ biorthogonal to ${({\psi }_{i})}_{i\in I}$. Define

$$\begin{eqnarray*}&&{{\rm{\Pi }}}_{i}:= {p}_{i}\displaystyle \frac{| {\phi }_{i}\rangle \langle {\phi }_{i}| }{\parallel | {\phi }_{i}\rangle \langle {\phi }_{i}| \parallel },\quad {{\rm{\Pi }}}_{?}:= 1-\displaystyle \sum _{i\in I}{{\rm{\Pi }}}_{i},\end{eqnarray*}$$

where ${\sum }_{i\in I}{p}_{i}=1$ and ${p}_{i}\gt 0$ for all $i\in I$. This is possible since I is countable. The operators ${{\rm{\Pi }}}_{j}$ constitutes a POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in I\bigsqcup \{?\}}$ because positivity of ${{\rm{\Pi }}}_{?}$ is guaranteed by an inequality for the operator norm, $\| {\sum }_{i\in I}{{\rm{\Pi }}}_{i}\| \leqslant \sum {p}_{i}=1$. It follows from the biorthogonality that Π distinguishes ${({\psi }_{i})}_{i\in I}$.

(3) (Uniform distinguishability implies the Riesz–Fischer property). Suppose Π uniformly distinguishes ${({\psi }_{i})}_{i\in I}$. Let $\psi ={\sum }_{i\in I}{\alpha }_{i}{\psi }_{i}$, with all ${\alpha }_{i}$ but finitely many being zero. One has

$$\begin{eqnarray*}{\| \displaystyle \sum _{i}{\alpha }_{i}{\psi }_{i}\| }^{2}=\langle \psi ,\psi \rangle & = & \displaystyle \sum _{k\in I}\langle \psi ,{{\rm{\Pi }}}_{k}\psi \rangle +\langle \psi ,{{\rm{\Pi }}}_{?}\psi \rangle \\ & \geqslant & \displaystyle \sum _{i,j,k\in I}\bar{{\alpha }_{i}}\langle {\psi }_{i},{{\rm{\Pi }}}_{k}{\psi }_{j}\rangle {\alpha }_{j}+0=q({\rm{\Pi }})\displaystyle \sum _{k}{| {\alpha }_{k}| }^{2},\end{eqnarray*}$$

where the inequality follows from positivity of ${{\rm{\Pi }}}_{?}$ and the last equality from lemma 1. Thus ${({\psi }_{i})}_{i\in I}$ is Riesz–Fischer with bound $q({\rm{\Pi }})$.

(4) (The Riesz–Fischer property implies uniform distinguishability). Suppose ${({\psi }_{i})}_{i\in I}$ is Riesz–Fischer with bound A. By proposition 2, there exists a biorthogonal family ${({\phi }_{i})}_{i\in I}$ which is Bessel with bound ${A}^{-1}$. It follows from (7) that for any ξ the sum ${\sum }_{i\in I}{| \langle {\phi }_{i},\xi \rangle | }^{2}$ converges. This in particular implies that only countably many $\langle {\phi }_{i},\xi \rangle$ are nonzero. With an appropriate order, the sequence ${\sum }_{i\leqslant k}{| \langle {\phi }_{i},\xi \rangle | }^{2}$ becomes Cauchy. From this fact and (6), one can show that

$$\begin{eqnarray*}&&X:= \displaystyle \sum _{i\in I}| {\phi }_{i}\rangle \langle {\phi }_{i}| \end{eqnarray*}$$

converges in the strong operator topology and defines a bounded liner map with bound $\parallel X\parallel \leqslant {A}^{-1}$. We define the POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in I\bigsqcup \{?\}}$ by

$$\begin{eqnarray*}&&{{\rm{\Pi }}}_{i}:= \displaystyle \frac{| {\phi }_{i}\rangle \langle {\phi }_{i}| }{\parallel X\parallel },\end{eqnarray*}$$

where positivity of ${{\rm{\Pi }}}_{?}$ is verified by

$$\begin{eqnarray*}&&{{\rm{\Pi }}}_{?}:= 1-\displaystyle \sum _{i\in I}{{\rm{\Pi }}}_{i}=1-\displaystyle \frac{X}{\parallel X\parallel }\geqslant 0.\end{eqnarray*}$$

This POVM uniformly distinguishes ${({\psi }_{i})}_{i\in I}$ with uniform success probability $q({\rm{\Pi }})=1/\parallel X\parallel \geqslant 1/{A}^{-1}=A$.

(5) (Perfect distinguishability implies orthonormality). Let Π distinguish ${({\psi }_{i})}_{i\in I}$. Then lemma 1 implies that

$$\begin{eqnarray*}&&\langle {\psi }_{i},{\psi }_{j}\rangle =\displaystyle \sum _{k\in I}\langle {\psi }_{i},{{\rm{\Pi }}}_{k}{\psi }_{j}\rangle +\langle {\psi }_{i},{{\rm{\Pi }}}_{?}{\psi }_{j}\rangle ={\delta }_{{ij}}{q}_{j}({\rm{\Pi }})+0={\delta }_{{ij}}.\end{eqnarray*}$$

Thus ${({\psi }_{i})}_{i\in I}$ is an orthonormal family.

(6) (Orthonormality implies perfect distinguishability). If ${({\psi }_{i})}_{i\in I}$ is an orthonormal family, the POVM Π defined by ${{\rm{\Pi }}}_{i}:= | {\psi }_{i}\rangle \langle {\psi }_{i}|$ and ${{\rm{\Pi }}}_{?}:= 1-{\sum }_{i\in I}{{\rm{\Pi }}}_{i}$ perfectly distinguishes ${({\psi }_{i})}_{i\in I}$.□

If a family ${({\psi }_{i})}_{i\in I}$ is given, one is interested in not only whether or not it is uniformly distinguishable but also the value of the largest possible success probability. The following theorem gives the value and the construction of the measurement.

Theorem 2. Suppose that a family ${({\psi }_{i})}_{i\in I}$ of pure states is uniformly distinguishable. By proposition 1 and theorem 1, it admits a unique biorthogonal family ${({\phi }_{i})}_{i\in I}$ contained in $\bar{{span}}\{{\psi }_{i}| i\in I\}$. Then among all POVMs which uniformly distinguish ${({\psi }_{i})}_{i\in I}$, the POVM ${\rm{\Pi }}={({{\rm{\Pi }}}_{j})}_{j\in I\bigsqcup \{?\}}$ defined by

$$\begin{eqnarray}{{\rm{\Pi }}}_{j}:= \left\{\begin{array}{ll}\displaystyle \frac{| {\phi }_{j}\rangle \langle {\phi }_{j}| }{\ \| \;{\displaystyle \sum }_{k\in I}| {\phi }_{k}\rangle \langle {\phi }_{k}| \;\| \ }, & j\in I,\\ 1-\displaystyle \sum _{k\in I}{{\rm{\Pi }}}_{k}, & j=?,\end{array}\right.\end{eqnarray} \tag{ 8 }$$

attains the maximum uniform success probability

$$\begin{eqnarray}&&q({\rm{\Pi }})=\displaystyle \frac{1}{\ \| \;{\displaystyle \sum }_{k\in I}| {\phi }_{k}\rangle \langle {\phi }_{k}| \;\| \ }.\end{eqnarray} \tag{ 9 }$$

Applying theorem 2 to the case of two pure states ${({\psi }_{i})}_{i=1,2}$, we can immediately obtain the maximum uniform success probability

$$\begin{eqnarray}&&q({\rm{\Pi }})=1-| \langle {\psi }_{1}| {\psi }_{2}\rangle | ,\end{eqnarray} \tag{ 10 }$$

by calculating the smallest nonzero eigenvalue of the operator ${\sum }_{i=\mathrm{1,2}}| {\psi }_{i}\rangle \langle {\psi }_{i}|$. This is essentially the result of Dieks [3], though he gave a prior uniform probability density ${({p}_{i})}_{i=1,2}\quad ={(1/2)}_{i=\mathrm{1,2}}$ to ${({\psi }_{i})}_{i=1,2}$ while we do not. For finitely many states, uniform prior probability density is canonical in the sense of having a maximum entropy. However, in the case of infinity many states, uniform probability density does not exist. This is the reason why we do not employ the prior probability in our discussion. Our definition of uniform distinguishability is meaningful even when the cardinality (number of elements) of I is infinite.

Prior to the proof for the theorem, we show a lemma which states that any unambiguous measurement of a given family of pure states is essentially the projection to its biorthogonal family. The lemma corresponds to equation (2.9) in [6], extended to the case of countable families ${({\psi }_{i})}_{i\in I}$.

Lemma 2. Assume that a POVM Π distinguishes a family ${({\psi }_{i})}_{i\in I}$ of pure states so that, by theorem 1 and proposition 1, ${({\psi }_{i})}_{i\in I}$ admits a unique biorthogonal family ${({\phi }_{i})}_{i\in I}$ contained in $\bar{{span}}\{{\psi }_{i}| i\in I\}$. Then Π satisfies

$$\begin{eqnarray*}&&P{{\rm{\Pi }}}_{i}{P}^{\ast }={q}_{i}({\rm{\Pi }})| {\phi }_{i}\rangle \langle {\phi }_{i}| ,\end{eqnarray*}$$

where P is the orthogonal projection of H onto $\bar{{span}}\{{\psi }_{i}| i\in I\}$.

Proof. Assume for a while $\bar{{span}}\{{\psi }_{i}| i\in I\}=H$ and therefore P = 1. For any $\psi =\sum {\alpha }_{i}{\psi }_{i}\in \mathrm{span}\{{\psi }_{i}| i\in I\}$, it follows from lemma 1 that

$$\begin{eqnarray*}&&\langle \psi ,({{\rm{\Pi }}}_{i}-{q}_{i}({\rm{\Pi }})| {\phi }_{i}\rangle \langle {\phi }_{i}| )\psi \rangle =0\end{eqnarray*}$$

holds for all $i\in I$. Thus, by continuity of the inner product, ${{\rm{\Pi }}}_{i}$, and $| {\phi }_{i}\rangle \langle {\phi }_{i}|$ , the equation above holds for all $\psi \in \bar{{span}}\{{\psi }_{i}| i\in I\}=H$. One therefore has ${{\rm{\Pi }}}_{i}={q}_{i}({\rm{\Pi }})| {\phi }_{i}\rangle \langle {\phi }_{i}|$. This proves the lemma in the case $\bar{{span}}\{{\psi }_{i}| i\in I\}=H$. When $\bar{{span}}\{{\psi }_{i}| i\in I\}\ne H$, define ${{\rm{\Pi }}}^{\prime }$ by

$$\begin{eqnarray*}{{\rm{\Pi }}}_{j}^{\prime }:= \left\{\begin{array}{ll}P{{\rm{\Pi }}}_{j}{P}^{\ast }, & j\in I,\\ P{{\rm{\Pi }}}_{?}P+(1-P), & j\quad =\quad ?.\end{array}\right.\end{eqnarray*}$$

Then ${{\rm{\Pi }}}^{\prime }$ is a POVM on the Hilbert space PH. The claim reduces to the case $\bar{{span}}\{{\psi }_{i}| i\in I\}=H$. □

Proof of theorem 2. As was shown in the proof of theorem 1, $\parallel {\sum }_{i\in I}| {\phi }_{i}\rangle \langle {\phi }_{i}| \parallel$ is finite and the POVM Π in the theorem is well-defined. Let ${{\rm{\Pi }}}^{\prime }$ be an arbitrary POVM which distinguishes ${({\psi }_{i})}_{i\in I}$ uniformly and let P be the orthogonal projection from H onto $\bar{{span}}\{{\psi }_{i}| i\in I\}$. It follows from from lemma 2 and ${{\rm{\Pi }}}_{?}^{\prime }\geqslant 0$ that

$$\begin{eqnarray*}&&P=P1{P}^{\ast }\geqslant \displaystyle \sum _{i\in I}P{{\rm{\Pi }}}_{i}^{\prime }{P}^{\ast }=q({{\rm{\Pi }}}^{\prime })\displaystyle \sum _{i\in I}| {\phi }_{i}\rangle \langle {\phi }_{i}| .\end{eqnarray*}$$

Thus one has $1\geqslant \parallel P\parallel \geqslant q({{\rm{\Pi }}}^{\prime })\| {\sum }_{i\in I}| {\phi }_{i}\rangle \langle {\phi }_{i}| \| =q({{\rm{\Pi }}}^{\prime })/q({\rm{\Pi }})$, i.e., $q({\rm{\Pi }})\geqslant q({{\rm{\Pi }}}^{\prime })$. □

In this section, we shall discuss distinguishability of coherent states represented by a lattice in the complex plane, which is called the von Neumann lattice. The coherent states may be defined for a particle in one dimension or any quantum system which allows a harmonic oscillator, so that they can represent photons, phonons, or other bosonic particles. We do not specify the physical system here and treat them in general, though we may sometimes use the terminology for a particle.

Let H be the Hilbert space which represents the states of a quantum system. Let a be the annihilation operator on H that satisfies ${{aa}}^{\ast }-{a}^{\ast }a=1$. Let $| 0\rangle$ be a state which satisfies $a| 0\rangle =0$. The state $| 0\rangle$ is unique up to phase factor and is called the vacuum. The coherent state $| z\rangle$, where $z\in {\mathbb{C}}$, is defined by [19, 20]

$$\begin{eqnarray}&&| z\rangle := \mathrm{exp}[{{za}}^{\ast }-\bar{z}a]| 0\rangle .\end{eqnarray} \tag{ 11 }$$

There are minimum uncertainty states for the position operator $Q={2}^{-1/2}(a+{a}^{\ast })$ and the momentum operator $P={(2i)}^{-1/2}(a-{a}^{\ast })$. This allows one to regard a coherent state $| z\rangle$ as the quantum state that corresponds to the classical state represented by a single point $z={2}^{-1/2}(q+{ip})\in {\mathbb{C}}$ in the phase space. It is easily verified by the equation $a| z\rangle =z| z\rangle$ that coherent states ${(| z\rangle )}_{z\in {\mathbb{C}}}$ is linearly independent. It follows that mutually different finite number of states ${(| {z}_{i}\rangle )}_{i=1,2,...,n}$ are uniformly distinguishable.

In the context of simultaneous measurement of position and momentum, von Neumann considered the following family of coherent states, which corresponds to a lattice on the phase space.

Definition 4. Let ${\omega }_{1},{\omega }_{2}\in {\mathbb{C}}$ be such that ${\rm{Im}}({\omega }_{2}/{\omega }_{1})\gt 0$. Let

$$\begin{eqnarray}&&{\mathsf{L}}({\omega }_{1},{\omega }_{2}):= \{{n}_{1}{\omega }_{1}+{n}_{2}{\omega }_{2}\in {\mathbb{C}}| \;{n}_{1},{n}_{2}\in {\mathbb{Z}}\}\subset {\mathbb{C}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\mathsf{vNL}}({\omega }_{1},{\omega }_{2}):= \{| {\rm{\Omega }}\rangle | \;{\rm{\Omega }}\in {\mathsf{L}}({\omega }_{1},{\omega }_{2})\}\subset H.\end{eqnarray} \tag{ 13 }$$

A family ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ is called a von Neumann lattice. The set $\{{t}_{1}{\omega }_{1}+{t}_{2}{\omega }_{2}| \;{t}_{1},{t}_{2}\in [0,1)\}\subset {\mathbb{C}}$ is called the fundamental region of ${\mathsf{L}}({\omega }_{1},{\omega }_{2})$ or of ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$. The area of the fundamental region is usually denoted by S.

Though in some literature the name von Neumann lattice is used only for the case $S=\pi$, we use the term for all S. They are called Weyl–Heisenberg systems in the field of time-frequency analysis or in Gabor analysis [11, 21]. von Neumann stated without proof that a von Neumann lattice is complete when $S\lesssim 1/4$ [10, p.217, p.407 in the English version]. It was about 40 years later that Perelomov [13] and Bargmann et al [14] gave the proof for this fact. Hereafter, we identify a set and a list of vectors in our discussion of von Neumann lattices, so that we may say, for example, that a von Neumann lattice is distinguishable, or that it is linearly independent.

We collect the properties of von Neumann lattices [12–14] in table 1. These properties are essentially attributed, via the Fock-Bargmann space [22], to the nature of entire functions in complex analysis.

Table 1.  Properties of von Neumann lattices: ${{\mathsf{vNL}}}^{(n)}$ denotes the set obtained by removing arbitrary n elements from ${\mathsf{vNL}}$. Facts marked by † and ‡ can be deduced from (38) in [13]. However, they have not been stated explicitly.

	State families	Complete	Minimal	Riesz–Fischer
$S\lt \pi$	${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$	Yes	No	No
	${{\mathsf{vNL}}}^{(n)}({\omega }_{1},{\omega }_{2}),\ 1\leqslant n\lt \infty$	Yes	No	No
$S=\pi$	${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$	Yes	No	No
	${{\mathsf{vNL}}}^{(1)}({\omega }_{1},{\omega }_{2})$	Yes	Yes	No${}^{\dagger }$
	${{\mathsf{vNL}}}^{(n)}({\omega }_{1},{\omega }_{2}),\ 2\leqslant n\lt \infty$	No	Yes	No${}^{\ddagger }$
$S\gt \pi$	${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$	No	Yes	Yes

Combining these properties and theorem 1, we arrive at our second main result.

Theorem 3. Let S be the area of the fundamental region of ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$, Then, the following hold.

• i.  
  When $S\lt \pi$, ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ is not distinguishable.
• ii.  
  When $S=\pi$, ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ is not distinguishable. However, the set ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ with more than one element removed is distinguishable, but is not uniformly distinguishable.
• iii.  
  When $S\gt \pi$, ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ is uniformly distinguishable, but is not perfectly distinguishable.

A finite subset of coherent states is always uniformly distinguishable, whereas von Neumann's lattice, which is infinite, behaves quite differently. The result shows that each level of distinguishability is directly related to the area $S={\rm{Im}}(\bar{{\omega }_{1}}\;{\omega }_{2})$ and does not depend on ${\omega }_{1}$ and ${\omega }_{2}$ separately. Furthermore, it can be shown that distinguishability of von Neumann's lattice is determined solely by the density of points in the complex plane, $1/S$, and is robust to deformation of the lattice (see proofs in [13] and [12]). The threshold $S=\pi$ corresponds to the area of the Planck constant h in the classical phase space (note that ${{\rm{d}}}^{2}z={(h/2\pi )}^{-1}({2}^{-1/2}{\rm{d}}{q})({2}^{-1/2}{\rm{d}}{p})$). Physically, the area h is the minimum unit of area of the phase space, which appeared e.g. in the Bohr–Sommerfeld quantum condition [23, (48.2)]. Theorem 3 reveals the measurement theoretical meaning of the Planck constant.

Finally, we give a simple and direct estimate for the uniform success probability, though it is not tight.

Theorem 4.  $| {\omega }_{1}| ,| {\omega }_{2}| \to \infty$ with $\mathrm{sin}\ \mathrm{arg}(\bar{{\omega }_{1}}\;{\omega }_{2})\gt \varepsilon$ for some $\varepsilon \gt 0$ forces maximal uniform success probability $q({\rm{\Pi }})$ of ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ approaches 1. More specifically, when ${\omega }_{1},{\omega }_{2}\in {\mathbb{C}}$ satisfies $S={\rm{Im}}(\bar{{\omega }_{1}}\;{\omega }_{2})\gt \pi$, there exists a POVM Π which distinguishes ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ uniformly with uniform success probability

$$\begin{eqnarray*}&&q({\rm{\Pi }})\geqslant 2-{\left(1+\displaystyle \frac{2\sqrt{\pi }}{\mathrm{sin}\mathrm{arg}(\bar{{\omega }_{1}}{\omega }_{2}){\mathrm{min}}_{i}| {\omega }_{i}| }\right)}^{2}.\end{eqnarray*}$$

Proof. Let ${{\rm{\Omega }}}_{{nm}}=n{\omega }_{1}+m{\omega }_{2}\in {\mathsf{L}}({\omega }_{1},{\omega }_{2})$. For any numerical sequence ${({\alpha }_{n,m})}_{(n,m)\in {{\mathbb{Z}}}^{2}}$ with only finitely many being nonzero, one has

$$\begin{eqnarray*} & & {\| \displaystyle \sum {\alpha }_{m,n}| {{\rm{\Omega }}}_{m,n}\rangle \| }^{2}\\ & = & \displaystyle \sum _{m,n}\;\displaystyle \sum _{k,{\ell }}\bar{{\alpha }_{k,{\ell }}}{\alpha }_{k+m,{\ell }+n}\langle {{\rm{\Omega }}}_{k,{\ell }}| {{\rm{\Omega }}}_{k+m,{\ell }+n}\rangle \\ & = & \displaystyle \sum _{k,{\ell }}{| {\alpha }_{k,{\ell }}| }^{2}+\displaystyle \sum _{(m,n)\ne (0,0)}\displaystyle \sum _{k,{\ell }}\bar{{\alpha }_{k,{\ell }}}{\alpha }_{k+m,{\ell }+n}\langle {{\rm{\Omega }}}_{k,{\ell }}| {{\rm{\Omega }}}_{k+m,{\ell }+n}\rangle .\end{eqnarray*}$$

The second sum in the last line can be estimated as

$$\begin{eqnarray*} & & \left|\displaystyle \sum _{(m,n)\ne (0,0)}\quad \displaystyle \sum _{k,{\ell }}\bar{{\alpha }_{k,{\ell }}}{\alpha }_{k+m,{\ell }+n}\langle {{\rm{\Omega }}}_{k,{\ell }}| {{\rm{\Omega }}}_{k+m,{\ell }+n}\rangle \right|\\ & \leqslant & \displaystyle \sum _{(m,n)\ne (0,0)}{{\rm{e}}}^{-{| {{\rm{\Omega }}}_{m,n}| }^{2}/2}\quad \displaystyle \sum _{k,{\ell }}| \bar{{\alpha }_{k,{\ell }}}{\alpha }_{k+m,{\ell }+n}| \\ & \leqslant & \displaystyle \sum _{(m,n)\ne (0,0)}{{\rm{e}}}^{-{| {{\rm{\Omega }}}_{m,n}| }^{2}/2}\quad \displaystyle \sum _{k,{\ell }}{| {\alpha }_{k,{\ell }}| }^{2},\end{eqnarray*}$$

where we have used the formula ${| \langle z| w\rangle | }^{2}={{\rm{e}}}^{-{| z-w| }^{2}}$ and the triangle inequality in the first line and the Cauchy-Schwarz inequality in the second. Therefore, we have

$$\begin{eqnarray*}&&{\| \sum {\alpha }_{m,n}| {{\rm{\Omega }}}_{m,n}\rangle \| }^{2}\geqslant A\displaystyle \sum _{k,{\ell }}{| {\alpha }_{k,{\ell }}| }^{2},\qquad A:= 1-\displaystyle \sum _{(m,n)\ne (0,0)}{{\rm{e}}}^{-{| {{\rm{\Omega }}}_{m,n}| }^{2}/2}.\end{eqnarray*}$$

In the case $A\gt 0$, ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ is Riesz–Fischer with bound A. So, by the theorem 1, ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$ is uniformly distinguishable with uniform success probability at least A.

Since $| \bar{\lambda }\lambda +\bar{\lambda }\nu | \geqslant | {\rm{Im}}(\bar{\lambda }\lambda +\bar{\lambda }\nu )| =| {\rm{Im}}(\bar{\lambda }\nu )|$, we have

$$\begin{eqnarray*}| {{\rm{\Omega }}}_{m,n}| & = & \displaystyle \frac{1}{| {\omega }_{1}m| }| \bar{{\omega }_{1}m}\;{\omega }_{1}m+\bar{{\omega }_{1}m}\;{\omega }_{2}n| \\ & \geqslant & \displaystyle \frac{1}{| {\omega }_{1}m| }| {\rm{Im}}\{\bar{{\omega }_{1}m}\;{\omega }_{2}n\}| =\displaystyle \frac{1}{| {\omega }_{1}m| }| {mn}| S\geqslant \displaystyle \frac{S}{\;{\mathrm{max}}_{i}\;| {\omega }_{i}| }\;| n| \end{eqnarray*}$$

when $m\ne 0$, and

$$\begin{eqnarray*}&&{| {{\rm{\Omega }}}_{m,n}| }^{2}\geqslant \displaystyle \frac{{S}^{2}}{\;{\mathrm{max}}_{i}{| {\omega }_{i}| }^{2}}\;{n}^{2}\end{eqnarray*}$$

for all $m,n\in {\mathbb{Z}}$. Hence

$$\begin{eqnarray*}1-A & \leqslant & \displaystyle \sum _{(m,n)\ne (0,0)}\mathrm{exp}\left[-\displaystyle \frac{{S}^{2}}{4\;{\mathrm{max}}_{i}{| {\omega }_{i}| }^{2}}({m}^{2}+{n}^{2})\right]\\ & = & -1+{\left(\displaystyle \sum _{k\in {\mathbb{Z}}}\mathrm{exp}\left[-\displaystyle \frac{{S}^{2}}{4{\mathrm{max}}_{i}{| {\omega }_{i}| }^{2}}{k}^{2}\right]\right)}^{2}\\ & \leqslant & -1+{\left(1+{\displaystyle \int }_{-\infty }^{\infty }\mathrm{exp}\left[-\displaystyle \frac{{S}^{2}}{4{\mathrm{max}}_{i}{| {\omega }_{i}| }^{2}}{x}^{2}\right]{\rm{d}}{x}\right)}^{2}\\ & = & -1+{\left(1+\displaystyle \frac{2\sqrt{\pi }{\mathrm{max}}_{i}| {\omega }_{i}| }{S}\right)}^{2}.\end{eqnarray*}$$

Since $S=| {\omega }_{1}| \;| {\omega }_{2}| \;\mathrm{sinarg}(\bar{{\omega }_{1}}\;{\omega }_{2})$, the desired estimate follows.□

Theorem 4 proves a part of theorem 3 directly. Theorem 4 justifies the intuition that, as the lattice becomes large, the von Neumann lattice approaches an orthogonal family.

It should be noted however that the uniform success probability cannot be estimated by S only. In particular, the condition $S\to \infty$ is not sufficient for the uniform success probability $q({\rm{\Pi }})$ to approach unity. It can be seen easily as follows. The uniform success probability $q({\rm{\Pi }})$ cannot be greater than the maximal uniform success probability of distinguishing two states $\{| 0\rangle ,| {\rm{\Omega }}\rangle \}$ in ${\mathsf{vNL}}({\omega }_{1},{\omega }_{2})$, where Ω is either ${\omega }_{1}$ or ${\omega }_{2}$. One therefore has, by (10),

$$\begin{eqnarray}&&q({\rm{\Pi }})\leqslant 1-| \langle 0| {\rm{\Omega }}\rangle | =1-\mathrm{exp}\left[-\displaystyle \frac{{| {\rm{\Omega }}| }^{2}}{2}\right]\end{eqnarray} \tag{ 14 }$$

for ${\rm{\Omega }}={\omega }_{1},{\omega }_{2}$. This proves that $q({\rm{\Pi }})$ cannot be estimated solely by S.

We examined distinguishability of countable pure states. We defined distinguishability of countable states as the possibility of unambiguous measurements on these states. There we classified the distinguishability into three types, namely distinguishability, uniform distinguishability, and perfect distinguishability. Distinguishability and uniform distinguishability, which are equivalent when the number of states is finite, split when the number becomes infinite. We then proved a criterion of distinguishability for countable pure states in theorem 1. The theorem establishes a relation between operational definitions of distinguishability and intrinsic properties of a family of state vectors in the Hilbert space. In addition, we gave the maximal uniform success probability and a POVM which attain it in theorem 2.

After developing a general criterion of distinguishability, we discussed distinguishability of von Neumann's lattice, which is a family of states which corresponds to a lattice of the phase space in classical mechanics. Besides its own interest in measurement theory, it serves as an excellent example for the general theory in the sense that all the subtleties arise and can be discussed completely. We showed in theorem 3 that the distinguishability of a von Neumann lattice depends only on the area of its fundamental region. We see a drastic change in distinguishability when the area becomes the Planck constant h. The result is robust to deformation of the lattice.

The Planck constant h is without doubt the most fundamental constant in quantum physics. It appears in the canonical commutation relation $[Q,P]={ih}/(2\pi )$ and characterizes quantum physics in many ways. One is the uncertainty relation of physical quantities. A simple, well-known inequality is Kennard's inequality [24, (27)], which gives a bound for the standard deviations σ of the observables Q and P,

$$\begin{eqnarray*}&&\sigma (Q)\sigma (P)\geqslant \displaystyle \frac{h}{4\pi }.\end{eqnarray*}$$

Modern interpretations of Heisenberg's noise–disturbance uncertainty relation and corresponding rigorous inequalities are also found in [25] (See also [26, section 6].) and [27]. Another important aspect of h is the Bohr–Sommerfeld quantum condition [23, (48.2)]

$$\begin{eqnarray*}&&\oint p\;{\rm{d}}{q}=h\left(n+\displaystyle \frac{1}{2}\right)\qquad n=0,1,2,....\end{eqnarray*}$$

This not only stimulated the discovery of quantum mechanics but also can be shown in quantum mechanics with the Wentzel–Kramers–Brillouin (WKB) approximation. This condition explains the familiar fact in statistical mechanics that a single quantum state occupies an area of h in the classical phase. Our analysis of von Neumann lattices revealed the significance of the Planck constant h through the context of state discrimination, thereby giving another measurement-theoretic meaning of h, and gives a rigorous version of justification for h to be the unit of the phase space.

We have not discussed criterions of distinguishability for mixed states. For finitely many mixed states, Feng et al [9] obtained a condition of distinguishability. We generalize their result to the case of countably many states and present it in a slightly different manner.

Proposition 3. Let ${({\rho }_{i})}_{i\in I}$ be a countable family of density operators on a Hilbert space H. Then, the following are equivalent.

• i.  
  ${({\rho }_{i})}_{i\in I}$ is distinguishable.
• ii.  
  For all $i\in I$,

  $$\begin{eqnarray*}&&(\bigcap _{k\ne i}\mathrm{ker}{\rho }_{k})\setminus \mathrm{ker}{\rho }_{i}\ne \varnothing .\end{eqnarray*}$$
• iii.  
  For all $i\in I$,

  $$\begin{eqnarray*}&&(\bigcap _{k\in I}\mathrm{ker}{\rho }_{k})\subsetneq (\bigcap _{k\ne i}\mathrm{ker}{\rho }_{k}).\end{eqnarray*}$$
• iv.  
  For all $i\in I$,

  $$\begin{eqnarray*}&&\bar{{span}}(\bigcup _{k\ne i}\bar{\mathrm{im}\;{\rho }_{k}})\subsetneq \bar{{span}}(\bigcup _{k\in I}\bar{\mathrm{im}\;{\rho }_{k}}).\end{eqnarray*}$$

Here, $\bar{L}$ denotes the norm closure of $L\subset H$, $\mathrm{ker}\rho =\{\xi \in H\quad | \quad \rho \xi =0\}$ and $\mathrm{im}\;\rho =\{\rho \xi \quad | \quad \xi \in H\}$ for a bounded operator $\rho$ on H.

Proof. Assume (i). Then, for distinct elements i and j of I, one has $0=\mathrm{tr}[{\rho }_{i}{{\rm{\Pi }}}_{j}]=\mathrm{tr}[{{\rm{\Pi }}}_{j}^{1/2}{\rho }_{i}{{\rm{\Pi }}}_{j}^{1/2}]$ hence ${\rho }_{i}{{\rm{\Pi }}}_{j}^{1/2}=0$. Therefore, for each $i\in I$, there exists ${\psi }_{i}$ such that ${\rho }_{k}({{\rm{\Pi }}}_{i}^{1/2}{\psi }_{i})=0$ for all $k\ne i$ and ${\rho }_{i}({{\rm{\Pi }}}_{i}^{1/2}{\psi }_{i})\ne 0$. This ensures (ii). That (ii) implies (i) is as in the proof of theorem 1. That (ii) is equivalent to (iii) is seen by a trivial set-theoretical identity $X\setminus (X\cap Y)=X\setminus Y$. The equivalence of (iii) and (iv) is due to $\bar{\mathrm{im}{\rho }_{i}}={(\mathrm{ker}{\rho }_{i})}^{\perp }$ and $\bar{{span}}({\bigcup }_{j}{K}_{j}^{\perp })={({\bigcap }_{j}{K}_{j})}^{\perp },$ where each K_j is a closed subspace of H, and ⊥ denotes the orthogonal complement. □

When ${({\rho }_{i})}_{i\in I}$ is finite family, (iv) reduces to

$$\begin{eqnarray*}&&\mathrm{span}(\bigcup _{k\ne i}\mathrm{im}\;{\rho }_{k})\subsetneq \mathrm{span}(\bigcup _{k\in I}\mathrm{im}\;{\rho }_{k})\end{eqnarray*}$$

for all $i\in I$. This is the condition that Feng et al [9] presented.

The criterion enables us to investigate the time evolution of distinguishability, i.e. the relation between distinguishability of ${({\rho }_{i})}_{i\in I}$ and that of ${({\rho }_{i}^{\prime })}_{i\in I}$. Here we assume each state ${\rho }_{i}$, at time t, evolves to ${\rho }_{i}^{\prime }$ at ${t}^{\prime }\gt t$. We already note in the remark below the definition 1 that distinguishability is invariant under unitary evolutions. Thus we should concern non-unitary evolutions of the system, which changes a pure state into a mixed state in general. Hence we need a criterion for mixed states.

Incidentally, we remark that ambiguous measurement, in contrary to the unambiguous one that we have discussed, is also of interest in a typical communication scenario. The optimal protocol depends heavily on the evaluation function, such as mutual information, in the distinction of mixed states (see e.g. [28, 29]). It may be interesting to examine to what degree a protocol based on an unambiguous measurement is useful in other contexts.

Distinction of coherent states is not only a subject of theoretical interest but also of a practical problem. A coherent state of light is easy to handle and is often used in optical communication. Let us consider the following simple example. The sender generates several coherent states and sends one of them, which travels through optical fibers. The receiver detects it and determines which coherent state was sent. The simplest case is to distinguish two states, the vacuum $| 0\rangle$ and another coherent state $| {\omega }_{1}\rangle$. A slightly more complicated problem is to distinguish nine states, i.e., the vacuum and the eight states enclosing the vacuum ${C}_{9}=\{\;| {\rm{\Omega }}\rangle | {\rm{\Omega }}=0,\pm {\omega }_{1},\pm {\omega }_{2},\pm {\omega }_{1}\pm {\omega }_{2}\}$. A still more complicated one is the distinction of 25 states in the set C₂₅, doubly surrounding the vacuum. In this manner, we consider the distinction of the states in the set ${C}_{1+4n(n-1)}$ which is a finite subset of a von Neumann lattice. It approaches the whole von Neumann lattice as $n\to \infty$. We denote by S the area of fundamental region of lattice corresponding to ${C}_{1+4n(n-1)}$ (we assume $S\gt 0$). The family ${C}_{1+4n(n-1)}$ is linearly independent so that by theorem 1 it is uniformly distinguishable with uniform success probability ${q}_{n}\gt 0$. The q_n satisfies ${q}_{1}\geqslant {q}_{2}\geqslant \cdots \;\gt \;0$. Thus there exists a finite limit ${\mathrm{lim}}_{n\to \infty }\quad {q}_{n}\geqslant q$. Here, q is the uniform success probability for the whole lattice which is positive when $S\gt \pi$ and vanishes otherwise. The behavior of q_n for smaller n may be of practical interest, and the asymptotic behavior for large n may be of theoretical interest.

This work was supported by JSPS Grant-in-Aid for Scientific Research No. 24540282 and by MEXT-Supported Program for the Strategic Research Foundation at Private Universities Topological Science.

In this appendix, we briefly discuss the case that states are not countable. We shall show that uncountably many states cannot be distinguishable when H is separable. Note that in the standard formulation of quantum mechanics (e.g. [10, II-1, Postulate E]) the Hilbert space is assumed to be separable.

We begin by extending the definition of distinguishability to the case of uncountable states. We cannot define POVM with ${\sum }_{j\in J}{{\rm{\Pi }}}_{j}=1$ when the set J of outcomes is uncountable since the sum exceeds countable additivity. We therefore go back to the measure-theoretical definition of the POVM. Let B(H) be the set of bounded operators on H.

Let $(J,{ \mathcal J })$ denote a measurable space, where J is a set and ${ \mathcal J }$ a σ-algebra on J. The map ${\rm{\Pi }}\quad :{ \mathcal J }\to B(H)$ is called a POVM on $(J,{ \mathcal J })$ if it satisfies the following conditions [15 section 3.1].

• i.  
  ${\rm{\Pi }}(J)=1$ and ${\rm{\Pi }}({\rm{\Delta }})\geqslant 0$ for all ${\rm{\Delta }}\in { \mathcal J }$.
• ii.  
  ${\rm{\Pi }}({\bigcup }_{k}{{\rm{\Delta }}}_{k})={\sum }_{k}{\rm{\Pi }}({{\rm{\Delta }}}_{k})$ in the sense of weak operator topology for all disjoint countable collection $\{{{\rm{\Delta }}}_{k}\}\subset { \mathcal J }$.

We shall redefine the distinguishability of states, or extend definition 1(i), to include families of uncountable states.

Definition 5. A POVM Π on $(J,{ \mathcal J })$ distinguishes the family ${({\rho }_{i})}_{i\in I}$ of states if the following conditions hold.

• i.  
  $J=I\bigsqcup \{?\}$, and $\{i\}\in { \mathcal J }$ holds for all $i\in I$.
• ii.  
  For all $i,j\in I$, $\mathrm{tr}[{\rm{\Pi }}(\{i\}){\rho }_{j}]$ is positive if i = j and vanishes if $i\ne j$.

The states ${({\rho }_{i})}_{i\in I}$ are distinguishable if there exists a POVM Π which distinguishes them.

We show a general relation between $\mathrm{dim}\ H$ and the number of input states that is necessary for distinguishability. Here, $\mathrm{dim}\ H$ is defined as the cardinality of an orthonormal basis of H, which is countable if and only if H is separable.

Theorem 5. Let ${({\rho }_{i})}_{i\in I}$ be a state family of a Hilbert space H. If ${({\rho }_{i})}_{i\in I}$ is distinguishable in the sense of definition 5, then $\mathrm{dim}\ H\geqslant | I|$, where $| I|$ denote the cardinality of the set I.

Proof. The proof is divided into two steps.

1 (In the case that all ${\rho }_{i}$ are pure). Let ${\rho }_{i}=| {\psi }_{i}\rangle \langle {\psi }_{i}| ,{\psi }_{i}\in H$. Then ${({\psi }_{i})}_{i\in I}$ is minimal, which can be shown in a similar manner to the proof of theorem 1. For $J\subset I$, let ${K}_{J}:= \bar{{span}}\{{\psi }_{i}\;| \;i\in J\}$. Minimality of ${({\psi }_{i})}_{i\in I}$ implies that for $k\notin J$, there is a normalized vector e_k which is orthogonal to K_J and ${\psi }_{k}\in {\mathbb{C}}{e}_{k}+{K}_{J}$. Using transfinite induction on I, we can construct a orthonormal family ${({e}_{i})}_{i\in I}$. Therefore, $\mathrm{dim}\ H\geqslant | I|$.

2 (General case). Assume a POVM Π distinguishes ${({\rho }_{i})}_{i\in I}$. Let ${{\rm{\Pi }}}_{j}$ denote ${\rm{\Pi }}(\{j\})$. As in the proof of proposition 4, ${{\rm{\Pi }}}_{j}^{1/2}{\rho }_{i}=0$ holds for all $i,j\in I$ with $i\ne j$ and ${{\rm{\Pi }}}_{i}^{1/2}{\rho }_{i}\ne 0$ for all $i\in I$. Hence, for each $i\in I$, there exists ${\phi }_{i}\in H$ such that ${{\rm{\Pi }}}_{i}^{1/2}{\rho }_{i}{\phi }_{i}\ne 0$. Define ${\psi }_{i}={\rho }_{i}{\phi }_{i}/\parallel {\rho }_{i}{\phi }_{i}\parallel$. Since

$$\begin{eqnarray*}&&\mathrm{tr}[{{\rm{\Pi }}}_{j}| {\psi }_{i}\rangle \langle {\psi }_{i}| ]=\displaystyle \frac{1}{{\parallel {\rho }_{i}{\psi }_{i}\parallel }^{2}}{\parallel {{\rm{\Pi }}}_{j}^{1/2}{\rho }_{i}{\phi }_{i}\parallel }^{2}\end{eqnarray*}$$

for all $i,j\in I$, Π distinguishes ${(| {\psi }_{i}\rangle \langle {\psi }_{i}| )}_{i\in I}$. Therefore, the claim follows from the first step.□

Theorem 5 shows that separability of H forces any distinguishable family of states to be countable. On the other hand, there exists a distinguishable family of uncountable states in a non-separable Hilbert space. A simple example is the following.

Example 1. Let H be a non-separable Hilbert space, and ${({e}_{i})}_{i\in I}$ be a complete orthonormal system of H, which is uncountable. Let $(I,{2}^{I})$ be a measurable space, where ${2}^{I}:= \{{\rm{\Delta }}\quad | \quad {\rm{\Delta }}\subset I\}$. Define ${\rm{\Pi }}\quad :{2}^{J}\longrightarrow B(H)$ by

$$\begin{eqnarray*}&&{\rm{\Pi }}({\rm{\Delta }}):= \displaystyle \sum _{i\in {\rm{\Delta }}}| {e}_{i}\rangle \langle {e}_{i}| ,\qquad {\rm{\Delta }}\in {2}^{I}.\end{eqnarray*}$$

The sum converges in the strong operator topology. Then Π is a well-defined POVM on $(I,{2}^{I})$ and (perfectly) distinguishes ${({e}_{i})}_{i\in I}$.

In the example above, Π satisfies 'uncountable additivity' ${\sum }_{j\in J}{\rm{\Pi }}(\{j\})=1={\rm{\Pi }}(J)$. However, when we define a measure ${\mu }_{\psi }\quad :{2}^{J}\longrightarrow [0,\infty )\subset {\mathbb{R}}$ for a vector $\psi \in H$ as ${\mu }_{\psi }({\rm{\Delta }})=\langle \psi ,{\rm{\Pi }}({\rm{\Delta }})\psi \rangle$, then 'uncountable additivity' of ${\mu }_{\psi }$ reduces to a trivial matter because $\sum \;\langle \psi ,{\rm{\Pi }}(\{i\})\psi \rangle ={\parallel \psi \parallel }^{2}\lt \infty$ and $\{j\in J| {\mu }_{\psi }(\{j\})\ne 0\}$ is countable.

We give a proof of proposition 1 for convenience of the reader. We do so by showing a slightly generalized proposition below. For a normed space X, let ${X}^{\ast }$ denote the topological dual of X, which consists of continuous functionals. In this case, we say ${({\psi }_{i})}_{i\in I}$ of X and ${({\phi }_{i})}_{i\in I}$ of ${X}^{\ast }$ are biorthogonal when ${\phi }_{i}{\psi }_{j}={\delta }_{i,j}$ for all $i,j\in I$.

Proposition 4. Let X be a normed space and ${({\psi }_{i})}_{i\in I}$ be a family of vectors in X. ${({\psi }_{i})}_{i\in I}$ is minimal if and only if ${({\psi }_{i})}_{i\in I}$ admits a biorthogonal family. If the condition holds and $\bar{{span}}\{{\psi }_{i}| i\in I\}=X$, then the biorthogonal family is unique.

Proof. When X = 0, the statement is trivial. We assume $X\ne 0$ in the following. Let $Y=\mathrm{span}\{{\psi }_{i}| i\in I\},{Y}_{i}=\mathrm{span}\{{\psi }_{j}| i\ne j\in I\}$ and these norm closure $\bar{Y},\bar{{Y}_{i}}$, respectably.

First, suppose ${({\psi }_{i})}_{i\in I}$ is minimal. Because ${({\psi }_{i})}_{i\in I}$ is linearly independent, one can define for each $i\in I$ a linear functional ${\phi }_{i}^{\prime }\quad :Y\longrightarrow {\mathbb{C}}$ by the condition ${\phi }_{i}^{\prime }{\psi }_{j}={\delta }_{i,j}$, where $i,j\in I$. We shall show ${\phi }_{i}^{\prime }$ is continuous on Y. One has

$$\begin{eqnarray*}&&\parallel {\phi }_{i}^{\prime }\parallel := \underset{\psi \in Y\setminus \{0\}}{\mathrm{sup}}\displaystyle \frac{| {\phi }_{i}^{\prime }\psi | }{\parallel \psi \parallel }=\displaystyle \frac{1}{\underset{\psi \in {\psi }_{i}+{Y}_{i}}{\mathrm{inf}}\parallel \psi \parallel }.\end{eqnarray*}$$

The denominator is positive because ${({\psi }_{i})}_{i\in I}$ is minimal. Therefore, $\parallel {\phi }_{i}^{\prime }\parallel \lt \infty$ and the linear functional ${\phi }_{i}^{\prime }$ is continuous on Y. Due to the Hahn–Banach theorem, ${\phi }_{i}^{\prime }$ admits a continuous extension to the whole space X. The extension, which we denote by ${\phi }_{i}$, belongs to ${X}^{\ast }$. By construction, ${({\phi }_{i})}_{i\in I}$ is a biorthogonal family for ${({\psi }_{i})}_{i\in I}$.

Second, let ${({\phi }_{i})}_{i\in I}$ be a biorthogonal family for ${({\psi }_{i})}_{i\in I}$. Suppose that ${({\psi }_{i})}_{i\in I}$ were not minimal. Then there would exist $i\in I$ such that ${\psi }_{i}\in \bar{{Y}_{i}}$, i.e., there would exist a sequence ${({\xi }_{n})}_{n\in {\mathbb{N}}}$ in Y_i such that ${\xi }_{n}\to {\psi }_{i}$. Since ${\phi }_{i}\in {X}^{\ast }$ is continuous and ${\phi }_{i}{Y}_{i}=0$, one would have $1={\phi }_{i}{\psi }_{i}={\phi }_{i}{\mathrm{lim}}_{n}{\xi }_{n}={\mathrm{lim}}_{n}\;{\phi }_{i}{\xi }_{n}={\mathrm{lim}}_{n}\;0=0,$ a contradiction.

For the proof of the uniqueness part of the proposition, assume $\bar{Y}=X$ and ${({\psi }_{i})}_{i\in I}$ has two biorthogonal families ${({\phi }_{i})}_{i\in I}$ and ${({\phi }_{i}^{\prime })}_{i\in I}$. Because the functional ${\phi }_{i}-{\phi }_{i}^{\prime }$ is continuous on X and vanishes on Y, one has ${\phi }_{i}={\phi }_{i}^{\prime }$ on $\bar{Y}=X$.□

When X = H is a Hilbert space, the Riesz theorem establishes a conjugate isomorphism ${H}^{\ast }\simeq H$ so that a biorthogonal family ${({\phi }_{i})}_{i\in I}$ of ${H}^{\ast }$ can be regarded as a family in H.