On the complexity of finding the maximum entropy compatible quantum state

TL;DR

This paper investigates the computational complexity of identifying maximum entropy quantum states compatible with given marginals, revealing QSZK-completeness and providing efficient algorithms for quantum Markov chains and trees.

Contribution

It establishes the QSZK-completeness of entropy comparison for compatible quantum states and develops polynomial-time algorithms for quantum Markov chains and trees.

Findings

Entropy comparison is QSZK-complete for 3-chains.

Efficient quantum circuits exist for maximum entropy states in quantum Markov chains.

Chow-Liu algorithm extension for quantum states.

Abstract

Herein we study the problem of recovering a density operator from a set of compatible marginals, motivated from limitations of physical observations. Given that the set of compatible density operators is not singular, we adopt Jaynes' principle and wish to characterize a compatible density operator with maximum entropy. We first show that comparing the entropy of compatible density operators is QSZK-complete, even for the simplest case of 3-chains. Then, we focus on the particular case of quantum Markov chains and trees and establish that for these cases, there exists a quantum polynomial circuit that constructs the maximum entropy compatible density operator. Finally, we extend the Chow-Liu algorithm to the same subclass of quantum states.