Witnesses of causal nonseparability: an introduction and a few case studies

It was recently realised that quantum theory allows for so-called causally nonseparable processes, which are incompatible with any definite causal order. This was first suggested on a rather abstract level by the formalism of process matrices, an extension of the quantum formalism which only assumes that quantum theory holds locally in some observers’ laboratories, but does not impose a global causal structure; it was then shown, on a more practical level, that the quantum switch—a new, already implementable resource for quantum computation that goes beyond causally ordered circuits—provided precisely a physical example of a causally nonseparable process. To demonstrate that a given process is causally nonseparable, the concept of witnesses of causal nonseparability was introduced. Here we present a shorter introduction to this concept, and concentrate on some explicit examples—by considering in particular different noise models for the quantum switch—to show how to construct and use such witnesses in practice.

with W A≺B and W B≺A two positive semidefinite matrices satisyfing Note that if W is already assumed to be a valid process matrix in L V (hence from Eq. (3a), it satisfies in particular [ , then assuming that W A≺B satisfies (S2a) automatically implies that W B≺A = W − W A≺B satisfies (S2d); similarly, assuming that W B≺A satisfies (S2c) automatically implies that W A≺B satisfies (S2b). Hence, to determine whether W ∈ L V is causally separable, it is enough to check whether it can be decomposed as in (S1) with W A≺B ≥ 0 and W B≺A ≥ 0 satisfying (S2a) and (S2c), resp. Defining the linear subspaces the cone of (nonnormalised) causally separable process matrices can then be characterised as 1 This characterization is indeed equivalent to that of Eq. (42) given in the Methods section. (Note furthermore that in Ref., 1 instead of using the Minkowski sum notation, we wrote equivalently W sep = conv(W A≺B ∪ W B≺A ), where conv denotes the convex hull.)

A.1.2 Tripartite case with
With similar arguments as in the bipartite case above, we find that in the particular tripartite case where Charlie has a trivial outgoing system (d C O = 1), the cone of (nonnormalised) causally separable process matrices can be characterised as This characterization is indeed equivalent to that of Eq. (46).

A.2 S and S V : Witnesses of causal nonseparability
As explained in the main text, the set of witnesses of causal nonseparability is simply the dual cone of W sep . It can be characterised by using the previous descriptions of W sep , and making use of the following duality relations for two nonempty closed convex cones K 1 , K 2 : 2

A.2.1 Bipartite case
Using (S5) and (S9), noting that the dual cone of a linear subspace L is its orthogonal complement L ⊥ and that the cone P of positive semidefinite matrices is self-dual, one can write, in the bipartite case, , which can be written as Combining this with (S10), we find that Furthermore, S = S P + S ⊥ thus characterised is in L V if and only if S = L V (S P + S ⊥ ) = L V (S P ). Hence, we also simply have (S13)

A.2.2 Tripartite case with d C O = 1
One could follow a similar reasoning as above to characterise S in the tripartite case with d C O = 1, starting from the characterisation of W sep given by Eq. (S7). However, because the map -and ultimately of S-as much as before.
It is thus somewhat simpler here to start directly from the characterisation of W sep given by Eq. (S6). With W A≺B≺C = P ∩ L A≺B≺C and W B≺A≺C = P ∩ L B≺A≺C , we get, using again the relations (S9), where L A≺B≺C and L B≺A≺C are the projectors onto the linear subspaces L A≺B≺C and L B≺A≺C , which are Restricting the witnesses to the subspace L V , one can then write by referring to the previous characterisation (S14) of S, and with the projector L V onto L V now given by (S17)

2/6 B Witnesses for the quantum switch
In this second part we give explicit witnesses of the causal nonseparability of the quantum switch. Although the results reported in the main text do not depend on the initial state |ψ of the target qubit, the specific form of the witnesses does; in the following we fix it to be |ψ = |0 . For ease of notations, we will provide the various witnesses in the general form for some terms S i and coefficients s i to be specified below. To verify that S is a valid witness, we will provide the explicit decomposition of S = S P ABC + S ⊥ ABC as in (S14) in the form for some terms T i and coefficients t i to be specified as well. This will allow the reader to check that L A≺B≺C (S ⊥ ABC ) = 0 and S P ABC ≥ 0, as required by (S14). Due to the symmetries of the quantum switch and its witnesses, the second decomposition S = S P BAC + S ⊥ BAC in (S14) can then be obtained as where F A↔B is the map that exchanges the roles of Alice and Bob, defined as

B.1 Optimal witness with respect to white noise
By solving the dual SDP problem (55) for W = W switch with CVX, we obtained numerically the witness S switch of the form (S18), with The operator S ⊥ ABC is given here by (S19), with T 1 = 1Z111 , T 2 = 111Z1 , T 3 = ZZ111 , T 4 = 11ZZ1 , T 5 = Z11Z1 , T 6 = 1Z1Z1 , T 7 = ZZ1Z1 , T 8 = Z1ZZ1 , T 9 = 1ZZZ1 , T 10 = ZZZZ1 , With the witness S switch thus defined, we find tr[S switch · W switch ] = −r * switch −1.576 < 0, as reported in the main text. Note that in order to measure the witness S switch , one can decompose each of its terms in a similar way as we did in Eqs. (27)-(28) of the main text for S η 1 ,η 2 in terms of CP maps, implement them and combine the statistics in the appropriate way.

B.2 A family of witnesses for
Instead, we provide here a family of witnesses S(v), parametrised by v. Namely, S(v) and the corresponding S ⊥ ABC (v) are given in the forms (S18) and (S19), with the terms S i and T j defined again as in Eqs. (S22) and (S24), now with the coefficients and t 6 = t 7 = t 9 = t 10 = 1 , More explicitly, this gives (when written in the order A I B I A O B O C I for ease of notation) One finds which give negative values-thus proving that W depol switch (v) and W deph switch (v) are causally nonseparable-for all v > 0. Supplementary Figures S1 and S2 represent the witnesses S(v), for various values of v, in the two-dimensional slices of the space of process matrices containing W switch , W depol , 1 • and W switch , W deph , 1 • , respectively. Note that the witnesses S(v) are not optimal to detect causal nonseparability, as they are not tangent to the set of causally separable processes. E.g., for v = 1, we find tr[S(1) · W switch ] = −1, allowing one to prove causal nonseparability of the noisy quantum switch W 1 • switch (v) (33) only down to v > 1/2 (to be compared to v * switch 0.3882 for the optimal witness). We could not find an analytical expression for optimal witnesses; nevertheless, the witnesses are good enough for our goal, which was to prove that W depol switch (v) and W deph switch (v) are causally nonseparable for all v > 0.

B.3 Restricting Alice and Bob's operations to unitaries
Here we show how to impose that Alice and Bob's operations are restricted to unitaries, and provide the witness thus obtained.

B.3.1 Constraints on the CJ representation of a unitary
Following the convention of, 3 the Choi-Jamiołkowski representation of a unitary operation U : where 1 is the identity operator on H X I , |1 ≡ |1 X I X I = j |j X I ⊗ |j X I ∈ H X I ⊗ H X I is a (nonnormalised) maximally entangled state, {|j X I } is an orthonormal basis of H X I , and T denotes matrix transposition in that basis. Note first that U is a completely positive and trace-preserving map; the condition tr X O M X I X O U = 1 X I that its CJ matrix satisfies, cf Eq. (1), can be written as (S31) Let us furthermore calculate: from which it also follows that (S33)

B.3.2 An explicit witness made of unitaries for Alice and Bob
Solving the SDP problem (20) with CVX-in its more explicit form (55)-after replacing the constraint S ∈ S V (or S = L V (S) in the more explicit form) by Eq. (41), we obtained numerically the witnessS and the corresponding operatorS ⊥ ABC of the forms (S18)-(S19), now with