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

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.