Quantum marginal problem and incompatibility

TL;DR

This paper explores the deep connection between quantum incompatibility phenomena, unifying the quantum marginal problem across states and channels, and provides solutions and criteria for specific cases and measures.

Contribution

It establishes a many-to-one correspondence between quantum marginal problems and incompatibility of channels, including measurement incompatibility, and develops new criteria and hierarchies for quantifying quantum memory.

Findings

Solved the marginal problem for Gaussian and Bell diagonal states.

Derived entropic criteria for channel compatibility.

Developed a hierarchy of semi-definite programs for quantum memory quantification.

Abstract

One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.