Appearance of causality in process matrices when performing fixed-basis measurements for two parties

TL;DR

This paper demonstrates that when two-party process matrices are measured in a fixed basis, they can be effectively described by a causally ordered process matrix, revealing how fixed-basis measurements influence perceived causality in quantum processes.

Contribution

It shows that fixed-basis measurements on process matrices produce an effective process matrix compatible with definite causality, extending previous results to include input measurements and output re-preparations.

Findings

Effective process matrices are indistinguishable from original ones under fixed-basis measurements.

Such effective matrices are compatible with definite causal order.

The results extend previous proofs to include input measurements and output re-preparations.

Abstract

The recently developed framework for quantum theory with no global causal order allows for quantum processes in which operations in local laboratories are neither causally ordered nor in a probabilistic mixture of definite causal orders. The causal relation between the laboratories is described by the process matrix. We show that, if the inputs of the laboratories are measured in a fixed basis, one can introduce an effective process matrix which is operationally indistinguishable from the original one. This effective process matrix can be obtained by applying the von Neumann- Lu\"ders update rule for nonselective measurements to the original process matrix and in the bipartite case it is compatible with a definite causal order. The latter extends the original Oreshkov et al. proof where one considers that both the measurement of the input and the re-preparation of the output are performed in a fixed basis.