Witnessing Causal Nonseparability: Theory and Experiment

Abstract

We develop a technique to witness a process that is not causally separable, in that it cannot be written as a fixed-order quantum circuit or a mixture of those. Using the process matrix formalism, we find a set of measurements within the process that proves causal non-separability. The problem of finding these measurements can be written as a SemiDefinite Program that can be solved efficiently. We apply our method in an experiment we performed in our labs, where we implemented a causally non-separable process and proved it to be so by performing unitary operations and final measurements.

Appendix: Process Matrix Characterization

Bipartite valid process matrices: Here we provide the derivation of the Eq. (3.5) that completely characterize the set of bipartite causally separable process matrices.

The set of valid process matrices is defined by the following requirements: the bipartite expression of the probabilities in Eq. (3.1) must be non-negative and sum up to one for all possible operations (including operations that act on ancillary systems on arbitrary input states, which we ignore at the moment although it leads to the same conclusion). These yield

$$\begin{aligned}&W \ge 0 \end{aligned}$$

(3.47)

$$\begin{aligned}&{{\,\mathrm{Tr}\,}}[(M^{A_IA_O}\otimes M^{B_IB_O}) W^{A_IA_OB_IB_O}] = 1\quad \forall M^{A_IA_O}\ge 0,\ M^{B_IB_O}\ge 0,\end{aligned}$$

(3.48)

$$\begin{aligned}&\text {s. t.}\quad _{A_O}M^{A_IA_O} = \mathbbm {1}^{A_IA_O}/d_{A_O},\ _{B_O}M^{B_IB_O} = \mathbbm {1}^{B_IB_O}/d_{B_O}. \end{aligned}$$

(3.49)

We remind the definition of two maps, we will use extensively

$$\begin{aligned}&_{x} W = \frac{\mathbbm {1}^{x}}{d_x} {{\,\mathrm{Tr}\,}}_{x}W\end{aligned}$$

(3.50)

$$\begin{aligned}&_{[1-x]}W = W -~_xW, \end{aligned}$$

(3.51)

Using the second map we defined above, we can see that for any two Hermitian operators X and Y, the operators $_{[1-{A_O}]}X + \mathbbm {1}^{A_IA_O}$ and $_{[1-{B_O}]}Y + \mathbbm {1}^{B_IB_O}$ satisfy the above normalization constraints in Eq. (3.49). Then we write the normalization condition of the probabilities of Eq. (3.48) as

$$\begin{aligned} {{\,\mathrm{Tr}\,}}[(_{[1-{A_O}]}X + \mathbbm {1}^{A_IA_O}\otimes (_{[1-{B_O}]}Y + \mathbbm {1}^{B_IB_O}W] = 1,\quad \forall X,Y \end{aligned}$$

(3.52)

For $X=Y=0$ this yields

$$\begin{aligned} {{\,\mathrm{Tr}\,}}(W) = d_{A_O}d_{B_O}. \end{aligned}$$

(3.53)

For $Y=0$ and $X=0$ in turn, implies

$$\begin{aligned} \begin{aligned} {{\,\mathrm{Tr}\,}}[(_{[1-{A_O}]}X\otimes \mathbbm {1}^{B_IB_O}W] = 0,\quad \forall X,\\ {{\,\mathrm{Tr}\,}}[(\mathbbm {1}^{A_IA_O}\otimes _{[1-{B_O}]}Y)W] = 0,\quad \forall Y, \end{aligned} \end{aligned}$$

(3.54)

because for $Y=0$ we have

$$\begin{aligned} {{\,\mathrm{Tr}\,}}[(_{[1-{A_O}]}X + \mathbbm {1}^{A_IA_O})\otimes \mathbbm {1}^{B_IB_O})W] = 1\Rightarrow \\ {{\,\mathrm{Tr}\,}}[(_{[1-{A_O}]}X\otimes \mathbbm {1}^{B_IB_O})W] + {{\,\mathrm{Tr}\,}}[(\mathbbm {1}^{A_IA_O})\otimes \mathbbm {1}^{B_IB_O})W] = 1\Rightarrow \\ {{\,\mathrm{Tr}\,}}[(_{[1-{A_O}]}X\otimes \mathbbm {1}^{B_IB_O})W] = 0 \end{aligned}$$

Back to the Eq. (3.54), they imply

$$\begin{aligned} {{\,\mathrm{Tr}\,}}[(_{[1-{A_O}]}X\otimes _{[1-{B_O}]}Y) W] = 0 \end{aligned}$$

(3.55)

Finally, thinking of the trace as a Hilbert-Schmidt inner product and the fact that the maps $_{[1-{A_O}]}$ and $_{[1-{B_O}]}$ are self-adjoint, the above conditions (3.54), (3.55) are equivalent to

$$\begin{aligned} \begin{aligned} _{[1-{A_O}]}({{\,\mathrm{Tr}\,}}_{B_IB_O}W) = 0\\ _{[1-{B_O}]}({{\,\mathrm{Tr}\,}}_{A_IA_O} W) = 0\\ _{[1-{A_O}] [1-{B_O}]}W = 0. \end{aligned} \end{aligned}$$

(3.56)

which we rewrite as

$$\begin{aligned} \begin{aligned} _{B_IB_O}W = _{A_OB_IB_O}W\\ _{A_IA_O}W = _{A_IA_OB_O}W\\ W = ~_{B_O}W +~_{A_O}W -~_{A_OB_O}W \end{aligned} \end{aligned}$$

(3.57)

Each of these conditions defines a linear subspace, whose intersection is a subspace on which all valid bipartite process matrices live, and we denote as

$$\begin{aligned} \mathcal {L}_V =\{W\in A_I\otimes A_O\otimes B_I\otimes B_O|W=L_V(W)\}, \end{aligned}$$

(3.58)

The projector onto this subspace, $L_V$ will be used in many occasions in this chapter, hence it is useful to find its expression. Firstly, we can write each of the equations in (3.57) as a projector onto a subspace

$$\begin{aligned} W = L_A(W),\ W = L_B(W),\ W = L_{AB}(W) \end{aligned}$$

(3.59)

where the projectors are

$$\begin{aligned} \begin{aligned} L_A = W - ~ _{B_IB_O}W + ~_{A_OB_IB_O}W, \\ L_B = W - ~ _{A_IA_O}W + ~_{A_IA_OB_O}W,\\ L_{AB} = ~_{B_O}W +~_{A_O}W - ~_{A_OB_O}W \end{aligned} \end{aligned}$$

(3.60)

Then the projector we are looking for, $L_V$, is the intersection of the subspaces of the above three projectors, and is given simply by their composition, i.e. $L_V(W) = L_A \circ L_B \circ L_{AB}$, which can be written as

$$\begin{aligned} L_V(W) = ~_{A_O}W + ~ _{B_O}W - ~_{A_OB_O}W - ~_{B_IB_O}W + ~_{A_OB_IB_O}W - ~_{A_IA_O}W + ~ _{A_IA_OB_O}W \end{aligned}$$

(3.61)

This completes the characterization of the set of bipartite process matrices: an operator $W\in A_I\otimes A_O\otimes B_I\otimes B_O$ is valid if and only if $W\ge 0$, ${{\,\mathrm{Tr}\,}}W = d_{A_O}d_{B_O}$ and $W=L_V(W)$.

Tripartite valid process matrices: Here we provide the characterization of the set of valid process matrices in the tripartite case and write the explicit expression of the projector onto the subspace they live. An analogous to the bipartite case argument leads to the following conclusion: 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 $W\ge 0$, ${{\,\mathrm{Tr}\,}}W = d_{A_O}d_{B_O}d_{C_O}$ and

$$\begin{aligned} \begin{aligned} W = L_X(W),\quad X = \{A,B,C\}\\ W = L_{XY}(W), \quad \{X,Y\}= \{A,B,C\}\\ W = L_{ABC}(W) \end{aligned} \end{aligned}$$

(3.62)

where the maps $L_A, L_B, L_C, L_{AB}, L_{AC}, L_{BC}, L_{ABC}$ are 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

$$\begin{aligned} \begin{aligned} L_{A}(W) = {}_{[1-(1-A_O)B_IB_OC_IC_O]}W \, , \\ L_{B}(W) = {}_{[1-(1-B_O)A_IA_OC_IC_O]}W \, , \\ L_{C}(W) = {}_{[1-(1-C_O)A_IA_OB_IB_O]}W \, , \\ L_{AB}(W) = {}_{[1-(1-A_O)(1-B_O)C_IC_O]}W \, , \\ L_{AC}(W) = {}_{[1-(1-A_O)(1-C_O)B_IB_O]}W \, , \\ L_{BC}(W) = {}_{[1-(1-B_O)(1-C_O)A_IA_O]}W \, , \\ L_{ABC}(W) = {}_{[1-(1-A_O)(1-B_O)(1-C_O)]}W \end{aligned} \end{aligned}$$

(3.63)

where we used the shorthand notation

$$\begin{aligned} {}_{[\sum _X \alpha _X X]} W = \sum _X \alpha _X \cdot {}_X W \end{aligned}$$

(3.64)

for a sum over products of subsystems X with coefficients $\alpha _X$ (and with ${}_1 W := W$).

The above constraints are equivalent to $W = L_V(W)$, where the map $L_V$ is obtained by composing the 7 maps in (3.62), and is expressed as

$$\begin{aligned} L_V(W) = {}_{\big [1 - (1 - A_O + A_I A_O) (1 - B_O + B_I B_O) (1 - C_O + C_I C_O)} {}_{+ \ A_I A_O B_I B_O C_I C_O \big ]} W \, \end{aligned}$$

(3.65)

which defines a projector onto the linear subspace

$$\begin{aligned} \mathcal {L}_V = \big \{W \in A_I \!\otimes \! A_O \!\otimes \! B_I \!\otimes \! B_O \!\otimes \! C_I \!\otimes \! C_O \, | \, W = L_V(W)\big \} \, . \end{aligned}$$

(3.66)

This completes the characterization of the valid tripartite process matrices. For the n-partite case, refer to the Appendix B of the relevant paper [1].