Maximum entropy methods for quantum state compatibility problems

TL;DR

This paper introduces a maximum entropy approach to efficiently solve quantum state compatibility problems, including the quantum marginal problem, using fewer parameters and providing witnesses for incompatibility.

Contribution

It applies MaxEnt to quantum compatibility, reducing parameter complexity and enabling effective hybrid quantum-classical algorithms for large systems.

Findings

MaxEnt method efficiently solves quantum compatibility with fewer parameters.

The approach provides witnesses for incompatible measurement results.

The method has a clear geometric interpretation and is computationally effective.

Abstract

Inferring a quantum system from incomplete information is a common problem in many aspects of quantum information science and applications, where the principle of maximum entropy (MaxEnt) plays an important role. The quantum state compatibility problem asks whether there exists a density matrix $\rho$ compatible with some given measurement results. Such a compatibility problem can be naturally formulated as a semidefinite programming (SDP), which searches directly for the existence of a $\rho$. However, for large system dimensions, it is hard to represent $\rho$ directly, since it needs too many parameters. In this work, we apply MaxEnt to solve various quantum state compatibility problems, including the quantum marginal problem. An immediate advantage of the MaxEnt method is that it only needs to represent $\rho$ via a relatively small number of parameters, which is exactly the number of the operators measured. Furthermore, in case of incompatible measurement results, our method will further return a witness that is a supporting hyperplane of the compatible set. Our method has a clear geometric meaning and can be computed effectively with hybrid quantum-classical algorithms.