Are all extensibly causal processes purifiable?

Cite this problem as Problem 43.

Problem

The most general situation compatible with the assumption that the operations performed in each local laboratory are described by the quantum formalism without assuming a global causal order between the operations can be represented in the “process matrix” formalism [1]. The probability that $N$ parties $A^1, ..., A^N$ observe the outcomes a_1, ..., a_N, for a choice of operations $x_1, ..., x_N$ is a multilinear function of the corresponding CP maps $\mathcal{M}_{a_1|x_1},...,\mathcal{M}_{a_N|x_N}$ as given by

p(\mathcal{M}_{a_1|x_1},...,\mathcal{M}_{a_N|x_N})= \mbox{Tr}[(M_{a_1|x_1} \otimes ... \otimes M_{a_N|x_N}) W],

where $M_{a_1|x_1}, ..., M_{a_N|x_N}$ are the CJ representations of the CP maps and $W \in A^1_I \otimes A^1_O \otimes ... \otimes A^N_I \otimes A^N_O$ is a process matrix with $A^i_I$ and $A^i_O$ being the input and the output Hilbert spaces of party $i$. The set of process matrices is defined by requiring that probabilities are well-defined [1,2].

The condition for a bipartite probability distribution to be “causal”, i.e., not to violate any causal inequality, is that it can be decomposed into a convex combination of a probability distribution which is no-signaling from Bob to Alice ($p^{A<B}$) and a probability distribution which is no-signaling from Alice to Bob ($p^{B<A}$): $p_{causal} = \lambda p^{A<B} + (1 - \lambda) p^{B<A}$. The generalization to arbitrary number of parties can be found in [2,3]. It is known that some process matrices can violate causal inequalities[1,4], thought no physical implementation of such processes are known. The processes are called causal if they do not violate causal inequalities, and they are called extensibly causal if they remain causal even under extension with input systems in an arbitrary joint quantum state [3].

The process matrices are called pure if after extending them with a global past state from the Hilbert space $P$ and a global future state from the Hilbert space $F$, they induce a unitary transformation from the past to the future whenever unitary transformations are also applied in the local laboratories [5]. Then a process $W \in P \otimes A^1_I \otimes A^1_O \otimes ... \otimes A^N_I \otimes A^N_O \otimes F $ is called purifiable if one can recover it from a  pure process $S \in P \otimes P' \otimes A^1_I \otimes A^1_O \otimes ... \otimes A^N_I \otimes A^N_O \otimes F \otimes F'$ by inputing the state $|0\rangle$ in $P'$ and tracing out the state from $F'$

W=\mbox{Tr}_{P'F'} [S^T (|0\rangle \langle 0|^{P'} \otimes \mathbb{I}^{F'})].

Here $P'$ and $F'$ are additional input and output Hilbert spaces of $S$ and $S^T$ is transposed process $S$.

The question is whether or not any extensibly causal process is purifiable.

Background

All processes that we have observed in laboratory and in nature are causal. Since there is no obvious reasons why the existence of any of them would collide with our physical laws, one might conjecture that all extensibly causal processes are physically realizable. On the other hand, one can conjecture that all physically realizable processes are purifiable as reversibility of the transformations between quantum states is accepted in all our physical theories, and has been a central ingredient in all of the reconstructions of quantum theory to date. The questions then arises if all extensibly causal processes are purifiable. A negative answer would imply that either not  all causal processes are physical, or that purifiability is not a necessary condition for the physicality of a process.

References

[1] O. Oreshkov, F. Costa, and C. Brukner, Quantum correlations with no causal order, Nature Communications 3, 1092 (2012).

[2] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and C. Brukner, Witnessing causal nonseparability, New J. Phys. 17, 102001 (2015).

[3] O. Oreshkov and C. Giarmatzi, Causal and causally separable processes, New J. Phys. {\bf 18}, 093020, (2016).

[4] C. Branciard, M. Araújo, F. Costa, A. Feix, and C. Brukner, The simplest causal inequalities and their violation, New J. Phys. 18, 013008 (2016).

[5] M. Araújo, A. Feix, M. Navascués, and C. Brukner, A purification postulate for quantum mechanics with no causal order, Quantum 1, 10 (2017).