Witnessing causal nonseparability

Each party A acts in a local quantum laboratory, which can be identified by an input Hilbert space ${{\mathcal{H}}}^{{A}_{I}}$ and an output Hilbert space ${{\mathcal{H}}}^{{A}_{O}}.$ The dimensions ${d}_{{A}_{I}}$ and ${d}_{{A}_{O}}$ of input and output spaces do not have to be equal, as ancillary systems can be added or discarded during an operation; we shall nevertheless assume throughout the paper that all Hilbert spaces are finite-dimensional. According to quantum theory, the most general local operation is described by a completely positive (CP), trace non-increasing map ${{\mathcal{M}}}^{A}\;:{A}_{I}\to {A}_{O}$ [15], where we write A_I, respectively A_O, for the space of Hermitian linear operators over the Hilbert space ${{\mathcal{H}}}^{{A}_{I}},$ resp. ${{\mathcal{H}}}^{{A}_{O}}.$ Examples of CP maps are deterministic operations, such as unitaries or quantum channels, or (generalized) measurements. In general, a label a, denoting the measurement outcome, is associated with the CP map ${{\mathcal{M}}}_{a}^{A}.$ The choice of operation (e.g. of measurement setting) is represented by an instrument [16], which is defined as the collection ${{\mathcal{J}}}^{A}={\{{{\mathcal{M}}}_{a}^{A}\}}_{a=1}^{m}$ of CP maps associated to all measurement outcomes, characterized by the property that ${\sum }_{a=1}^{m}{{\mathcal{M}}}_{a}^{A}$ is CP and trace-preserving (CPTP). An instrument generalizes the notion of positive operator-valued measure (POVM) to include the transformations applied to the system; it reduces to a POVM for one-dimensional output spaces. When the choice of operation is described by a classical variable x, we will express such a dependence explicitly as ${{\mathcal{J}}}_{x}^{A}={\left\{{{\mathcal{M}}}_{a| x}^{A}\right\}}_{a=1}^{m}.$

A convenient representation of CP maps is given by the Choi–Jamiołkowski (CJ) isomorphism [17]. For a CP map ${{\mathcal{M}}}_{a}^{A}\;:{A}_{I}\to {A}_{O},$ its corresponding CJ matrix is defined here as

$$\begin{eqnarray}&&{M}_{a}^{{A}_{I}{A}_{O}}:= {\left[{\mathcal{I}}\otimes {{\mathcal{M}}}_{a}^{A}\left(| {\mathbb{1}}\rangle \rangle \langle \langle {\mathbb{1}}| \right)\right]}^{T}\in {A}_{I}\otimes {A}_{O},\end{eqnarray} \tag{ 1 }$$

where ${\mathcal{I}}$ is the identity map, $| {\mathbb{1}}\rangle \rangle \equiv {| {\mathbb{1}}\rangle \rangle }^{{A}_{I}{A}_{I}}:= {\displaystyle \sum }_{j}{\left|j\right\rangle }^{{A}_{I}}\otimes {\left|j\right\rangle }^{{A}_{I}}\in {{\mathcal{H}}}^{{A}_{I}}\otimes {{\mathcal{H}}}^{{A}_{I}}$ is a (non-normalized) maximally entangled state, and T denotes matrix transposition with respect to the chosen orthonormal basis $\{{\left|j\right\rangle }^{{A}_{I}}\}$ of ${{\mathcal{H}}}^{{A}_{I}}.$ Some useful properties of the CJ isomorphism are given in appendix A.1. A map is CP if and only if its CJ representation is positive semidefinite, while the trace-preserving condition is equivalent to ${\mathrm{tr}}_{{A}_{O}}{M}^{{A}_{I}{A}_{O}}={{\mathbb{1}}}^{{A}_{I}}$ (where ${\mathrm{tr}}_{{A}_{O}}$ denotes the partial trace over A_O, and ${{\mathbb{1}}}^{{A}_{I}}$ is the identity matrix in A_I). An instrument is therefore equivalently represented as a set

$$\begin{eqnarray}&&{\left\{{M}_{a}^{{A}_{I}{A}_{O}}\right\}}_{a=1}^{m},\quad {M}_{a}^{{A}_{I}{A}_{O}}\geqslant 0,\quad {\mathrm{tr}}_{{A}_{O}}\displaystyle \sum _{a=1}^{m}{M}_{a}^{{A}_{I}{A}_{O}}={{\mathbb{1}}}^{{A}_{I}}.\end{eqnarray} \tag{ 2 }$$

As discussed in [13], requiring that quantum mechanics holds locally implies that the probability that the N parties Aⁱ observe the outcomes ${a}_{1},\ldots ,{a}_{N},$ for a choice of operations ${x}_{1},\ldots ,{x}_{N},$ is a multilinear function $P\left({{\mathcal{M}}}_{{a}_{1}| {x}_{1}}^{{A}^{1}},\ldots ,{{\mathcal{M}}}_{{a}_{N}| {x}_{N}}^{{A}^{N}}\right)$ of the corresponding CP maps ${{\mathcal{M}}}_{{a}_{1}| {x}_{1}}^{{A}^{1}},\ldots ,{{\mathcal{M}}}_{{a}_{N}| {x}_{N}}^{{A}^{N}}.$ Using the CJ representation, it was shown that these probabilities can then be expressed as

$$\begin{eqnarray}&&P\left({M}_{{a}_{1}| {x}_{1}}^{{A}^{1}},\ldots ,{M}_{{a}_{N}| {x}_{N}}^{{A}^{N}}\right)=\mathrm{tr}\left[\left({M}_{{a}_{1}| {x}_{1}}^{{A}_{I}^{1}{A}_{O}^{1}}\otimes \ldots \otimes {M}_{{a}_{N}| {x}_{N}}^{{A}_{I}^{N}{A}_{O}^{N}}\right)W\right],\end{eqnarray} \tag{ 3 }$$

for some Hermitian operator $W\in {A}_{I}^{1}\otimes {A}_{O}^{1}\otimes \ldots \otimes {A}_{I}^{N}\otimes {A}_{O}^{N}$ called a process matrix, which describes the general quantum resource connecting the local laboratories.

The set of valid process matrices is defined by requiring that probabilities are well-defined—that is, they must be non-negative and must sum up to one—for all possible operations, including operations that involve, in each laboratory, local interactions with ancillary systems that may be entangled with the other laboratories. As we show in appendix B, these conditions are equivalent to

$$\begin{eqnarray}&&W\geqslant 0,\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\mathrm{tr}W={d}_{O},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&W={L}_{V}(W),\end{eqnarray} \tag{ 6 }$$

where ${d}_{O}={d}_{{A}_{O}^{1}}\ldots {d}_{{A}_{O}^{N}},$ and L_V is a projector onto the linear subspace ${{\mathcal{L}}}_{V}\subset {A}_{I}^{1}\otimes {A}_{O}^{1}\otimes \ldots \otimes {A}_{I}^{N}\otimes {A}_{O}^{N}$ defined in appendix B. We will denote the closed convex cone of non-normalized processes defined by (4) and (6) by ${\mathcal{W}}.$

In the case of two parties A (Alice) and B (Bob), see figure 2, these conditions on $W\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}$ reduce to

$$\begin{eqnarray}&&W\geqslant 0,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\mathrm{tr}W={d}_{O},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{}_{{B}_{I}{B}_{O}}W={}_{{A}_{O}{B}_{I}{B}_{O}}W,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{}_{{A}_{I}{A}_{O}}W={}_{{A}_{I}{A}_{O}{B}_{O}}W,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&W={}_{{B}_{O}}W+{}_{{A}_{O}}W-{}_{{A}_{O}{B}_{O}}W,\end{eqnarray} \tag{ 11 }$$

where (here and throughout the paper) the operator ${}_{X}\cdot$ denotes the CPTP map consisting in tracing out the subsystem X and replacing it by the normalized identity operator, formally defined as

$$\begin{eqnarray}&&{}_{X}W=\displaystyle \frac{{{\mathbb{1}}}^{X}}{{d}_{X}}\otimes {\mathrm{tr}}_{X}W.\end{eqnarray} \tag{ 12 }$$

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2.2.1. Non-signalling and one-way-signalling process matrices

Two important particular cases of process matrices may shed light on the above definition. The first case is when the process matrix does not allow for any signalling, and the second one is when it allows for signalling only in one fixed direction between the parties. They are discussed in more details in appendix A.2.

The first case is described by process matrices W satisfying

$$\begin{eqnarray}&&W={}_{{A}_{O}^{1}\ldots {A}_{O}^{N}}W={\rho }^{{A}_{I}^{1}\ldots {A}_{I}^{N}}\otimes {{\mathbb{1}}}^{{A}_{O}^{1}\ldots {A}_{O}^{N}},\end{eqnarray} \tag{ 13 }$$

where ${\rho }^{{A}_{I}^{1}\ldots {A}_{I}^{N}}$ is a density matrix representing an ordinary quantum state. In this case, the probability rule (3) reduces to the standard Born rule

$$\begin{eqnarray}&&P\left({M}_{{a}_{1}| {x}_{1}}^{{A}^{1}},\ldots ,{M}_{{a}_{N}| {x}_{N}}^{{A}^{N}}\right)=\mathrm{tr}\left[\left({E}_{{a}_{1}| {x}_{1}}^{{A}_{I}^{1}}\otimes \ldots \otimes {E}_{{a}_{N}| {x}_{N}}^{{A}_{I}^{N}}\right)\rho \right],\end{eqnarray} \tag{ 14 }$$

where ${E}_{{a}_{i}| {x}_{i}}^{{A}_{I}^{i}}:= {\mathrm{tr}}_{{A}_{O}^{i}}{M}_{{a}_{i}| {x}_{i}}^{{A}_{I}^{i}{A}_{O}^{i}}$ are POVM elements.

The second case, of which the first one is a particular case, is described by process matrices W satisfying

$$\begin{eqnarray}W & = & {}_{{A}_{O}^{N}}W,\\ {}_{{A}_{I}^{N}{A}_{O}^{N}}W & = & {}_{{A}_{O}^{N-1}{A}_{I}^{N}{A}_{O}^{N}}W,\\ & \vdots & \\ {}_{{A}_{I}^{2}{A}_{O}^{2}\ldots {A}_{I}^{N}{A}_{O}^{N}}W & = & {}_{{A}_{O}^{1}{A}_{I}^{2}{A}_{O}^{2}\ldots {A}_{I}^{N}{A}_{O}^{N}}W.\end{eqnarray} \tag{ 15 }$$

These conditions, first found in [14, 18], mean that party Aⁱ can only signal to party A^j if $i\lt j.$ The process is therefore compatible with the causal order ${A}^{1}\;\prec \;{A}^{2}\;\prec \ldots \prec \;{A}^{N}.$ When this is the case, we write as a mnemonic

$$\begin{eqnarray}&&W={W}^{{A}^{1}\prec {A}^{2}\prec \ldots \prec {A}^{N}}.\end{eqnarray} \tag{ 16 }$$

Process matrices of this form (and the obvious permutations) are called causally ordered. As shown in [14, 18], they correspond to standard (causally ordered) quantum circuits and can be implemented as quantum channels with memory between the parties.

2.2.2. Bipartite causally separable processes

According to equation (15), a bipartite causally ordered process matrix ${W}^{A\prec B}$ compatible with the order $A\;\prec \;B$ satisfies ${W}^{A\prec B}={}_{{B}_{O}}{W}^{A\prec B}$ and ${}_{{B}_{I}{B}_{O}}{W}^{A\prec B}={}_{{A}_{O}{B}_{I}{B}_{O}}{W}^{A\prec B}.$ Note that the latter relation corresponds to equation (9) above, and is therefore automatically satisfied if ${W}^{A\prec B}\in {{\mathcal{L}}}_{V}.$ Thus, a given matrix ${W}^{A\prec B}\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}$ is a valid causally ordered process matrix compatible with the order $A\;\prec \;B$ if and only if ${W}^{A\prec B}\geqslant 0,$ $\mathrm{tr}{W}^{A\prec B}={d}_{O},$

$$\begin{eqnarray}&&{W}^{A\prec B}\in {{\mathcal{L}}}_{V}\quad {\rm{and}}\quad {W}^{A\prec B}={}_{{B}_{O}}{W}^{A\prec B}.\end{eqnarray} \tag{ 17 }$$

The analogous condition holds for the order $B\;\prec \;A.$

Note that a non-signalling process matrix W must be compatible with both orders $A\;\prec \;B$ and $B\;\prec \;A.$ It must therefore satisfy $W={}_{{B}_{O}}W={}_{{A}_{O}}W,$ or equivalently $W={}_{{A}_{O}{B}_{O}}W;$ we indeed recover the form of equation (13).

Following [13], we say that a bipartite process matrix W is causally separable if it can be decomposed as a convex combination of causally ordered processes, i.e., if it is of the form

$$\begin{eqnarray}&&{W}^{{\rm{sep}}}\;=\;q\;{W}^{A\prec B}\;+\;(1-q)\;{W}^{B\prec A},\end{eqnarray} \tag{ 18 }$$

with $0\leqslant q\leqslant 1.$ Ignoring the normalization constraint, the set of causally separable process matrices is a convex cone, which we denote by ${{\mathcal{W}}}^{{\rm{sep}}}.$ A process matrix that cannot be decomposed as in (18) is called causally nonseparable.

2.2.3. Tripartite causally separable processes

In this paper we will define tripartite causal separability only for processes where the output space of the third party C (Charlie) is trivial, i.e., ${d}_{{C}_{O}}=1$ (see figure 3). As C cannot signal to the other parties, every process of this kind if compatible with C being last. Thus, only two causal orders are relevant in this case: $A\;\prec \;B\;\prec \;C$ and $B\;\prec \;A\;\prec \;C.$ The conditions for process matrices being compatible with these orders are, according to equation (15),

$$\begin{eqnarray}&&{W}^{A\prec B\prec C}={}_{{C}_{O}}{W}^{A\prec B\prec C},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{}_{{C}_{I}{C}_{O}}{W}^{A\prec B\prec C}={}_{{B}_{O}{C}_{I}{C}_{O}}{W}^{A\prec B\prec C},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{}_{{B}_{I}{B}_{O}{C}_{I}{C}_{O}}{W}^{A\prec B\prec C}={}_{{A}_{O}{B}_{I}{B}_{O}{C}_{I}{C}_{O}}{W}^{A\prec B\prec C},\end{eqnarray} \tag{ 21 }$$

and

$$\begin{eqnarray}&&{W}^{B\prec A\prec C}={}_{{C}_{O}}{W}^{B\prec A\prec C},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{}_{{C}_{I}{C}_{O}}{W}^{B\prec A\prec C}={}_{{A}_{O}{C}_{I}{C}_{O}}{W}^{B\prec A\prec C},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{}_{{A}_{I}{A}_{O}{C}_{I}{C}_{O}}{W}^{B\prec A\prec C}={}_{{B}_{O}{A}_{I}{A}_{O}{C}_{I}{C}_{O}}{W}^{B\prec A\prec C}.\end{eqnarray} \tag{ 24 }$$

Since these three conditions together define a linear subspace, we can write them more succinctly as

$$\begin{eqnarray}&&{W}^{A\prec B\prec C}={L}_{A\prec B\prec C}({W}^{A\prec B\prec C}),\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{W}^{B\prec A\prec C}={L}_{B\prec A\prec C}({W}^{B\prec A\prec C}),\end{eqnarray} \tag{ 26 }$$

where ${L}_{A\prec B\prec C}$ and ${L}_{B\prec A\prec C}$ are the projectors onto the aforementioned subspaces.

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Therefore, when C's output space is trivial, we will call a tripartite process matrix ${W}^{{\rm{sep}}}$ causally separable if it is of the form

$$\begin{eqnarray}&&{W}^{{\rm{sep}}}\;=\;q\;{W}^{A\prec B\prec C}\;+\;(1-q)\;{W}^{B\prec A\prec C},\end{eqnarray} \tag{ 27 }$$

with $0\leqslant q\leqslant 1.$ Ignoring the normalization constraint, this defines a convex cone ${{\mathcal{W}}}_{3C}^{{\rm{sep}}}.$ We will use this definition in section 5 to show that a recently introduced tripartite quantum resource, which yields information-processing advantages with respect to causally ordered processes [7, 8], is causally nonseparable.

The generalization of the notion of causal separability to a larger number of parties, with arbitrary dimensions of the output spaces, is not trivial. The reason is that one can consider situations in which an agent, through her local operations, could modify a classical variable that determines the causal order of agents in her future. In such a 'classical switch', operations would still be causally ordered in each run of an experiment, but it would not be possible to write the corresponding process matrix as a mixture of causally ordered ones. As this issue does not affect the cases treated here, we shall not consider it further. A more detailed analysis is presented in [19].

In this section we develop mathematical tools to identify, in the bipartite case, which process matrices are causally separable and which are not. In analogy with entanglement witnesses [20], we call a Hermitian operator S a causal witness (or witness, simply) if ⁶

$$\begin{eqnarray}&&\mathrm{tr}[S\;{W}^{{\rm{sep}}}]\geqslant 0\end{eqnarray} \tag{ 28 }$$

for every causally separable process matrix ${W}^{{\rm{sep}}}.$ This definition is motivated by the separating hyperplane theorem [21]: since the set of causally separable processes is closed and convex, for every causally nonseparable process matrix ${W}_{{\rm{ns}}}$ there exists a causal witness ${S}_{{W}_{{\rm{ns}}}}$ such that $\mathrm{tr}[{S}_{{W}_{{\rm{ns}}}}{W}_{{\rm{ns}}}]\lt 0.$

To construct a witness for a given causally nonseparable process, we will start by characterizing the set of all causal witnesses in terms of linear constraints on a convex cone. This will allow us to cast the problem of finding a witness as an SDP problem. First, note that (28) is equivalent to

$$\begin{eqnarray}&&\mathrm{tr}[S\;{W}^{A\prec B}]\geqslant 0\quad \forall {W}^{A\prec B},\end{eqnarray} \tag{ 29a }$$

$$\begin{eqnarray}&&\mathrm{tr}[S\;{W}^{B\prec A}]\geqslant 0\quad \forall {W}^{B\prec A}.\end{eqnarray} \tag{ 29b }$$

Let us focus on condition (29a). Using equation (17) and noting that for any valid process matrix W, ${}_{{B}_{O}}W$ is a valid causally ordered process matrix compatible with the order $A\;\prec \;B,$ one finds that (29a) is equivalent to

$$\begin{eqnarray}&&\mathrm{tr}\left[S({}_{{B}_{O}}W)\right]\geqslant 0\quad \forall W\in {{\mathcal{L}}}_{V},W\geqslant 0.\end{eqnarray} \tag{ 30 }$$

Thinking of the trace as the Hilbert–Schmidt inner product and noting that the map ${}_{{B}_{O}}\cdot$ is self-adjoint, we have that

$$\begin{eqnarray}&&\mathrm{tr}\left[S({}_{{B}_{O}}W)\right]=\mathrm{tr}\left[({}_{{B}_{O}}S)\;W\right],\end{eqnarray} \tag{ 31 }$$

and it is sufficient that ${}_{{BO}}S\geqslant 0$ for the right-hand side to be non-negative for all valid W. An analogous argument shows that ${}_{{A}_{O}}S\geqslant 0$ is sufficient to satisfy condition (29b). We conclude that for S to be a causal witness, it is sufficient that

$$\begin{eqnarray}&&{}_{{B}_{O}}S\geqslant 0\quad {\rm{and}}\quad {}_{{A}_{O}}S\geqslant 0.\end{eqnarray} \tag{ 32 }$$

Note also that adding an operator ${S}^{\perp }$ belonging to the orthogonal complement ${{\mathcal{L}}}_{V}^{\perp }$ of ${{\mathcal{L}}}_{V}$ to any witness S gives another valid witness, since $\mathrm{tr}[(S+{S}^{\perp })W]=\mathrm{tr}[{SW}]$ for any valid process matrix W. It turns out that this suffices to completely characterize the set of causal witnesses, as stated in the following theorem:

Theorem 1. A Hermitian operator $S\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}$ is a causal witness if and only if S can be written as

$$\begin{eqnarray}&&S={S}_{P}+{S}^{\perp },\end{eqnarray} \tag{ 33 }$$

where S_P and ${S}^{\perp }$ are Hermitian operators such that

$$\begin{eqnarray}&&{}_{{B}_{O}}{S}_{P}\geqslant 0,\quad {}_{{A}_{O}}{S}_{P}\geqslant 0,\quad {L}_{V}({S}^{\perp })=0.\end{eqnarray} \tag{ 34 }$$

The rather technical proof of this theorem is relegated to appendix C. This theorem provides a characterization of the closed convex cone of causal witnesses ${\mathcal{S}}.$

Since ${S}^{\perp }$ does not change the expectation value $\mathrm{tr}\left[{SW}\right],$ it can freely be chosen to be for instance

$$\begin{eqnarray}&&{S}^{\perp }={L}_{V}({S}_{P})-{S}_{P},\end{eqnarray} \tag{ 35 }$$

so that $S={L}_{V}({S}_{P}).$ This has the effect of restricting witnesses to the subspace of valid processes ${{\mathcal{L}}}_{V},$ which have the following characterization:

Corollary 2. A Hermitian operator $S\in {{\mathcal{L}}}_{V}$ is a causal witness if and only if there exists a Hermitian operator ${S}_{P}\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}$ such that $S={L}_{V}({S}_{P}),$ ${}_{{B}_{O}}{S}_{P}\geqslant 0,$ and ${}_{{A}_{O}}{S}_{P}\geqslant 0.$

This restricted set of causal witnesses is also a closed convex cone, which we denote by ${{\mathcal{S}}}_{V}={\mathcal{S}}\cap {{\mathcal{L}}}_{V}.$

One could define witnesses as belonging to ${{\mathcal{S}}}_{V}$ instead of ${\mathcal{S}},$ since both sets are as powerful in detecting causal nonseparability. However, some physically motivated witnesses, such as those presented in section 5.2 (for the tripartite case), do not belong to ${{\mathcal{S}}}_{V},$ which is why we use the more general definition that witnesses belong to ${\mathcal{S}}.$

The previous characterization of the convex cone of causal witnesses allows one to efficiently check the causal nonseparability of any process matrix W through algorithms for SDP [22]. They output a causal witness if W is causally nonseparable, and an explicit decomposition in terms of causally ordered process matrices otherwise.

The idea is simply to minimize $\mathrm{tr}[SW]$ over the cone of causal witnesses⁷ ${{\mathcal{S}}}_{V},$ and check whether we obtain a negative value or not. Note that in order to make $\mathrm{tr}[S\;W]$ lower bounded (to avoid getting a value $-\infty$ for causally nonseparable process matrices) a normalization constraint on the witnesses has to be imposed. This normalization is arbitrary—any constraint that makes ${{\mathcal{S}}}_{V}$ compact suffices—and different normalization choices give rise to different interpretations for the value of $\mathrm{tr}[SW].$ We shall normalize the witnesses by imposing that $\mathrm{tr}\left[S\;{\rm{\Omega }}\right]\leqslant 1$ for every (normalized) process matrix Ω, for $-\mathrm{tr}[SW]$ can then be interpreted as a measure of causal nonseparability, as we shall see later in this subsection. In order to be able to use it in the SDP problem we still need to write this normalization as a conic constraint. To do so, we extend the constraint $\mathrm{tr}\left[S\;{\rm{\Omega }}\right]\leqslant 1$ to non-normalized process matrices by linearity:

$$\begin{eqnarray}&&\mathrm{tr}\left[S\;{\rm{\Omega }}\right]\leqslant \mathrm{tr}[{\rm{\Omega }}]/{d}_{O},\end{eqnarray} \tag{ 36 }$$

which is equivalent to

$$\begin{eqnarray}&&\mathrm{tr}\left[\left({\mathbb{1}}/{d}_{O}-S\right)\;{\rm{\Omega }}\right]\geqslant 0\end{eqnarray} \tag{ 37 }$$

for all ${\rm{\Omega }}\in {\mathcal{W}}.$ Recalling that S is assumed to be in ${{\mathcal{S}}}_{V}\subset {{\mathcal{L}}}_{V},$ this means that ${\mathbb{1}}/{d}_{O}-S\in {{\mathcal{W}}}_{V}^{*}:= {{\mathcal{W}}}^{*}\cap {{\mathcal{L}}}_{V},$ where ${{\mathcal{W}}}^{*}$ is the dual cone of ${\mathcal{W}}$—that is, the cone of Hermitian operators that have non-negative trace with process matrices.

To test the causal nonseparability of a given process matrix W, we are thus led to define the following SDP problem:

$$\begin{eqnarray}&&\mathrm{min}\mathrm{tr}[{SW}]\;\;{\rm{s.t.}}\quad S\in {{\mathcal{S}}}_{V},\quad {\mathbb{1}}/{d}_{O}-S\ \in {{\mathcal{W}}}_{V}^{*},\end{eqnarray} \tag{ 38 }$$

which is written explicitly in terms of positive semidefinite constraints in appendix D.

If the solution of the SDP problem (38) leads to a negative expectation value of S, one can conclude that W is causally nonseparable, since SDP algorithms can be guaranteed⁸ to find the optimal solution [22]. In such a case, the optimal solution ${S}^{*}$ provides an explicit witness to verify the causal nonseparability of W. On the other hand, if $\mathrm{tr}[{S}^{*}W]=0,$ one concludes that W is causally separable, and an explicit decomposition of W into causally ordered processes is given by the SDP problem dual to (38) (this can be seen explicitly from the representation of the SDP problem (39) given in appendix D). As shown in appendix E, this dual is

$$\begin{eqnarray}&&\mathrm{min}\mathrm{tr}[{\rm{\Omega }}]/{d}_{O}\;\;{\rm{s.t.}}\quad W+{\rm{\Omega }}\in {{\mathcal{W}}}^{{\rm{sep}}},\quad {\rm{\Omega }}\in {\mathcal{W}},\end{eqnarray} \tag{ 39 }$$

where ${{\mathcal{W}}}^{{\rm{sep}}}$ is the cone of non-normalized causally separable process matrices, as previously defined. Furthermore, the optimal value $\mathrm{tr}[{{\rm{\Omega }}}^{*}]/{d}_{O}$ of problem (39) is related to the optimal value $\mathrm{tr}[{S}^{*}W]$ of problem (38) through

$$\begin{eqnarray}&&\mathrm{tr}[{{\rm{\Omega }}}^{*}]/{d}_{O}=-\mathrm{tr}[{S}^{*}W].\end{eqnarray} \tag{ 40 }$$

This gives an operational meaning to $-\mathrm{tr}[{S}^{*}W].$ As shown in appendix E, this quantity corresponds to the minimal $\lambda \geqslant 0$ such that

$$\begin{eqnarray}&&\displaystyle \frac{1}{1+\lambda }\left(W+\lambda \;\tilde{{\rm{\Omega }}}\right)\end{eqnarray} \tag{ 41 }$$

is causally separable, optimized over all valid, normalized processes $\tilde{{\rm{\Omega }}}.$ In other words, it quantifies the resistance of W to the worst-case noise. This is an analogue of the measure of entanglement called generalized robustness, which quantifies the resistance of the entanglement of a quantum state to worst-case noise [23]. It turns out that for our case the interpretation of $-\mathrm{tr}[{S}^{*}W]$ as a measure of causal nonseparability is also tenable, as it respects some simple axioms that we propose in appendix F. For this reason, we define the generalized robustness of a process W as

$$\begin{eqnarray}&&{R}_{{\rm{g}}}(W)=-\mathrm{tr}[{S}^{*}W].\end{eqnarray} \tag{ 42 }$$

Again in analogy with the case of entanglement measures, one can also define the random robustness [24] of W as is its resistance to 'white noise', which can be defined as the process that sends maximally mixed states to each laboratory, independently of the local operations:

$$\begin{eqnarray}&&{{\mathbb{1}}}^{\circ }:= \displaystyle \frac{{\mathbb{1}}}{{d}_{{A}_{I}}{d}_{{B}_{I}}}.\end{eqnarray} \tag{ 43 }$$

The optimal witness with respect to random robustness can be found by solving an SDP problem analogous to (38):

$$\begin{eqnarray}&&\mathrm{min}\mathrm{tr}({SW})\;\;{\rm{s.t.}}\quad S\in {{\mathcal{S}}}_{V},\quad \mathrm{tr}[S{{\mathbb{1}}}^{\circ }]\leqslant 1,\end{eqnarray} \tag{ 44 }$$

whose dual is

$$\begin{eqnarray}&&\mathrm{min}\lambda \;\;{\rm{s.t.}}\quad \lambda \geqslant 0,\quad W+\lambda {{\mathbb{1}}}^{\circ }\in {{\mathcal{W}}}^{{\rm{sep}}},\end{eqnarray} \tag{ 45 }$$

and random robustness itself is defined as

$$\begin{eqnarray}&&{R}_{{\rm{r}}}(W)=-\mathrm{tr}[{S}^{*}W],\end{eqnarray} \tag{ 46 }$$

where $\mathrm{tr}({S}^{*}W)$ is now the optimal value of the problem (44). This quantity can be used to compare witnesses in scenarios where white noise is an appropriate noise model. It cannot, however, be interpreted as a proper measure of causal nonseparability, as it does not respect all the axioms we propose in appendix F—more specifically, it is not monotonous under local operations.

A geometrical interpretation of the results of this section is shown in figure 4.

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Once a causal witness S has been obtained for a given causally nonseparable process matrix W, a natural question is how to 'measure' it, i.e., how to access the quantity $\mathrm{tr}[S\;W]$—and, in particular, check its sign—experimentally.

To do so, note that as $S\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}$ is a Hermitian operator, it can always be decomposed as a linear combination of the form⁹

$$\begin{eqnarray}&&S=\displaystyle \sum _{x,y,a,b}{\gamma }_{x,y,a,b}\ {M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y}^{{B}_{I}{B}_{O}},\end{eqnarray} \tag{ 47 }$$

where ${\gamma }_{x,y,a,b}$ are real coefficients and ${M}_{a| x}^{{A}_{I}{A}_{O}}$ and ${M}_{b| y}^{{B}_{I}{B}_{O}}$ are positive semidefinite matrices that can be interpreted as the CJ representation of CP trace non-increasing maps (see section 2).

Expanding $\mathrm{tr}[S\;W],$

$$\begin{eqnarray}&&\mathrm{tr}[S\;W]=\displaystyle \sum _{x,y,a,b}{\gamma }_{x,y,a,b}\ \mathrm{tr}\left[\left({M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y}^{{B}_{I}{B}_{O}}\right)\;W\right],\end{eqnarray} \tag{ 48 }$$

where according to the generalized Born rule (3), the terms $\mathrm{tr}\left[\left({M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y}^{{B}_{I}{B}_{O}}\right)\;W\right]$ represent the probabilities $P\left({M}_{a| x}^{{A}_{I}{A}_{O}},{M}_{b| y}^{{B}_{I}{B}_{O}}\right)$ that the maps ${M}_{a| x}^{{A}_{I}{A}_{O}}$ and ${M}_{b| y}^{{B}_{I}{B}_{O}}$ are realized. We assume that these CP maps can be implemented even if the causal order of the parties is not well-defined. The quantity $\mathrm{tr}[{SW}]$ can thus in principle be implemented experimentally by estimating the probabilities $P\left({M}_{a| x}^{{A}_{I}{A}_{O}},{M}_{b| y}^{{B}_{I}{B}_{O}}\right)$ and combining them as in equation (48).

The decomposition (47) is not unique. Furthermore, as noted before we can add to any witness S a term ${S}^{\perp }$ such that ${L}_{V}({S}^{\perp })=0$ without changing its validity or its trace with any valid process. Hence, it actually suffices to find a decomposition for $S+{S}^{\perp }$ for some arbitrary ${S}^{\perp },$ implement the corresponding maps, and combine their statistics as above.

Let us now illustrate the above considerations on an explicit example. Reference [13] introduced the following process matrix, for a case where all incoming and outgoing systems of A and B are two-dimensional (qubit) systems (i.e., ${d}_{{A}_{I}}={d}_{{A}_{O}}={d}_{{B}_{I}}={d}_{{B}_{O}}=2$ ):

$$\begin{eqnarray}&&{W}_{{\rm{OCB}}}=\displaystyle \frac{1}{4}\left[{\mathbb{1}}+\displaystyle \frac{{{\mathbb{1}}}^{{A}_{I}}{Z}^{{A}_{O}}{Z}^{{B}_{I}}{{\mathbb{1}}}^{{B}_{O}}+{Z}^{{A}_{I}}{{\mathbb{1}}}^{{A}_{O}}{X}^{{B}_{I}}{Z}^{{B}_{O}}}{\sqrt{2}}\right],\end{eqnarray} \tag{ 49 }$$

where Z and X are the Pauli matrices, and tensor products are implicit. One can easily check that ${W}_{{\rm{OCB}}}\geqslant 0,$ that $\mathrm{tr}[{W}_{{\rm{OCB}}}]=4={d}_{O},$ and that ${W}_{{\rm{OCB}}}$ satisfies equations (9)–(11), which ensures that it is indeed a valid process matrix. It was shown that ${W}_{{\rm{OCB}}}$ allows for a violation of a causal inequality (see section 6), which implies that it is causally nonseparable.

The concept of causal witnesses introduced here allows us to prove the causal nonseparability of ${W}_{{\rm{OCB}}}$ more directly. Solving the SDP problem (44) with YALMIP [25] and the solver MOSEK [26], we obtained, up to numerical precision, the optimal witness with respect to random robustness

$$\begin{eqnarray}&&{S}_{{\rm{OCB}}}=\displaystyle \frac{1}{4}\left[{\mathbb{1}}-\left({{\mathbb{1}}}^{{A}_{I}}{Z}^{{A}_{O}}{Z}^{{B}_{I}}{{\mathbb{1}}}^{{B}_{O}}+{Z}^{{A}_{I}}{{\mathbb{1}}}^{{A}_{O}}{X}^{{B}_{I}}{Z}^{{B}_{O}}\right)\right].\end{eqnarray} \tag{ 50 }$$

Applying it to ${W}_{{\rm{OCB}}},$ we find that $-\mathrm{tr}[{S}_{{\rm{OCB}}}\;{W}_{{\rm{OCB}}}]={R}_{{\rm{r}}}({W}_{{\rm{OCB}}})=\sqrt{2}-1\gt 0$ (where ${R}_{{\rm{r}}}({W}_{{\rm{OCB}}})$ is the random robustness as defined in equation (46)). This proves that ${W}_{{\rm{OCB}}}$ is causally nonseparable.

This also implies that the process matrices of the form

$$\begin{eqnarray}&&{W}_{{\rm{OCB}}}(\lambda )=\displaystyle \frac{1}{1+\lambda }\;\left({W}_{{\rm{OCB}}}+\lambda \;{{\mathbb{1}}}^{\circ }\right),\end{eqnarray} \tag{ 51 }$$

are causally nonseparable for

$$\begin{eqnarray}&&\lambda \lt {R}_{{\rm{r}}}({W}_{{\rm{OCB}}})=\sqrt{2}-1\end{eqnarray} \tag{ 52 }$$

(and their causal nonseparability is then witnessed by ${S}_{{\rm{OCB}}}$). For $\lambda \geqslant \sqrt{2}-1,$ ${W}_{{\rm{OCB}}}(\lambda )$ is causally separable; the solution of the SDP problem (45) provides an explicit decomposition for ${W}_{{\rm{OCB}}}\left({R}_{{\rm{r}}}({W}_{{\rm{OCB}}})\right)$ (as can be seen when writing (45) in a form similar to equation (D.3)), from which we can derive an explicit decomposition for all ${W}_{{\rm{OCB}}}(\lambda )$ for $\lambda \geqslant \sqrt{2}-1,$ as

$$\begin{eqnarray}&&{W}_{{\rm{OCB}}}(\lambda )=\displaystyle \frac{1}{2}{W}_{{\rm{OCB}}}^{A\prec B}(\lambda )+\displaystyle \frac{1}{2}{W}_{{\rm{OCB}}}^{B\prec A}(\lambda ),\end{eqnarray} \tag{ 53 }$$

where

$$\begin{eqnarray}&&{W}_{{\rm{OCB}}}^{A\prec B}(\lambda ):= \displaystyle \frac{1}{4}\left[{\mathbb{1}}+\displaystyle \frac{\sqrt{2}}{1+\lambda }\ {{\mathbb{1}}}^{{A}_{I}}{Z}^{{A}_{O}}{Z}^{{B}_{I}}{{\mathbb{1}}}^{{B}_{O}}\right],\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&{W}_{{\rm{OCB}}}^{B\prec A}(\lambda ):= \displaystyle \frac{1}{4}\left[{\mathbb{1}}+\displaystyle \frac{\sqrt{2}}{1+\lambda }\ {Z}^{{A}_{I}}{{\mathbb{1}}}^{{A}_{O}}{X}^{{B}_{I}}{Z}^{{B}_{O}}\right]\end{eqnarray} \tag{ 55 }$$

are causally ordered process matrices. (Note that for $\lambda \lt \sqrt{2}-1,$ ${W}_{{\rm{OCB}}}^{A\prec B}(\lambda )$ and ${W}_{{\rm{OCB}}}^{B\prec A}(\lambda )$ as defined above would not be positive semidefinite, which explains why equation (53) then fails to provide a valid causally separable decomposition of ${W}_{{\rm{OCB}}}(\lambda ).$)

To measure the witness ${S}_{{\rm{OCB}}}$ and obtain the quantity $\mathrm{tr}[{S}_{{\rm{OCB}}}\cdot W]$ experimentally, one can for instance decompose it in the following way: define, for $x,y,y^{\prime} ,a,b=0,1,$ the CJ matrices

$$\begin{eqnarray}&&{M}_{a| x}^{{A}_{I}{A}_{O}}:= {\left(\displaystyle \frac{{\mathbb{1}}+{(-1)}^{a}Z}{2}\right)}^{{A}_{I}}\otimes {\left(\displaystyle \frac{{\mathbb{1}}+{(-1)}^{x}Z}{2}\right)}^{{A}_{O}},\end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}&&{M}_{b| y,y^{\prime} =0}^{{B}_{I}{B}_{O}}:= {\left(\displaystyle \frac{{\mathbb{1}}+{(-1)}^{b}X}{2}\right)}^{{B}_{I}}\otimes {\left(\displaystyle \frac{{\mathbb{1}}+{(-1)}^{y+b}Z}{2}\right)}^{{B}_{O}},\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&{M}_{b| y,y^{\prime} =1}^{{B}_{I}{B}_{O}}:= {\left(\displaystyle \frac{{\mathbb{1}}+{(-1)}^{b}Z}{2}\right)}^{{B}_{I}}\otimes \displaystyle \frac{{{\mathbb{1}}}^{{B}_{O}}}{2},\end{eqnarray} \tag{ 58 }$$

which represent measure-and-prepare maps (see appendix A.1). One can then check that

$$\begin{eqnarray}&&{S}_{{\rm{OCB}}}=3\cdot {{\mathbb{1}}}^{\circ }-4\;{G}_{{\rm{OCB}}}\end{eqnarray} \tag{ 59 }$$

with

$$\begin{eqnarray}&&{G}_{{\rm{OCB}}}=\displaystyle \frac{1}{8}\displaystyle \sum _{x,y,a,b}\left[{\delta }_{a,y}\ {M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y,y^{\prime} =0}^{{B}_{I}{B}_{O}}+{\delta }_{b,x}\ {M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y,y^{\prime} =1}^{{B}_{I}{B}_{O}}\right],\end{eqnarray} \tag{ 60 }$$

where ${\delta }_{j,k}$ is the Kronecker delta. Thus, one can compute $\mathrm{tr}[{S}_{{\rm{OCB}}}\cdot W]$ by performing the maps above on W and combining the probabilities $P\left({M}_{a| x}^{{A}_{I}{A}_{O}},{M}_{b| y,y^{\prime} }^{{B}_{I}{B}_{O}}\right)=\mathrm{tr}[{M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y,y^{\prime} }^{{B}_{I}{B}_{O}}\cdot W]$ as follows:

$$\begin{eqnarray}\mathrm{tr}[{S}_{{\rm{OCB}}}\cdot W] & = & 3-4\;\mathrm{tr}[{G}_{{\rm{OCB}}}\cdot W]\\ & = & 3-4\cdot \displaystyle \frac{1}{8}\displaystyle \sum _{x,y,a,b}\left[{\delta }_{a,y}\ P\left({M}_{a| x}^{{A}_{I}{A}_{O}},{M}_{b| y,y^{\prime} =0}^{{B}_{I}{B}_{O}}\right)\right.+\left.{\delta }_{b,x}\ P\left({M}_{a| x}^{{A}_{I}{A}_{O}},{M}_{b| y,y^{\prime} =1}^{{B}_{I}{B}_{O}}\right)\right].\end{eqnarray} \tag{ 61 }$$

As one may recognize, the choice of CP maps in (56)–(58) is the same¹⁰ as that considered in [13], so that the experimental procedure proposed here to measure the witness ${S}_{{\rm{OCB}}}$ would be the same as that suggested in [13] to violate a causal inequality. The labels $x,y,y^{\prime} ,a,b$ can be considered as inputs and outputs for the above maps (which indeed satisfy ${}_{{A}_{O}}\left({\sum }_{a}{M}_{a| x}^{{A}_{I}{A}_{O}}\right)={{\mathbb{1}}}^{{A}_{I}{A}_{O}}/{d}_{{A}_{O}}$ and ${}_{{B}_{O}}\left({\sum }_{b}{M}_{b| y,y^{\prime} }^{{B}_{I}{B}_{O}}\right)={{\mathbb{1}}}^{{B}_{I}{B}_{O}}/{d}_{{B}_{O}}$ for all $x,y,y^{\prime}$). As it turns out, in the causal inequality of [13] the probabilities $P(a,b| x,y,y^{\prime} )=P\left({M}_{a| x}^{{A}_{I}{A}_{O}},{M}_{b| y,y^{\prime} }^{{B}_{I}{B}_{O}}\right)$ are actually combined in precisely the same way as above—namely, $\mathrm{tr}[{G}_{{\rm{OCB}}}\cdot W]$ above can be identified with the probability ${p}_{{\rm{succ}}}$ of winning the corresponding 'causal game'

$$\begin{eqnarray}&&{p}_{{\rm{succ}}}=\displaystyle \frac{1}{2}\left[P(a=y| y^{\prime} =0)+P(b=x| y^{\prime} =1)\right],\end{eqnarray} \tag{ 62 }$$

when the inputs $x,y,y^{\prime} =0,1$ are given with equal probabilities.

Remarkably, in this particular case the bounds of the causal witness ${S}_{{\rm{OCB}}}$ and of the causal inequality (62) coincide, i.e., $\mathrm{tr}[{S}_{{\rm{OCB}}}\cdot W]\geqslant 0$ if and only if ${p}_{{\rm{succ}}}=\mathrm{tr}[{G}_{{\rm{OCB}}}\cdot W]\leqslant 3/4,$ where 3/4 is the upper bound on ${p}_{{\rm{succ}}}$ for any causal correlation (as defined in section 6 below). Furthermore, the noise threshold below which the noisy process matrix ${W}_{{\rm{OCB}}}(\lambda )$ (53) can violate the causal inequality is the same as the threshold ${R}_{{\rm{r}}}({W}_{{\rm{OCB}}})$ below which ${W}_{{\rm{OCB}}}(\lambda )$ is causally nonseparable, as already noted in [28]. This is however not a general property of causal witnesses and causal inequalities: similarly to the case of entanglement vs. quantum nonlocality and of entanglement witnesses vs. Bell inequalities [27], there exist causally nonseparable process matrices that cannot yield any violation of any causal inequality—while there always exists a causal witness that detects their causal nonseparability. We will come back to this issue in section 6 below, with an explicit example in the tripartite case.

It has recently been suggested that quantum computation can be extended beyond the framework of quantum circuits, which enforces a fixed order between the execution of quantum gates. The main idea is that the order in which gates are performed can be coherently controlled by a quantum system. The new resource that allows for such a control is the quantum switch, first proposed in [6]. It works as follows: consider a two-qubit system, composed of a control and of a target qubit. Two parties A and B act on the target qubit with the unitaries U_A, U_B respectively. If the control qubit is prepared in the state $\left|0\right\rangle ,$U_A is applied to the target before U_B, while if the control is in state $\left|1\right\rangle$ the two unitaries are applied in the reversed order. The global unitary, acting on both the target and control qubits, is thus

$$\begin{eqnarray}&&V({U}_{A},{U}_{B})=| 0\rangle \langle 0| \otimes {U}_{B}{U}_{A}+| 1\rangle \langle 1| \otimes {U}_{A}{U}_{B},\end{eqnarray} \tag{ 63 }$$

where the first factor in each tensor product acts on the control system and the second factor acts on the target. For an initial state $\displaystyle \frac{\left|0\right\rangle +\left|1\right\rangle }{\sqrt{2}}\otimes \left|\psi \right\rangle$ of the control-target system, one gets, after applying V, the state $\displaystyle \frac{1}{\sqrt{2}}\left(\left|0\right\rangle \otimes {U}_{B}{U}_{A}\left|\psi \right\rangle +\left|1\right\rangle \otimes {U}_{A}{U}_{B}\left|\psi \right\rangle \right),$ which can be interpreted as having applied the two unitaries on the target in a 'superposition of orders'¹¹ .

Note that if the control system is discarded, one is left with the mixed state

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\left({U}_{B}{U}_{A}| \psi \rangle \langle \psi | {U}_{A}^{\dagger }{U}_{B}^{\dagger }+{U}_{A}{U}_{B}| \psi \rangle \langle \psi | {U}_{B}^{\dagger }{U}_{A}^{\dagger }\right).\end{eqnarray} \tag{ 64 }$$

This can be produced by randomly exchanging the order in which U_A and U_B are applied and thus can be seen as an equal mixture of causally ordered processes. To make the situation more interesting, we shall be led to introduce a third party, C, who can perform measurements on the control qubit (and possibly also on the target qubit) in order to define a causally nonseparable process (using the definition (27)) using quantum control of causal order.

For our purposes, we can formally represent the quantum switch (with fixed input state) as a tripartite process matrix: the two parties A and B perform an arbitrary CP map each on the target qubit, while C performs an arbitrary two-qubit POVM measurement on the resulting control-target state (with no outgoing system). The dimensions of input and output systems of the local laboratories are therefore

$$\begin{eqnarray}&&{d}_{{A}_{I}}={d}_{{A}_{O}}={d}_{{B}_{I}}={d}_{{B}_{O}}=2,\quad {d}_{{C}_{I}}=4,\quad {d}_{{C}_{O}}=1.\end{eqnarray} \tag{ 65 }$$

For clarity, we shall divide C's input space as ${C}_{I}={C}_{I}^{c}\otimes {C}_{I}^{t},$ where ${C}_{I}^{c}$ and ${C}_{I}^{t}$ refer to the control and target qubits, respectively (with therefore ${d}_{{C}_{I}^{c}}={d}_{{C}_{I}^{t}}=2$).

In order to describe the process matrix of the quantum switch, we are first going to make use of the 'pure' version of the formalism, described in appendix A.2. An identity channel from a party's output space A_O to another party's input space B_I is described, as a process matrix, by the projector onto the 'process vector' ${| {\mathbb{1}}\rangle \rangle }^{{A}_{O}{B}_{I}}={\sum }_{j=\mathrm{0,1}}{\left|j\right\rangle }^{{A}_{O}}{\left|j\right\rangle }^{{B}_{I}}.$ The situation where A receives a state $\left|\psi \right\rangle ,$ performs an arbitrary operation on it, and sends the output directly to B through an identity channel, who in turn sends the output of his operation to ${C}_{I}^{t},$ is represented by the process vector ${\left|\psi \right\rangle }^{{A}_{I}}{| {\mathbb{1}}\rangle \rangle }^{{A}_{O}{B}_{I}}{| {\mathbb{1}}\rangle \rangle }^{{B}_{O}{C}_{I}^{t}},$ see appendix A.2. Then the quantum switch, with the control qubit initially in the state $\frac{\left|0\right\rangle +\left|1\right\rangle }{\sqrt{2}}$ and the target qubit in the state $\left|\psi \right\rangle ,$ is represented by the process matrix $| w\rangle \langle w| ,$ where

$$\begin{eqnarray}&&\left|w\right\rangle =\displaystyle \frac{1}{\sqrt{2}}\left({\left|\psi \right\rangle }^{{A}_{I}}{| {\mathbb{1}}\rangle \rangle }^{{A}_{O}{B}_{I}}{| {\mathbb{1}}\rangle \rangle }^{{B}_{O}{C}_{I}^{t}}{\left|0\right\rangle }^{{C}_{I}^{c}}\right.+\left.{\left|\psi \right\rangle }^{{B}_{I}}{| {\mathbb{1}}\rangle \rangle }^{{B}_{O}{A}_{I}}{| {\mathbb{1}}\rangle \rangle }^{{A}_{O}{C}_{I}^{t}}{\left|1\right\rangle }^{{C}_{I}^{c}}\right).\end{eqnarray} \tag{ 66 }$$

This can be checked by noting that

$$\begin{eqnarray}&&{\langle \langle {U}_{A}^{*}| }^{{A}_{I}{A}_{O}}{\langle \langle {U}_{B}^{*}| }^{{B}_{I}{B}_{O}}\cdot \left|w\right\rangle \\ &&\quad =\displaystyle \frac{1}{\sqrt{2}}\left({\left|0\right\rangle }^{{C}_{I}^{c}}\otimes {\left({U}_{B}{U}_{A}\left|\psi \right\rangle \right)}^{{C}_{I}^{t}}+{\left|1\right\rangle }^{{C}_{I}^{c}}\otimes {\left({U}_{A}{U}_{B}\left|\psi \right\rangle \right)}^{{C}_{I}^{t}}\right),\end{eqnarray} \tag{ 67 }$$

where ${| {U}_{A}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}:= {\mathbb{1}}\otimes {U}_{A}^{*}| {\mathbb{1}}\rangle \rangle$ is the 'pure' CJ representation of U_A, and similarly for ${| {U}_{B}^{*}\rangle \rangle }^{{B}_{I}{B}_{O}}$ (and with $\langle \langle {U}^{*}| ={| {U}^{*}\rangle \rangle }^{\dagger \;}$); see appendix A.1. Note that the resulting state is the same as that obtained by applying (63) to an initial state $\displaystyle \frac{\left|0\right\rangle +\left|1\right\rangle }{\sqrt{2}}\otimes \left|\psi \right\rangle .$

Note that the process (66) itself is clearly causally nonseparable¹² , since (i) it is a superposition of a pure process only compatible with the order $A\;\prec \;B\;\prec \;C$ and a pure process only compatible with the order $B\;\prec \;A\;\prec \;C$ and (ii) it is a projector onto a pure vector, thus it cannot be written as a nontrivial mixture of causally ordered processes.

From equation (66), one finds (using the facts that ${\mathrm{tr}}_{{C}_{I}^{t}}(| {\mathbb{1}}\rangle \rangle {\langle \langle {\mathbb{1}}| }^{{B}_{O}{C}_{I}^{t}})={{\mathbb{1}}}^{{B}_{O}}$ and ${\mathrm{tr}}_{{C}_{I}^{t}}(| {\mathbb{1}}\rangle \rangle {\langle \langle {\mathbb{1}}| }^{{A}_{O}{C}_{I}^{t}})={{\mathbb{1}}}^{{A}_{O}}$) that by tracing out C, one gets

$$\begin{eqnarray}&&{\mathrm{tr}}_{{C}_{I}}\;| w\rangle \langle w| ={\mathrm{tr}}_{{C}_{I}^{c}{C}_{I}^{t}}\;| w\rangle \langle w| =\displaystyle \frac{1}{2}{W}^{A\prec B}+\displaystyle \frac{1}{2}{W}^{B\prec A},\end{eqnarray} \tag{ 68 }$$

where

$$\begin{eqnarray}&&{W}^{A\prec B}=| \psi \rangle {\langle \psi | }^{{A}_{I}}\otimes | {\mathbb{1}}\rangle \rangle {\langle \langle {\mathbb{1}}| }^{{A}_{O}{B}_{I}}\otimes {{\mathbb{1}}}^{{B}_{O}},\end{eqnarray} \tag{ 69 }$$

$$\begin{eqnarray}&&{W}^{B\prec A}=| \psi \rangle {\langle \psi | }^{{B}_{I}}\otimes | {\mathbb{1}}\rangle \rangle {\langle \langle {\mathbb{1}}| }^{{B}_{O}{A}_{I}}\otimes {{\mathbb{1}}}^{{A}_{O}},\end{eqnarray} \tag{ 70 }$$

are (bipartite) causally ordered process matrices; ${\mathrm{tr}}_{{C}_{I}}\;| w\rangle \langle w|$ indeed describes the situation of equation (64).

For some information-processing tasks, the quantum switch is known to provide an advantage over causally ordered processes [7, 8], even when C ignores the target system and only measures the control system. We will thus restrict our attention to witnesses of the form ${S}^{{C}_{I}^{c}}\otimes {{\mathbb{1}}}^{{C}_{I}^{t}},$ which can simplify the analysis and the experimental implementation. The reduced process we will be dealing with is the partial trace of the quantum switch (66) over the target system:

$$\begin{eqnarray}&&{W}_{{\rm{switch}}}={\mathrm{tr}}_{{C}_{I}^{t}}| w\rangle \langle w| .\end{eqnarray} \tag{ 71 }$$

Note that the proof of causal nonseparability based on the purity of the switch does not extend to the reduced switch (71), since it is not an extremal process. We will therefore use the framework of causal witnesses to show that the reduced switch is also causally nonseparable.

To find the optimal generalized robustness witness for the quantum switch we need to solve SDP problem (76) providing ${W}_{{\rm{switch}}}$ from equation (71) as an argument. Solving it using YALMIP and the solver MOSEK we obtain a witness ${S}_{{\rm{optimal}}}$ numerically; the generalized robustness of the quantum switch is found to be

$$\begin{eqnarray}&&{R}_{{\rm{g}}}({W}_{{\rm{switch}}})=-\mathrm{tr}[{S}_{{\rm{optimal}}}{W}_{{\rm{switch}}}]\approx 0.5454.\end{eqnarray} \tag{ 78 }$$

Later in this section we will compare this number to that obtained from non-optimal witnesses. For this purpose, we shall use the amount of worst-case noise tolerated by a witness, i.e., the amount of worst-case noise that can be added to the quantum switch before the witness can no longer detect its causal nonseparability. It should be clear that, when the said witness is optimal, this number reduces to the generalized robustness of the quantum switch.

In [7] Chiribella proposed an information-processing task for which the quantum switch had an advantage over causally ordered processes. We want to understand what this advantage means, and how it relates to causal nonseparability. For that we shall present a slightly modified version of his task and show how it can be understood as a causal witness.

Our version of the task is as follows: Alice (party A) receives a qubit in her lab, applies a unitary U_A to it, and sends it away. Bob (party B) receives a qubit in his lab, applies a unitary U_B to it, and sends it away. We assume that in each run of the experiment, U_A and U_B either commute or anticommute. Charlie (party C) receives a qubit in his lab, and makes a measurement on it to decide whether U_A and U_B commute or anticommute.

To construct a causal witness in relation to this task, we start with the CJ representation of the actions of the parties: Alice applying a unitary U_A, Bob applying a unitary U_B, and Charlie obtaining the result ± when measuring in the $\left|\pm \right\rangle =\displaystyle \frac{\left|0\right\rangle \pm \left|1\right\rangle }{\sqrt{2}}$ basis. Using the CJ representations $| {U}_{A}^{*}\rangle \rangle$ and $| {U}_{B}^{*}\rangle \rangle$ of U_A and U_B (see appendix A.1), the corresponding operator is

$$\begin{eqnarray}&&{G}_{\pm }^{{U}_{A},{U}_{B}}=| {U}_{A}^{*}\rangle \rangle \langle \langle {U}_{A}^{*}| \otimes | {U}_{B}^{*}\rangle \rangle \langle \langle {U}_{B}^{*}| \otimes | \pm \rangle \langle \pm | .\end{eqnarray} \tag{ 79 }$$

The witness corresponding to the task is obtained by averaging over the cases where Charlie obtains + when Alice and Bob apply commuting unitaries, and the cases where Charlie obtains − when Alice and Bob apply anticommuting unitaries:

$$\begin{eqnarray}&&{G}_{{\rm{Chiribella}}}=\displaystyle \frac{1}{2}\int {\rm{d}}{\mu }_{[\;,\;]}\;{G}_{+}^{{U}_{A},{U}_{B}}+\displaystyle \frac{1}{2}\int {\rm{d}}{\mu }_{\{\;,\;\}}\;{G}_{-}^{{U}_{A},{U}_{B}},\end{eqnarray} \tag{ 80 }$$

where ${\rm{d}}{\mu }_{[\;,\;]}$ is a measure over commuting unitaries, and ${\rm{d}}{\mu }_{\{\;,\;\}}$ is a measure over anticommuting unitaries (we assume here that the cases where U_A and U_B commute and anticommute each appear with probability $\frac{1}{2}$). The probability of success in this task when the parties are using a strategy described by a process matrix W is then

$$\begin{eqnarray}&&{p}_{{\rm{succ}}}=\mathrm{tr}[{G}_{{\rm{Chiribella}}}W].\end{eqnarray} \tag{ 81 }$$

It is easy to check that for any choice of measures ${\rm{d}}{\mu }_{[\;,\;]},{\rm{d}}{\mu }_{\{\;,\;\}}$ the probability of success is 1 when $W={W}_{{\rm{switch}}}.$ The maximal probability of success for causally separable processes, however, depends crucially on the measures ${\rm{d}}{\mu }_{[,]}$ and ${\rm{d}}{\mu }_{\{,\}}.$ If we were to choose, for example, measures that only produce pairs of Pauli matrices, then there is a causally separable circuit¹³ that can decide the commutativity or anticommutativity with probability 1.

To avoid this problem we will first choose measures that can produce any pair of commuting or anticommuting unitaries (modulo global phases). Specifically, we choose the commuting measure ${\rm{d}}{\mu }_{[\;,\;]}$ to pick up commuting unitaries of the form

$$\begin{eqnarray}{U}_{A}=U\left(\begin{array}{cc}1 & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{1}}\end{array}\right){U}^{\dagger }\quad {\rm{and}}\quad {U}_{B}=U\left(\begin{array}{cc}1 & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{2}}\end{array}\right){U}^{\dagger },\end{eqnarray} \tag{ 82 }$$

where U is uniformly distributed according to the Haar measure, and ${\theta }_{i}$ are uniformly distributed in the interval $[0,2\pi ].$ For the anticommuting measure ${\rm{d}}{\mu }_{\{\;,\;\}},$ we will use ${U}_{A}={{VXV}}^{\dagger }$ and ${U}_{B}={{VZV}}^{\dagger },$ where V is also a Haar-random unitary (and X and Z are the Pauli matrices)¹⁴ .

With these measures ${G}_{{\rm{Chiribella}}}$ turns out to be a valid causal witness, as the maximal probability of success for causally separable processes ${p}_{{\rm{succ}}}^{{\rm{sep}}}$ is bounded below one. To calculate it we need to solve the following SDP problem:

$$\begin{eqnarray}&&\mathrm{max}\mathrm{tr}[{G}_{{\rm{Chiribella}}}W]\;{\rm{s.t.}}\quad \mathrm{tr}W={d}_{O},\quad W\in {{\mathcal{W}}}_{3C}^{{\rm{sep}}}.\end{eqnarray} \tag{ 83 }$$

Solving it with YALMIP and MOSEK, we obtain

$$\begin{eqnarray}&&{p}_{{\rm{succ}}}^{{\rm{sep}}}\approx 0.9288.\end{eqnarray} \tag{ 84 }$$

The amount of worst-case noise that ${G}_{{\rm{Chiribella}}}$ can tolerate is 0.0766, which is much worse than the 0.5454 tolerated by ${S}_{{\rm{optimal}}}.$

An issue with ${G}_{{\rm{Chiribella}}}$ is that it would take an infinite number of measurements to estimate each term of the sum in (80). Furthermore ${\rm{d}}{\mu }_{[\;,\;]}$ and ${\rm{d}}{\mu }_{\{\;,\;\}}$ were chosen arbitrarily, while it would be preferable to have a justification for the choice of a particular measure. Both problems are solved by restricting the unitaries U_A and U_B to come from a finite set. In this way we only need perform a finite number of measurements to estimate the witness, and it is possible to optimize the measures over commuting and anticommuting unitaries through SDP problems.

The best witness we found is obtained by choosing the following ten unitaries:

$$\begin{eqnarray}&&{\mathcal{G}}=\{{\mathbb{1}},X,Y,Z,\displaystyle \frac{X+Y}{\sqrt{2}},\displaystyle \frac{X-Y}{\sqrt{2}},\displaystyle \frac{X+Z}{\sqrt{2}},\displaystyle \frac{X-Z}{\sqrt{2}},\displaystyle \frac{Y+Z}{\sqrt{2}},\displaystyle \frac{Y-Z}{\sqrt{2}}\}\end{eqnarray} \tag{ 85 }$$

(Y being the third Pauli matrix), and defining the witness to be

$$\begin{eqnarray}&&{G}_{{\rm{finite}}}=\displaystyle \sum _{i,j=1}^{10}{q}_{{ij}}^{[\;,\;]}\;{G}_{+}^{{U}_{i},{U}_{j}}+{q}_{{ij}}^{\{\;,\;\}}\;{G}_{-}^{{U}_{i},{U}_{j}},\end{eqnarray} \tag{ 86 }$$

where ${U}_{k}\in {\mathcal{G}},$ and ${q}_{{ij}}^{[\;,\;]},{q}_{{ij}}^{\{\;,\;\}}$ are the input probability distributions over commuting and anticommuting unitaries, normalized such that ${\displaystyle \sum }_{i,j}({q}_{{ij}}^{[\;,\;]}+{q}_{{ij}}^{\{\;,\;\}})=1.$

To obtain the weights ${q}_{{ij}}^{[\;,\;]},{q}_{{ij}}^{\{\;,\;\}}$ and ${p}_{{\rm{succ}}}^{{\rm{sep}}}$ we solved an SDP problem presented in appendix H. We obtained

$$\begin{eqnarray}&&{p}_{{\rm{succ}}}^{{\rm{sep}}}\approx 0.8690,\end{eqnarray} \tag{ 87 }$$

and tolerance to worst-case noise 0.1507, which is higher than ${G}_{{\rm{Chiribella}}}$'s 0.0766, but still lower than ${S}_{{\rm{optimal}}}$'s 0.5454.

We want to emphasize that the witnesses obtained in this subsection are equivalent to the ones defined through (72) in the beginning of the present section—the only difference being the arbitary choice of the causal bound being $\geqslant 0$ vs $\leqslant \;{p}_{{\rm{succ}}}^{{\rm{sep}}}.$ More precisely, let G be a witness such that

$$\begin{eqnarray}&&\mathrm{tr}[G\;{W}^{{\rm{sep}}}]\leqslant {p}_{{\rm{succ}}}^{{\rm{sep}}}\end{eqnarray} \tag{ 88 }$$

for every (normalized) causally separable ${W}^{{\rm{sep}}}$ and

$$\begin{eqnarray}&&{T}_{0}\leqslant \mathrm{tr}(G\;W)\leqslant {T}_{1}\end{eqnarray} \tag{ 89 }$$

for every (normalized) process matrix W. Then

$$\begin{eqnarray}&&S=\displaystyle \frac{1}{{p}_{{\rm{succ}}}^{{\rm{sep}}}-{T}_{0}}\left({p}_{{\rm{succ}}}^{{\rm{sep}}}\displaystyle \frac{{\mathbb{1}}}{{d}_{O}}-G\right)\end{eqnarray} \tag{ 90 }$$

is a valid generalized robustness witness. Furthermore, if S is the optimal witness for some process matrix W that saturates the upper bound $\mathrm{tr}(G\;W)={T}_{1},$ it follows that

$$\begin{eqnarray}&&{R}_{{\rm{g}}}(W)=-\mathrm{tr}[{SW}]=\displaystyle \frac{{T}_{1}-{p}_{{\rm{succ}}}^{{\rm{sep}}}}{{p}_{{\rm{succ}}}^{{\rm{sep}}}-{T}_{0}}.\end{eqnarray} \tag{ 91 }$$

When G is either ${G}_{{\rm{finite}}}$ or ${G}_{{\rm{Chiribella}}},$ we have that ${T}_{0}=0$ and ${T}_{1}=1.$ And even though they are not optimal witnesses for ${W}_{{\rm{switch}}},$ the relationship between ${p}_{{\rm{succ}}}^{{\rm{sep}}}$ and resistance to worst-case noise is valid for them, i.e., for both ${G}_{{\rm{finite}}}$ and ${G}_{{\rm{Chiribella}}}$ the resistance to worst-case noise is equal to $1/{p}_{{\rm{succ}}}^{{\rm{sep}}}-1,$ as given by (91).

We still consider a multipartite scenario in which a set of N parties ${\{{A}^{i}\}}_{i=1}^{N}$ are located in different, separated laboratories. Each party can perform operations and obtain measurement outcomes. Contrary to the previous case however, we do not consider here any particular physical description of what happens in each lab; the 'settings' for the operations in the different laboratories and the measurement outcomes are labelled by some classical variables x_i and a_i (with $1\leqslant i\leqslant N$), respectively; for simplicity we assume that the x_i's and a_i's take a finite number of values. Defining the vector of settings $\vec{x}=({x}_{1},...{x}_{N})$ and the vector of outcomes $\vec{a}=({a}_{1},...,{a}_{N}),$ the device-independent description of the correlations established in such an experiment is encoded in the conditional probability $P(\vec{a}| \vec{x}).$

Causal inequalities [13] are constraints on $P(\vec{a}| \vec{x})$ derived from the assumption that there exists an underlying causal structure defining the order between parties. To be more precise, let us represent the causal order in which the parties act by a permutation σ, defined such that party i acts before party j if and only if $\sigma (i)\lt \sigma (j).$ This leads to a total ordering of the parties, namely ${A}^{\sigma (1)}\;\prec \;{A}^{\sigma (2)}\;\prec \ldots \prec \;{A}^{\sigma (N)}.$ We then say that a probability distribution $P(\vec{a}| \vec{x})$ is compatible with the causal order σ if no party signals to those before her¹⁵ , namely if for every i the marginal distribution

$$\begin{eqnarray}&&P({a}_{\sigma (1)},\ldots ,{a}_{\sigma (i)}| \vec{x}):= \displaystyle \sum _{\overset{{a}_{\sigma (j)}}{j\gt i}}P(\vec{a}| \vec{x})\end{eqnarray} \tag{ 92 }$$

does not depend on the inputs ${x}_{\sigma (j)}$ with $j\gt i$; i.e.

$$\begin{eqnarray}&&P({a}_{\sigma (1)},\ldots ,{a}_{\sigma (i)}| {x}_{\sigma (1)},\ldots ,{x}_{\sigma (i)},{x}_{\sigma (i+1)},\ldots ,{x}_{\sigma (N)})\\ &&=P({a}_{\sigma (1)},\ldots ,{a}_{\sigma (i)}| {x}_{\sigma (1)},\ldots ,{x}_{\sigma (i)},x{^{\prime} }_{\sigma (i+1)},\ldots ,x{^{\prime} }_{\sigma (N)})\\ &&\forall {x}_{\sigma (j)},x{^{\prime} }_{\sigma (j)}.\end{eqnarray} \tag{ 93 }$$

A probability distribution that is compatible with at least one causal order σ is said to be causally ordered.

More generally, we allow the parties to share randomness to agree on a specific order of sending signals between them before the inputs of the game are given to them. This allows for convex combinations of causally ordered probability distributions:

$$\begin{eqnarray}&&P(\vec{a}| \vec{x})=\displaystyle \sum _{\sigma }\;{q}_{\sigma }\;{P}_{\sigma }(\vec{a}| \vec{x}),\quad {q}_{\sigma }\geqslant 0,\quad \displaystyle \sum _{\sigma \;}{q}_{\sigma }=1,\end{eqnarray} \tag{ 94 }$$

where each ${P}_{\sigma }$ is compatible with a fixed order σ. These are still not the most general correlations compatible with the assumption of a definite causal structure, as one party could control the causal order of a set of parties in its future [19, 29, 30]. Correlations compatible with this most general scenario of definite causal order are called simply causal. In the bipartite case, the set of causal correlations forms a convex polytope, delimited by a finite number of facets that define causal inequalities [31]. The explicit definition of causal correlations in the general N-partite case is, however, rather cumbersome, and for the purposes of this article it will be enough to consider probability distributions of the form (94), which is a sufficient (although not necessary) condition for causal separability.

As causally separable processes can only generate causal correlations, the violation of a causal inequality can also be used to detect the causal nonseparability of a process. While causal witnesses are device-dependent and can only detect causal nonseparability if each party trusts her operation's implementation, causal inequalities are completely device-independent: even if each party distrusts her laboratory, they can still detect causal nonseparability from the statistics of their experimental outcomes, if those violate a causal inequality. While for every causally nonseparable process there is causal witness that will detect its nonseparability, there are causally nonseparable processes cannot be used to violate any causal inequalities: in the next subsection we will prove that the quantum switch provides such an example. There is an analogy here with entanglement witnesses, which allow for a device-dependent way of detecting entanglement, and Bell inequalities, which provide a device-independent entanglement certification—'nonlocality' [27]. The important difference is that states violating Bell inequalities are physically implementable, while no example of a physically implementable process violating causal inequalities is known.

One might first wonder if the quantum switch allows for a causal inequality violation between A and B (such as the bipartite causal inequalities of [13, 31]); this is however clearly not the case since, as pointed out before, ignoring (i.e., tracing out) the third party C makes the process matrix of the quantum switch causally separable.

One might still hope that the quantum switch can be used to violate a tripartite inequality (see e.g. [30]), explicitly involving party C; as it turns out, this is also impossible, as a consequence of the following theorem¹⁶ .

Theorem 4. Consider $N+1$ parties $\left\{{A}^{1},...,{A}^{N},C\right\}$ with settings $\left\{{x}_{1},...,{x}_{N},z\right\}$ and outcomes $\left\{{a}_{1},...{a}_{N},c\right\}.$ If the marginal distribution

$$\begin{eqnarray}&&P\left(\vec{a}| \vec{x},z\right):= \displaystyle \sum _{c}P\left(\vec{a},c| \vec{x},z\right)\end{eqnarray} \tag{ 95 }$$

is such that

• (1)  
  $P\left(\vec{a}| \vec{x},z\right)=P\left(\vec{a}| \vec{x}\right)$—i.e., it does not depend on z: C does not signal to any other (group of) parties;
• (2)  
  $P\left(\vec{a}| \vec{x}\right)={\displaystyle \sum }_{\sigma }\;{q}_{\sigma }\;{P}_{\sigma }\left(\vec{a}| \vec{x}\right),$ where ${q}_{\sigma }\geqslant 0,$ ${\displaystyle \sum }_{\sigma }{q}_{\sigma }=1,$ and the probability distributions P_σ are causally ordered,

then the full $(N+1)$-partite probability distribution $P\left(\vec{a},c| \vec{x},z\right)$ is causal.

Proof. Using Bayes' rule and the assumptions of the theorem, we can write

$$\begin{eqnarray}&&P\left(\vec{a},c| \vec{x},z\right)=P\left(\vec{a}| \vec{x},z\right)\;P\left(c| \vec{a},\vec{x},z\right),\end{eqnarray} \tag{ 96 }$$

$$\begin{eqnarray}&&=\;\displaystyle \sum _{\sigma }\;{q}_{\sigma }\;{P}_{\sigma }\left(\vec{a}| \vec{x}\right)\;P\left(c| \vec{a},\vec{x},z\right),\end{eqnarray} \tag{ 97 }$$

$$\begin{eqnarray}&&=\;\displaystyle \sum _{\sigma }\;{q}_{\sigma }\;{\tilde{P}}_{\sigma }\left(\vec{a},c| \vec{x},z\right),\end{eqnarray} \tag{ 98 }$$

where ${\tilde{P}}_{\sigma }\left(\vec{a},c| \vec{x},z\right):= {P}_{\sigma }\left(\vec{a}| \vec{x}\right)\;P\left(c| \vec{a},\vec{x},z\right)$ is compatible with the order ${A}^{\sigma (1)}\;\prec \ldots \prec \;{A}^{\sigma (N)}\;\prec \;C;$ this shows that $P\left(\vec{a},c| \vec{x},z\right)$ is causal. □

To see that the correlations generated by the quantum switch (equation (66)) respect assumptions 1 and 2 of the previous theorem, let us calculate the marginal probability distribution defined in equation (95) through the generalized Born rule (3), when the three parties perform operations ${M}_{a| x}^{{A}_{I}{A}_{O}},{M}_{b| y}^{{B}_{I}{B}_{O}}$ and ${M}_{c| z}^{{C}_{I}}:$

$$\begin{eqnarray}P(a,b| x,y,z) & = & \displaystyle \sum _{c}\;\mathrm{tr}\left[{M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y}^{{B}_{I}{B}_{O}}\otimes {M}_{c| z}^{{C}_{I}}\cdot | w\rangle \langle w| \right]\\ & = & \mathrm{tr}\left[{M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y}^{{B}_{I}{B}_{O}}\otimes \left(\displaystyle \sum _{c}{M}_{c| z}^{{C}_{I}}\right)\cdot | w\rangle \langle w| \right].\end{eqnarray} \tag{ 99 }$$

Since the third party C has no output space (${d}_{{C}_{O}}=1$), then for any instrument $\{{M}_{c| z}^{{C}_{I}}\}$ we have ${\displaystyle \sum }_{c}{M}_{c| z}^{{C}_{I}}={{\mathbb{1}}}^{{C}_{I}},$ so that

$$\begin{eqnarray}&&P(a,b| x,y,z)=\mathrm{tr}\left[{M}_{a| x}^{{A}_{I}{A}_{O}}\otimes {M}_{b| y}^{{B}_{I}{B}_{O}}\cdot {W}^{{AB}}\right]\end{eqnarray} \tag{ 100 }$$

with

$$\begin{eqnarray}&&{W}^{{AB}}:= {\mathrm{tr}}_{{C}_{I}}\;| w\rangle \langle w| .\end{eqnarray} \tag{ 101 }$$

This implies that $P(a,b| x,y,z)$ does not depend on z, as required. As argued before, tracing out C from the process matrix representing the quantum switch leads to a causally separable process matrix of the form ${W}^{{AB}}=\displaystyle \frac{1}{2}{W}^{A\prec B}+\displaystyle \frac{1}{2}{W}^{B\prec A}$ with causally ordered process matrices ${W}^{A\prec B}$ and ${W}^{B\prec A},$ which can only generate causally ordered probability distributions ${P}_{A\prec B}$ and ${P}_{B\prec A}.$ Hence, $P(a,b| x,y,z)$ can be decomposed as $\displaystyle \frac{1}{2}{P}_{A\prec B}(a,b| x,y,z)+\displaystyle \frac{1}{2}{P}_{B\prec A}(a,b| x,y,z),$ so that the second assumption of theorem 4 is also satisfied.

Therefore, the quantum switch represents an example of a causally nonseparable process that can only generate causal correlations, and hence cannot be used to violate any causal inequality¹⁷ . Note that the quantum switch (as defined in equation (66)) is a pure process, so this result shows a stark dissimilarity between entangled states and causally nonseparable processes: while every pure entangled state violates a Bell inequality [32], the analogous result does not hold for process matrices.

It is noteworthy that all the examples of causally nonseparable processes for which a physical interpretation is known, including those generated by space–time superpositions [33], fall into this category. This raises the question of whether causally nonseparable processes that do violate causal inequalities can be physically implemented at all.

A.1.1. Pure CJ isomorphism

It is convenient to distinguish two versions of the CJ isomorphism: one for maps over density matrices and one for linear operators on pure state. The latter—the 'pure CJ isomorphism'—can be represented via the 'double-ket' notation [35, 36]. For a linear operator $A\;:{{\mathcal{H}}}^{{A}_{I}}\to {{\mathcal{H}}}^{{A}_{O}},$ we define¹⁸

$$\begin{eqnarray}&&{| {A}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}:= {\mathbb{1}}\otimes {A}^{*}| {\mathbb{1}}\rangle \rangle ,\end{eqnarray} \tag{ A.1 }$$

where $| {\mathbb{1}}\rangle \rangle \equiv {| {\mathbb{1}}\rangle \rangle }^{{A}_{I}{A}_{I}}:= {\displaystyle \sum }_{j}{\left|j\right\rangle }^{{A}_{I}}\otimes {\left|j\right\rangle }^{{A}_{I}}\in {{\mathcal{H}}}^{{A}_{I}}\otimes {{\mathcal{H}}}^{{A}_{I}}$ (with also, of course, the usual notation $\langle \langle {\mathbb{1}}| ={| {\mathbb{1}}\rangle \rangle }^{\dagger }$), and the complex conjugation ${}^{*}$ is defined with respect to the chosen orthonormal basis $\{{\left|j\right\rangle }^{{A}_{I}}\}$ of ${{\mathcal{H}}}^{{A}_{I}}.$ The inverse map is given by

$$\begin{eqnarray}&&A\left|\psi \right\rangle ={\left[{\left\langle \psi \right|}^{{A}_{I}}\otimes {{\mathbb{1}}}^{{A}_{O}}\cdot {| {A}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}\right]}^{*}.\end{eqnarray} \tag{ A.2 }$$

We say that $| {A}^{*}\rangle \rangle$ is the CJ representation (or CJ vector) of A. The cumbersome complex conjugation in the definition allows us to have a simpler representation for the process matrix.

A.1.2. Maximally entangled states and unitaries

Consider here the case where the input and output spaces have equal dimensions, ${d}_{{A}_{I}}={d}_{{A}_{O}}.$ The state obtained by applying a local unitary to one subsystem of a maximally entangled state is also maximally entangled. in reverse, it is possible to generate any (bipartite) maximally entangled state by applying a local unitary to one subsystem of a reference maximally entangled state. Therefore, the CJ vector ${| {U}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}={\mathbb{1}}\otimes {U}^{*}| {\mathbb{1}}\rangle \rangle$is maximally entangled if and only if U is a unitary. More explicitly, an operator

$$\begin{eqnarray}&&U\;=\;\displaystyle \sum _{{jk}}\;{u}_{{jk}}\;\left|j\right\rangle \left\langle k\right|\end{eqnarray} \tag{ A.3 }$$

is unitary if and only if ${\displaystyle \sum }_{l}{u}_{{jl}}{u}_{{kl}}^{*}={\displaystyle \sum }_{l}{u}_{{lk}}^{*}{u}_{{lj}}={\delta }_{j,k}$ for all $j,k.$ One can check that this is also a necessary and sufficient condition for which

$$\begin{eqnarray}&&{| {U}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}={\displaystyle \sum }_{{jk}}{u}_{{jk}}^{*}{\left|k\right\rangle }^{{A}_{I}}{\left|j\right\rangle }^{{A}_{O}}\end{eqnarray} \tag{ A.4 }$$

is maximally entangled.

A.1.3. Measurement-preparation

Another useful linear operator is $| \psi \rangle \langle \phi | ,$ which describes the observation of an outcome $\left|\phi \right\rangle$ in a projective measurement, followed by the repreparation of a state $\left|\psi \right\rangle .$ Plugging this into the definition (A.1), we find the CJ representation

$$\begin{eqnarray}&&{| \;{\left(| \psi \rangle \langle \phi | \right)}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}={\left|\phi \right\rangle }^{{A}_{I}}\otimes {\left({\left|\psi \right\rangle }^{*}\right)}^{{A}_{O}}.\end{eqnarray} \tag{ A.5 }$$

Reciprocally, every pure product CJ vector represents a measurement-preparation operation.

An important particular case is when $\left|\psi \right\rangle =\left|\phi \right\rangle ,$ which corresponds to the ideal non-demolition von Neumann measurement:

$$\begin{eqnarray}&&{| \;{\left(| \phi \rangle \langle \phi | \right)}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}={\left|\phi \right\rangle }^{{A}_{I}}\otimes {\left({\left|\phi \right\rangle }^{*}\right)}^{{A}_{O}}.\end{eqnarray} \tag{ A.6 }$$

A.1.4. Mixed CJ operators

For the general case of a linear map ${{\mathcal{M}}}^{A}\;:{A}_{I}\to {A}_{O},$ we define the CJ isomorphism as

$$\begin{eqnarray}&&{M}^{{A}_{I}{A}_{O}}:= {\left[{\mathcal{I}}\otimes {{\mathcal{M}}}^{A}(| {\mathbb{1}}\rangle \rangle \langle \langle {\mathbb{1}}| )\right]}^{T}.\end{eqnarray} \tag{ A.7 }$$

It is easy to verify that the definition (A.7) reduces to (A.1) for operators of the form ${{\mathcal{M}}}^{A}(\rho )=A\rho {A}^{\dagger },$ i.e. that, in such a case

$$\begin{eqnarray}&&M=| {A}^{*}\rangle \rangle \langle \langle {A}^{*}| \;\end{eqnarray} \tag{ A.8 }$$

(with $| {A}^{*}\rangle \rangle \equiv {| {A}^{*}\rangle \rangle }^{{A}_{I}{A}_{O}}$ and $\langle \langle {A}^{*}| ={| {A}^{*}\rangle \rangle }^{\dagger }$).

According to Choi's theorem [37], a linear map ${{\mathcal{M}}}^{A}\;:{A}_{I}\to {A}_{O}$ is CP if and only if its CJ matrix is positive semidefinite, ${M}^{{A}_{I}{A}_{O}}\geqslant 0.$ A characterization of the trace-preserving condition can be found using the inverse CJ isomorphism

$$\begin{eqnarray}&&{{\mathcal{M}}}^{A}(\rho )={\left[{\mathrm{tr}}_{{A}_{I}}[{\rho }^{{A}_{I}}\otimes {{\mathbb{1}}}^{{A}_{O}}\cdot {M}^{{A}_{I}{A}_{O}}]\right]}^{T}.\end{eqnarray} \tag{ A.9 }$$

By taking the trace of both sides of the equation, it can be readily verified that the map ${{\mathcal{M}}}^{A}$ is trace-preserving if and only if

$$\begin{eqnarray}&&{\mathrm{tr}}_{{A}_{O}}{M}^{{A}_{I}{A}_{O}}={{\mathbb{1}}}^{{A}_{I}}.\end{eqnarray} \tag{ A.10 }$$

Note that a CP map can be part of an instrument only if it is trace-non-increasing, a condition that translates to

$$\begin{eqnarray}&&{{\mathbb{1}}}^{{A}_{I}}-{\mathrm{tr}}_{{A}_{O}}{M}^{{A}_{I}{A}_{O}}\geqslant 0.\end{eqnarray} \tag{ A.11 }$$

A useful example is the CPTP map ${{\mathcal{M}}}^{A}(\sigma )=\rho \mathrm{tr}\sigma ,$ which corresponds to the preparation of a (normalized) state ρ independently of the input state σ. Its CJ representation is found to be

$$\begin{eqnarray}&&{M}^{{A}_{I}{A}_{O}}={{\mathbb{1}}}^{{A}_{I}}\otimes {({\rho }^{T})}^{{A}_{O}}.\end{eqnarray} \tag{ A.12 }$$

A second relevant case is the CP (not trace-preserving) map that gives the probability of observing a POVM element E in a measurement: ${{\mathcal{M}}}^{A}(\rho )=\mathrm{tr}[E\rho ]$ (here ${d}_{{A}_{O}}=1$). Its CJ representation is simply

$$\begin{eqnarray}&&{M}^{{A}_{I}}={E}^{{A}_{I}}.\end{eqnarray} \tag{ A.13 }$$

Finally, the situation where a POVM element E is measured on the state σ in A_I and a state ρ is prepared in A_O corresponds to the CP map ${{\mathcal{M}}}^{A}(\sigma )=\rho \mathrm{tr}\left[E\sigma \right],$ which has CJ representation

$$\begin{eqnarray}&&{M}^{{A}_{I}{A}_{O}}={E}^{{A}_{I}}\otimes {({\rho }^{T})}^{{A}_{O}}.\end{eqnarray} \tag{ A.14 }$$

Here we discuss in more detail some examples and properties of process matrices.

A.2.1. Quantum states

Consider a bipartite process matrix of the form

$$\begin{eqnarray}&&{W}^{{A}_{I}{A}_{O}{B}_{I}{B}_{O}}={\rho }^{{A}_{I}{B}_{I}}\otimes {{\mathbb{1}}}^{{A}_{O}{B}_{O}}.\end{eqnarray} \tag{ A.15 }$$

According to the generalized Born rule, equation (3), the probability for the two parties A and B to perform trace non-increasing CP maps with CJ matrices ${M}^{{A}_{I}{A}_{O}}$ and ${M}^{{B}_{I}{B}_{O}},$ respectively, is given by

$$\begin{eqnarray}P\left({M}^{{A}_{I}{A}_{O}},{M}^{{B}_{I}{B}_{O}}\right) & = & \mathrm{tr}\left[\left({M}^{{A}_{I}{A}_{O}}\otimes {M}^{{B}_{I}{B}_{O}}\right)W\right],\\ & = & \mathrm{tr}\left[\left({E}^{{A}_{I}}\otimes {E}^{{B}_{I}}\right){\rho }^{{A}_{I}{B}_{I}}\right],\end{eqnarray} \tag{ A.16 }$$

where ${E}^{{A}_{I}}:= {\mathrm{tr}}_{{A}_{O}}{M}^{{A}_{I}{A}_{O}}$ and ${E}^{{B}_{I}}:= {\mathrm{tr}}_{{B}_{O}}{M}^{{B}_{I}{B}_{O}}.$ These operators are positive semidefinite and, because of equation (A.11), they can be completed to form a POVM. Thus equation (A.16) corresponds to the probability of observing the POVM element ${E}^{{A}_{I}}\otimes {E}^{{B}_{I}}$ given the state ${\rho }^{{A}_{I}{B}_{I}};$ in other words, the process matrix (A.15) describes a bipartite state. Notice that a process matrix of this form does not allow signalling in either direction and therefore, being compatible with both $A\;\prec \;B$ and $B\;\prec \;A,$ it is causally separable. This is irrespective of the state ρ, which can be entangled or separable. Note also the difference between the process matrix (A.15) and the CJ representation of state preparation, equation (A.12).

A.2.2. Channels

Consider a bipartite situation where a party A only performs state preparations, while the second party B only performs measurements. In this case, the local laboratory of A is characterized by a trivial input space, ${d}_{{A}_{I}}=1,$ while B has a trivial output space, ${d}_{{B}_{O}}=1.$ The process matrix shared by A and B, which represents here a quantum channel, is then defined on the space ${A}_{O}\otimes {B}_{I}\;\ni \;W.$ The probability that B observes a POVM element E when A prepares a state ρ is given by

$$\begin{eqnarray}&&P\left(E| \rho \right)=\mathrm{tr}\left[{\left({\rho }^{T}\right)}^{{A}_{O}}\otimes {E}^{{B}_{I}}\;\cdot \;{W}^{{A}_{O}{B}_{I}}\right],\end{eqnarray} \tag{ A.17 }$$

where we used (A.12) and (A.13) for the local operations. This is equivalent to saying that B measures E in the state ${\mathrm{tr}}_{{A}_{O}}\left[{\left({\rho }^{T}\right)}^{{A}_{O}}\otimes {{\mathbb{1}}}^{{B}_{I}}\cdot {W}^{{A}_{O}{B}_{I}}\right]={\left[{\mathrm{tr}}_{{A}_{O}}\left[{\rho }^{{A}_{O}}\otimes {{\mathbb{1}}}^{{B}_{I}}\cdot {\left({W}^{T}\right)}^{{A}_{O}{B}_{I}}\right]\right]}^{T}.$ Comparing this with the inverse CJ transformation (A.9), we find that the process matrix W corresponds to a channel with CJ representation W^T. In other words, a channel ${\mathcal{C}}$ from A_O to B_I is represented by the process matrix

$$\begin{eqnarray}&&{W}^{{A}_{O}{B}_{I}}={\mathcal{I}}\otimes {\mathcal{C}}(| {\mathbb{1}}\rangle \rangle \langle \langle {\mathbb{1}}| )\end{eqnarray} \tag{ A.18 }$$

(with here $| {\mathbb{1}}\rangle \rangle \equiv {| {\mathbb{1}}\rangle \rangle }^{{A}_{O}{A}_{O}}$). Note that the CJ representation of a channel, equation (A.7), differs by a transposition from the corresponding process matrix (A.18).

A.2.3. Reduced process matrices

Given a multipartite process $W={W}^{{A}_{I}^{1}{A}_{O}^{1}...{A}_{I}^{N}{A}_{O}^{N}}$ and a CPTP map for the jth party with CJ matrix ${M}^{{A}_{I}^{j}{A}_{O}^{j}},$ we define the reduced process matrix for the remaining $N-1$ parties, given ${M}^{{A}_{I}^{j}{A}_{O}^{j}},$ as

$$\begin{eqnarray}&&\bar{W}({M}^{{A}_{I}^{j}{A}_{O}^{j}}):= {\mathrm{tr}}_{{A}_{I}^{j}{A}_{O}^{j}}\left[\left({{\mathbb{1}}}^{{A}_{I}^{1}{A}_{O}^{1}}\otimes \ldots {M}^{{A}_{I}^{j}{A}_{O}^{j}}\otimes \ldots {{\mathbb{1}}}^{{A}_{I}^{N}{A}_{O}^{N}}\right)\cdot W\right].\end{eqnarray} \tag{ A.19 }$$

With the usual generalized Born rule (3), the reduced process matrix gives the probability for the remaining $N-1$ parties to measure arbitrary CP maps, given that the jth party performs ${M}^{{A}_{I}^{j}{A}_{O}^{j}}.$ The explicit dependence of $\bar{W}$ on ${M}^{{A}_{I}^{j}{A}_{O}^{j}}$ accounts for the possibility of signalling: the remaining parties observe different probability distributions depending on the choice of CPTP map performed by party j. As an example, consider a process matrix of the form (A.18). If A prepares a state ρ, the reduced process matrix for B is

$$\begin{eqnarray}{\bar{W}}^{{B}_{I}}(\rho ) & = & {\mathrm{tr}}_{{A}_{O}}\left[{\left({\rho }^{T}\right)}^{{A}_{O}}\otimes {{\mathbb{1}}}^{{B}_{I}}\;\cdot \;{W}^{{A}_{O}{B}_{I}}\right]\\ & = & \displaystyle \sum _{{jk}}\;\left\langle k\right|{\rho }^{T}\left|j\right\rangle \ {\mathcal{C}}\left(\left|j\right\rangle \left\langle k\right|\right)={\mathcal{C}}\left(\rho \right).\end{eqnarray} \tag{ A.20 }$$

Thus, for a process that represents a channel from A to B, the reduced process for B, given that A prepares ρ, is simply the channel applied to ρ, as should be expected.

A.2.4. Pure process matrices

In some cases, the process matrix turns out to be a rank-one projector: $W=| w\rangle \langle w|$ for some 'process vector' $\left|w\right\rangle .$ If the CJ operators representing the local operations are also rank-one projectors, as is the case for unitaries and projective measurements followed by pure repreparations, it is convenient to work at the level of vectors and of probability amplitudes: given the local operations ${A}_{1},...,{A}_{N}$ represented by the CJ vectors ${| {A}_{1}^{*}\rangle \rangle }^{{A}_{I}^{1}{A}_{O}^{1}},...,{| {A}_{N}^{*}\rangle \rangle }^{{A}_{I}^{N}{A}_{O}^{N}},$ the overall probability amplitude is given (up to global phase, which we choose to be 0) by

$$\begin{eqnarray}&&{\langle \langle {A}_{1}^{*}| }^{{A}_{I}^{1}{A}_{O}^{1}}\otimes ...\otimes {\langle \langle {A}_{N}^{*}| }^{{A}_{I}^{N}{A}_{O}^{N}}\cdot {\left|w\right\rangle }^{{A}_{I}^{1}{A}_{O}^{1}...{A}_{I}^{N}{A}_{O}^{N}}.\end{eqnarray} \tag{ A.21 }$$

The probability is then obtained as the modulus square of the amplitude and conforms to the general expression (3). Given that party j performs the unitary U_j, the reduced process is clearly given by the partial scalar product

$$\begin{eqnarray}&&{{\mathbb{1}}}^{{A}_{I}^{1}{A}_{O}^{1}}\otimes ...{\langle \langle {U}_{j}^{*}| }^{{A}_{I}^{j}{A}_{O}^{j}}\otimes ...{{\mathbb{1}}}^{{A}_{I}^{N}{A}_{O}^{N}}\cdot {\left|w\right\rangle }^{{A}_{I}^{1}{A}_{O}^{1}...{A}_{I}^{N}{A}_{O}^{N}}.\end{eqnarray} \tag{ A.22 }$$

The process matrix describing a unitary channel U from A_O to B_I is of particular interest. Using (A.18), we find that it is given by

$$\begin{eqnarray}&&{\left|w\right\rangle }^{{A}_{O}{B}_{I}}={\mathbb{1}}\otimes U| {\mathbb{1}}\rangle \rangle ={| U\rangle \rangle }^{{A}_{O}{B}_{I}}.\end{eqnarray} \tag{ A.23 }$$

Note again the difference between this expression and the CJ representation (A.1). Generalizing this to a sequence of parties ${A}_{1},...,{A}_{N},$ with the output of party j connected to the input of party $j+1$ via the unitary U_j, we find

$$\begin{eqnarray}&&{\left|w\right\rangle }^{{A}_{O}^{1}...{A}_{I}^{N}}={| {U}_{1}\rangle \rangle }^{{A}_{O}^{1}{A}_{I}^{2}}\otimes ...\otimes {| {U}_{N}\rangle \rangle }^{{A}_{O}^{N-1}{A}_{I}^{N}}.\end{eqnarray} \tag{ A.24 }$$

Recall that a given operator $W\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}$ is a valid process matrix if and only if it yields, through the generalized Born rule (3), only well-defined probabilities—that is, the probabilities must be non-negative and must sum up to 1.

Non-negativity

As recalled previously, a map is CP if and only if its CJ representation is positive semidefinite. Including the possibility that A and B's operations involve interactions with a (possibly entangled) ancillary system in a state ${\rho }^{{A}_{I}^{\prime} {B}_{I}^{\prime} },$ the non-negativity of probabilities is thus equivalent to¹⁹

$$\begin{eqnarray}&&\mathrm{tr}\left[\left({M}^{{A}_{I}^{\prime} {A}_{I}{A}_{O}}\otimes {M}^{{B}_{I}^{\prime} {B}_{I}{B}_{O}}\right)\cdot \left({\rho }^{{A}_{I}^{\prime} {B}_{I}^{\prime} }\otimes {W}^{{A}_{I}{A}_{O}{B}_{I}{B}_{O}}\right)\right]\geqslant 0\\ &&\forall \ {M}^{{A}_{I}^{\prime} {A}_{I}{A}_{O}}\geqslant 0,{M}^{{B}_{I}^{\prime} {B}_{I}{B}_{O}}\geqslant 0,{\rho }^{{A}_{I}^{\prime} {B}_{I}^{\prime} }\geqslant 0.\end{eqnarray} \tag{ B.1 }$$

For the case where the ancillary spaces ${A}_{I}^{\prime}$ and ${B}_{I}^{\prime}$ are isomorphic to ${A}_{I}\otimes {A}_{O},$ and ${M}^{{A}_{I}^{\prime} {A}_{I}{A}_{O}}$ and ${\rho }^{{A}_{I}^{\prime} {B}_{I}^{\prime} }$ are both projectors onto the maximally entangled state ${| {\mathbb{1}}\rangle \rangle }^{{A}_{I}{A}_{O}/{A}_{I}{A}_{O}}:= {\displaystyle \sum }_{j,k}{\left|j,k\right\rangle }^{{A}_{I}{A}_{O}}\otimes {\left|j,k\right\rangle }^{{A}_{I}{A}_{O}}$ (where $\{{\left|j,k\right\rangle }^{{A}_{I}{A}_{O}}\}$ is an orthonormal basis of ${{\mathcal{H}}}^{{A}_{I}}\otimes {{\mathcal{H}}}^{{A}_{O}}$), we find that the trace in (B.1) is equal to $\mathrm{tr}[{M}^{{B}_{I}^{\prime} {B}_{I}{B}_{O}}\cdot {W}^{{A}_{I}{A}_{O}{B}_{I}{B}_{O}}].$ Requiring that its value is non-negative for all ${M}^{{B}_{I}^{\prime} {B}_{I}{B}_{O}}\geqslant 0$ implies that W must be positive semidefinite.

Reciprocally, $W\geqslant 0$ clearly implies that (B.1) is satisfied. Hence, the non-negativity of probabilities is equivalent to W being positive semidefinite, equation (4).

Normalization

The fact that probabilities must sum up to 1 for all instruments is equivalent to the constraint that the probability of realization of any CPTP map is 1. Now, recall that a CP map ${{\mathcal{M}}}^{A}\;:{A}_{I}\to {A}_{O}$ is trace-preserving if and only if its CJ matrix ${M}^{{A}_{I}{A}_{O}}$ satisfies ${\mathrm{tr}}_{{A}_{O}}{M}^{{A}_{I}{A}_{O}}={{\mathbb{1}}}^{{A}_{I}}$—or equivalently, using the notation of equation (12), ${}_{{A}_{O}}{M}^{{A}_{I}{A}_{O}}={{\mathbb{1}}}^{{A}_{I}{A}_{O}}/{d}_{{A}_{O}}.$ Ignoring here for simplicity the possible use of an ancillary system (which leads to the same conclusion²⁰ ),the normalization of probabilities is thus equivalent to

$$\begin{eqnarray}&&\mathrm{tr}\left[\left({M}^{{A}_{I}{A}_{O}}\otimes {M}^{{B}_{I}{B}_{O}}\right)\cdot {W}^{{A}_{I}{A}_{O}{B}_{I}{B}_{O}}\right]=1\\ &&\forall \ {M}^{{A}_{I}{A}_{O}}\geqslant 0,{M}^{{B}_{I}{B}_{O}}\geqslant 0,\\ &&{\rm{s.t.}}\ \ {}_{{A}_{O}}{M}^{{A}_{I}{A}_{O}}={{\mathbb{1}}}^{{A}_{I}{A}_{O}}/{d}_{{A}_{O}},{}_{{B}_{O}}{M}^{{B}_{I}{B}_{O}}={{\mathbb{1}}}^{{B}_{I}{B}_{O}}/{d}_{{B}_{O}}.\end{eqnarray} \tag{ B.2 }$$

First of all, note that the positivity of ${M}^{{A}_{I}{A}_{O}}$ and ${M}^{{B}_{I}{B}_{O}}$ is irrelevant here, since the set of positive semidefinite operators is a full dimensional subset of the space of Hermitian operators²¹ . The only relevant conditions are the normalization constraints. Defining the maps ${}_{[1-{A}_{O}]}M=M-{}_{{A}_{O}}M$ and ${}_{[1-{B}_{O}]}M=M-{}_{{B}_{O}}M,$ and noting that for any Hermitian operators x and y, the operators ${}_{[1-{A}_{O}]}x+{\mathbb{1}}/{d}_{{A}_{O}}$ and ${}_{[1-{B}_{O}]}y+{\mathbb{1}}/{d}_{{B}_{O}}$ satisfy the above normalization constraints (where from now on we are omitting the superscripts to reduce cluttering), we find that equation (B.2) is equivalent to

$$\begin{eqnarray}&&\mathrm{tr}\left[\left({}_{[1-{A}_{O}]}x+{\mathbb{1}}/{d}_{{A}_{O}}\right)\otimes \left({}_{[1-{B}_{O}]}y+{\mathbb{1}}/{d}_{{B}_{O}}\right)W\right]=1\quad \forall \ x,y.\end{eqnarray} \tag{ B.3 }$$

For $x=y=0,$ this yields the normalization condition of equation (5),

$$\begin{eqnarray}&&\mathrm{tr}[W]={d}_{{A}_{O}}{d}_{{B}_{O}}.\end{eqnarray} \tag{ B.4 }$$

For y = 0 and x = 0, respectively, this in turn implies

$$\begin{eqnarray}&&\mathrm{tr}\left[\left({}_{[1-{A}_{O}]}x\otimes {\mathbb{1}}\right)W\right]=0\quad \forall x,\end{eqnarray} \tag{ B.5 }$$

$$\begin{eqnarray}&&\mathrm{tr}\left[\left({\mathbb{1}}\otimes {}_{[1-{B}_{O}]}y\right)W\right]=0\quad \forall y,\end{eqnarray} \tag{ B.6 }$$

which then imply

$$\begin{eqnarray}&&\mathrm{tr}\left[\left({}_{[1-{A}_{O}]}x\otimes {}_{[1-{B}_{O}]}y\right)W\right]=0\quad \forall x,y.\end{eqnarray} \tag{ B.7 }$$

Reciprocally, equations (B.4)–(B.7) clearly imply (B.3), so that these are equivalent to equation (B.2).

Thinking of the trace as the Hilbert–Schmidt inner product

$$\begin{eqnarray*}&&\langle M,W \rangle =\mathrm{tr}[M\cdot W]\end{eqnarray*}$$

(for Hermitian operators $M,W$) and noting that the maps ${}_{[1-{A}_{O}]}\cdot$ and ${}_{[1-{B}_{O}]}\cdot$ are self-adjoint, the conditions (B.5)–(B.7) are equivalent to

$$\begin{eqnarray}&&{}_{[1-{A}_{O}]}\left({\mathrm{tr}}_{{B}_{I}{B}_{O}}W\right)=0,\end{eqnarray} \tag{ B.8 }$$

$$\begin{eqnarray}&&{}_{[1-{B}_{O}]}\left({\mathrm{tr}}_{{A}_{I}{A}_{O}}W\right)=0,\end{eqnarray} \tag{ B.9 }$$

$$\begin{eqnarray}&&{}_{[1-{A}_{O}][1-{B}_{O}]}W=0,\end{eqnarray} \tag{ B.10 }$$

which we can rewrite as

$$\begin{eqnarray}&&{}_{{B}_{I}{B}_{O}}W={}_{{A}_{O}{B}_{I}{B}_{O}}W,\end{eqnarray} \tag{ B.11 }$$

$$\begin{eqnarray}&&{}_{{A}_{I}{A}_{O}}W={}_{{A}_{I}{A}_{O}{B}_{O}}W,\end{eqnarray} \tag{ B.12 }$$

$$\begin{eqnarray}&&W={}_{{B}_{O}}W+{}_{{A}_{O}}W-{}_{{A}_{O}{B}_{O}}W,\end{eqnarray} \tag{ B.13 }$$

which are conditions (9)–(11).

Note that each condition (B.8)–(B.10) defines a linear subspace, and the intersection of these three linear subspaces is the smallest subspace that contains all valid bipartite process matrices, which we denote by²²

$$\begin{eqnarray}&&{{\mathcal{L}}}_{V}=\left\{W\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}\;| \;W={L}_{V}(W)\right\},\end{eqnarray} \tag{ B.14 }$$

The projector onto this subspace, L_V, shall be used quite often in the paper, so it is useful to find an explicit expression for it. To do that, first we rewrite conditions (B.11)–(B.13) explicitly as projections onto subspaces, i.e., as

$$\begin{eqnarray}&&W={L}_{A}(W),W={L}_{B}(W),W={L}_{{AB}}(W),\end{eqnarray} \tag{ B.15 }$$

where the projectors L_A, L_B, and L_AB are given by

$$\begin{eqnarray}&&{L}_{A}(W)=W-{}_{{B}_{I}{B}_{O}}W+{}_{{A}_{O}{B}_{I}{B}_{O}}W,\end{eqnarray} \tag{ B.16 }$$

$$\begin{eqnarray}&&{L}_{B}(W)=W-{}_{{A}_{I}{A}_{O}}W+{}_{{A}_{I}{A}_{O}{B}_{O}}W,\end{eqnarray} \tag{ B.17 }$$

$$\begin{eqnarray}&&{L}_{{AB}}(W)={}_{{B}_{O}}W+{}_{{A}_{O}}W-{}_{{A}_{O}{B}_{O}}W,\end{eqnarray} \tag{ B.18 }$$

Since the three projectors above commute, the projector onto the intersection of their subspaces L_V is given simply by the composition of L_A, L_B, and L_AB, i.e.,

$$\begin{eqnarray}&&{L}_{V}(W)={L}_{A}\;\circ \;{L}_{B}\;\circ \;{L}_{{AB}}(W),\end{eqnarray} \tag{ B.19 }$$

which, after simplification, can be written as

$$\begin{eqnarray}&&{L}_{V}(W)={}_{{A}_{O}}W+{}_{{B}_{O}}W-{}_{{A}_{O}{B}_{O}}W-{}_{{B}_{I}{B}_{O}}W+{}_{{A}_{O}{B}_{I}{B}_{O}}W-{}_{{A}_{I}{A}_{O}}W+{}_{{A}_{I}{A}_{O}{B}_{O}}W.\end{eqnarray} \tag{ B.20 }$$

Summing up, we conclude that an operator $W\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}$ is a valid bipartite process matrix if and only if $W\geqslant 0,$ $\mathrm{tr}W={d}_{{A}_{O}}{d}_{{B}_{O}},$ and $W={L}_{V}(W),$ as in equations (4)–(6).

A similar reasoning leads to the conclusion that an operator $W\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}\otimes {C}_{I}\otimes {C}_{O}$ is a valid tripartite process matrix if and only if $W\geqslant 0,$ $\mathrm{tr}W={d}_{{A}_{O}}{d}_{{B}_{O}}{d}_{{C}_{O}},$ and

$$\begin{eqnarray}W & = & {L}_{A}(W),\quad W={L}_{B}(W),\quad W={L}_{C}(W),\\ W & = & {L}_{{AB}}(W),\quad W={L}_{{AC}}(W),\quad W={L}_{{BC}}(W),\\ W & = & {L}_{{ABC}}(W),\end{eqnarray} \tag{ B.21 }$$

where the maps L_A, L_B, L_C, L_AB, L_AC, L_BC, and L_ABC are now commuting projectors onto linear subspaces of ${A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}\otimes {C}_{I}\otimes {C}_{O},$ defined by

$$\begin{eqnarray*}{L}_{A}(W) & = & {}_{[1-(1-{A}_{O}){B}_{I}{B}_{O}{C}_{I}{C}_{O}]}W,\\ {L}_{B}(W) & = & {}_{[1-(1-{B}_{O}){A}_{I}{A}_{O}{C}_{I}{C}_{O}]}W,\\ {L}_{C}(W) & = & {}_{[1-(1-{C}_{O}){A}_{I}{A}_{O}{B}_{I}{B}_{O}]}W,\\ {L}_{{AB}}(W) & = & {}_{[1-(1-{A}_{O})(1-{B}_{O}){C}_{I}{C}_{O}]}W,\\ {L}_{{AC}}(W) & = & {}_{[1-(1-{A}_{O})(1-{C}_{O}){B}_{I}{B}_{O}]}W,\\ {L}_{{BC}}(W) & = & {}_{[1-(1-{B}_{O})(1-{C}_{O}){A}_{I}{A}_{O}]}W,\\ {L}_{{ABC}}(W) & = & {}_{[1-(1-{A}_{O})(1-{B}_{O})(1-{C}_{O})]}W,\end{eqnarray*}$$

where we used the shorthand notation

$$\begin{eqnarray}&&{}_{[\displaystyle \sum _{X}{\alpha }_{X}X]}W=\displaystyle \sum _{X}{\alpha }_{X}\cdot {}_{X}W\end{eqnarray} \tag{ B.22 }$$

for a sum over products of subsystems X with coefficients ${\alpha }_{X}$ (and with ${}_{1}W:= W$).

The constraints in (B.21) are equivalent to

$$\begin{eqnarray}&&W={L}_{V}(W),\end{eqnarray} \tag{ B.23 }$$

where the map L_V is obtained here by composing the 7 maps L_A, L_B, L_C, L_AB, L_AC, L_BC, and L_ABC. One finds in this tripartite case, after simplification

$$\begin{eqnarray}{L}_{V}(W) & = & {}_{\left[1-(1-{A}_{O}+{A}_{I}{A}_{O})(1-{B}_{O}+{B}_{I}{B}_{O})(1-{C}_{O}+{C}_{I}{C}_{O})\right.}{}_{+\ {A}_{I}{A}_{O}{B}_{I}{B}_{O}{C}_{I}{C}_{O}\phantom{\rule[-0000]{0ex}{2.35ex}}]\phantom{]}}W,\end{eqnarray} \tag{ B.24 }$$

which defines a projector onto the linear subspace

$$\begin{eqnarray}&&{{\mathcal{L}}}_{V}=\left\{W\in {A}_{I}\otimes {A}_{O}\otimes {B}_{I}\otimes {B}_{O}\otimes {C}_{I}\otimes {C}_{O}\;| \;W={L}_{V}(W)\right\}.\end{eqnarray} \tag{ B.25 }$$

The generalization to the N-partite case is rather straightforward. We find that an operator $W\in {A}_{I}^{1}\otimes {A}_{O}^{1}\otimes \ldots \otimes {A}_{I}^{N}\otimes {A}_{O}^{N}$ is a valid N-partite process matrix if and only if $W\geqslant 0,$ $\mathrm{tr}W={d}_{{A}_{O}^{1}}\ldots {d}_{{A}_{O}^{N}},$ and for all ${2}^{N}-1$ non-empty subsets ${\mathcal{X}}$ of $\{1,\ldots ,N\},$

$$\begin{eqnarray}&&W={L}_{{\mathcal{X}}}(W):= {}_{\left[1-\displaystyle \prod _{i\in {\mathcal{X}}}(1-{A}_{O}^{i})\displaystyle \prod _{i\notin {\mathcal{X}}}{A}_{I}^{i}{A}_{O}^{i}\right]}W.\end{eqnarray} \tag{ B.26 }$$

Note that the ${2}^{N}-1$ maps ${L}_{{\mathcal{X}}}$ are commuting projectors onto linear subspaces of ${A}_{I}^{1}\otimes {A}_{O}^{1}\otimes \ldots \otimes {A}_{I}^{N}\otimes {A}_{O}^{N}.$ The constraints (B.26) are equivalent to

$$\begin{eqnarray}&&W={L}_{V}(W),\end{eqnarray} \tag{ B.27 }$$

where the map L_V is obtained this time by composing the ${2}^{N}-1$ maps ${L}_{{\mathcal{X}}}.$ More explicitly, one finds in the N-partite case the general expression

$$\begin{eqnarray}&&{L}_{V}(W)={}_{\left[1\ -\ {\displaystyle \prod }_{i}(1-{A}_{O}^{i}+{A}_{I}^{i}{A}_{O}^{i})\ +\ {\displaystyle \prod }_{i}{A}_{I}^{i}{A}_{O}^{i}\right]}W,\end{eqnarray} \tag{ B.28 }$$

which again defines a projector onto the linear subspace

$$\begin{eqnarray}&&{{\mathcal{L}}}_{V}=\left\{W\in {A}_{I}^{1}\otimes {A}_{O}^{1}\otimes \ldots \otimes {A}_{I}^{N}\otimes {A}_{O}^{N}\ | \ W={L}_{V}(W)\right\}.\end{eqnarray} \tag{ B.29 }$$

Proof. For any subset ${\mathcal{X}}$ of $\{1,\ldots ,N\},$ define ${P}_{{\mathcal{X}}}={\prod }_{i\in {\mathcal{X}}}(1-{A}_{O}^{i}){\prod }_{i\notin {\mathcal{X}}}{A}_{I}^{i}{A}_{O}^{i}.$ For ${\mathcal{X}}\ne {\mathcal{X}}^{\prime} ,$ note that there exists (at least one) i₀ such that the product ${P}_{{\mathcal{X}}}{P}_{{\mathcal{X}}^{\prime} }$ contains the factor $(1-{A}_{O}^{{i}_{0}}){A}_{I}^{{i}_{0}}{A}_{O}^{{i}_{0}}.$ Now, ${}_{[(1-{A}_{O}^{{i}_{0}}){A}_{O}^{{i}_{0}}]}W=0,$ so that ${}_{[{P}_{{\mathcal{X}}}{P}_{{\mathcal{X}}^{\prime} }]}W=0.$

Developing L_V, we thus find ${L}_{V}(W)={}_{[{\prod }_{{\mathcal{X}}\ne \varnothing }(1-{P}_{{\mathcal{X}}})]}W={}_{[1-{\sum }_{{\mathcal{X}}\ne \varnothing }{P}_{{\mathcal{X}}}]}W={}_{[1-{\sum }_{{\mathcal{X}}}{P}_{{\mathcal{X}}}+{\prod }_{i}{A}_{I}^{i}{A}_{O}^{i}]}W.$ Now, one can write ${\sum }_{{\mathcal{X}}}{P}_{{\mathcal{X}}}={\sum }_{{k}_{1}=0}^{1}\ldots {\sum }_{{k}_{N}=0}^{1}$ ${(1-{A}_{O}^{1})}^{{k}_{1}}$ ${({A}_{I}^{1}{A}_{O}^{1})}^{1-{k}_{1}}\ldots$ ${(1-{A}_{O}^{N})}^{{k}_{N}}$ ${({A}_{I}^{N}{A}_{O}^{N})}^{1-{k}_{N}}={\prod }_{i}{\sum }_{{k}_{i}=0}^{1}$ ${(1-{A}_{O}^{i})}^{{k}_{i}}$ ${({A}_{I}^{i}{A}_{O}^{i})}^{1-{k}_{i}}={\prod }_{i}$ $(1-{A}_{O}^{i}+{A}_{I}^{i}{A}_{O}^{i}),$ from which equation (B.28) follows.□