Extensible causality and process purification
- Field
- Topic
Problem
Is every finite-dimensional extensibly causal process matrix purifiable? Call an \(N\)-laboratory process matrix \(W\) extensibly causal if \(W\otimes\rho_R\) generates only causal correlations for every ancillary input system \(R\), every joint ancillary state \(\rho_R\), and every choice of local instruments. Call \(W\) purifiable if there exist auxiliary global past and future systems, a fixed pure state on the auxiliary past, and a pure process \(S\) such that inserting the fixed state into \(S\) and discarding the auxiliary future yields \(W\). If \(\mathrm{EC}_N\) and \(\mathrm{Pur}_N\) denote the corresponding classes, the question is whether
holds for every \(N\) and every choice of finite local dimensions. In particular, does the inclusion in Eq. (1) hold for two laboratories?
Source
Araújo, Feix, Navascués, and Brukner explicitly identify the possible inclusion of extensibly causal processes in the purifiable class as open [AFN+17].
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Extensible causality rules out activation of noncausal correlations by entangled ancillary inputs. Ordinary causality is insufficient: there are causal processes that become noncausal after such an extension [OG16].
Necessary algebraic conditions for purifiability exclude several process matrices that violate causal inequalities, but remain inconclusive for the noisy processes that motivated Eq. (1) [AFN+17].
For two laboratories, every purifiable process is extensibly causal, so \(\mathrm{Pur}_2\subseteq\mathrm{EC}_2\) [YQS+21]. This is the converse of the open implication in Eq. (1); it neither proves the inclusion nor establishes equality.
Comment
Araújo, Feix, Navascués, and Brukner explicitly identify Eq. (1) as an unresolved possibility and give the noisy process built from their Eq. (37) as a concrete test case. The open direction is extensible causality implying purifiability, not the proved two-laboratory converse and not causal separability.