Causal influence versus signalling for interacting quantum channels

TL;DR

This paper investigates the relationship between causal influence and signalling in quantum channels, demonstrating their differences and establishing a continuity theorem for causal effects of unitary channels.

Contribution

It provides a formal analysis of causal influence versus signalling in quantum channels and proves a continuity theorem linking small causal influence to limited signalling.

Findings

Causal influence and signalling can differ in quantum channels.

A direct computation for the Cnot gate illustrates the mismatch.

Small causal influence implies limited signalling in unitary channels.

Abstract

A causal relation between quantum agents, say Alice and Bob, is necessarily mediated by an interaction. Modelling the last one as a reversible quantum channel, an intervention of Alice can have causal influence on Bob's system, modifying correlations between Alice and Bob's systems. Causal influence between quantum systems necessarily allows for signalling. Here we prove a mismatch between causal influence and signalling via direct computation of the two quantities for the Cnot gate. Finally we show a continuity theorem for causal effects of unitary channels: a channel has small causal influence iff it allows for small signalling.