Gibbs States and the Consistency of Local Density Matrices

TL;DR

This paper proves that locally consistent quantum states can be represented as Gibbs states, extending the maximum-entropy principle to non-commuting observables, with implications for quantum state reconstruction.

Contribution

It demonstrates that consistent local density matrices can be globally represented as Gibbs states, generalizing previous results to non-commuting observables with a new proof approach.

Findings

Local density matrices consistent with a global state can be expressed as Gibbs states.

The result extends to expectation values of non-commuting observables.

Provides a new proof using properties of the partition function.

Abstract

Suppose we have an n-qubit system, and we are given a collection of local density matrices rho_1,...,rho_m, where each rho_i describes some subset of the qubits. We say that rho_1,...,rho_m are "consistent" if there exists a global state sigma (on all n qubits) whose reduced density matrices match rho_1,...,rho_m. We prove the following result: if rho_1,...,rho_m are consistent with some state sigma > 0, then they are also consistent with a state sigma' of the form sigma' = (1/Z) exp(M_1+...+M_m), where each M_i is a Hermitian matrix acting on the same qubits as rho_i, and Z is a normalizing factor. (This is known as a Gibbs state.) Actually, we show a more general result, on the consistency of a set of expectation values <T_1>,...,<T_r>, where the observables T_1,...,T_r need not commute. This result was previously proved by Jaynes (1957) in the context of the maximum-entropy principle; here we provide a somewhat different proof, using properties of the partition function.