Efficiently Compressible Density Operators via Entropy Maximization

Serena Giorgio,Paulo Mateus

doi:10.3390/proceedings2019012039

Abstract

We address the problem of efficiently and effectively compress density operators (DOs), by providing an efficient procedure for learning the most likely DO, given a chosen set of partial information. We explore, in the context of quantum information theory, the generalisation of the maximum entropy estimator for DOs, when the direct dependencies between the subsystems are provided. As a preliminary analysis, we restrict the problem to tripartite systems when two marginals are known. When the marginals are compatible with the existence of a quantum Markov chain (QMC) we show that there exists a recovery procedure for the maximum entropy estimator, and moreover, that for these states many well-known classical results follow. Furthermore, we notice that, contrary to the classical case, two marginals, compatible with some tripartite state, might not be compatible with a QMC. Finally, we provide a new characterisation of quantum conditional independence in light of maximum entropy updating. At this level, all the Hilbert spaces are considered finite dimensional.

Highlights

Quantum tomography allows to reconstruct a unique quantum state by performing a complete set of measurements on multiple copies of a quantum system
We say that a set of marginal quantum states ρ AB, ρ BC are compatible if there exists a tripartite state ρ ABC such that Tr A (ρ ABC ) = ρ BC and
Addressing the problem of inferring the tripartite density operators (DOs) that maximises the von Neumann entropy given two bipartite marginals, we find out that the classical results of algebraic recoverability can be extended to the set of DO addressed as quantum Markov chain

Summary

Introduction

Quantum tomography allows to reconstruct a unique quantum state by performing a complete set of measurements on multiple copies of a quantum system. Given a multipartite quantum system, the problem of inferring the DO that maximises the von Neumann entropy from its set of bipartite marginals is expected to be efficiently solvable, at least when it coincides with a purely classical tree.

Results

Conclusion