Quantum Channel Marginal Problem

TL;DR

This paper introduces a systematic framework for quantum channel marginal problems, connecting local and global quantum dynamics, and provides a semidefinite programming solution to determine their compatibility.

Contribution

It formulates quantum channel marginal problems, develops a semidefinite programming approach, and explores their implications for quantum incompatibility and operational advantages.

Findings

Provided a semidefinite program for compatibility testing.

Constructed channel incompatibility witnesses.

Demonstrated a gap between classical and quantum dynamical marginal problems.

Abstract

Given a set of local dynamics, are they compatible with a global dynamics? We systematically formulate these questions as quantum channel marginal problems. These problems are strongly connected to the generalization of the no-signaling conditions to quantized inputs and outputs and can be understood as a general toolkit to study notions of quantum incompatibility. In fact, they include as special cases channel broadcasting, channel extendibility, measurement compatibility, and state marginal problems. After defining the notion of compatibility between global and local dynamics, we provide a solution to the channel marginal problem that takes the form of a semidefinite program. Using this formulation, we construct channel incompatibility witnesses, discuss their operational interpretation in terms of an advantage for a state-discrimination task, prove a gap between classical and quantum dynamical marginal problems and show that the latter is irreducible to state marginal problems.