Disentangling signalling and causal influence

TL;DR

This paper investigates the distinction between signalling and causal influence in quantum interactions, introducing functions to quantify these effects, analyzing their properties, and exploring their behavior in specific quantum gates and scenarios.

Contribution

It defines and analyzes measures of signalling and causal influence in quantum channels, demonstrating their properties and differences, especially in finite versus asymptotic regimes.

Findings

Finite gap between signalling and causal influence for CNOT gate.

Monotonicity of influence measures under tensor product, not sequential composition.

Asymptotic equivalence of signalling and causal influence in infinite parallel uses.

Abstract

The causal effects activated by a quantum interaction are studied, modelling the last one as a bipartite unitary channel. The two parties, say Alice and Bob, can use the channel to exchange messages -- i.e. to signal. On the other hand, the most general form of causal influence includes also the possibility for Alice, via a local operation on her system, to modify Bob's correlations and viceversa. The presence or absence of these two effects are equivalent, but when they both occur, they can differ in their magnitude. We study the properties of two functions that quantify the amount of signalling and causal influence conveyed by an arbitrary unitary channel. The functions are proved to be continuous and monotonically increasing with respect to the tensor product of channels. Monotonicity is instead disproved in the case of sequential composition. Signalling and causal influence are analytically computed for the quantum SWAP and CNOT gates, in the single use scenario, in the $n$-parallel uses scenario, and in the asymptotic regime. A finite gap is found between signalling and causal influence for the quantum CNOT, thus proving the existence of extra causal effects that cannot be explained in terms of communication only. However, the gap disappears in the asymptotic limit of an infinite number of parallel uses, leaving room for asymptotic equivalence between signalling and causal influence.