Gibbs States and the Consistency of Local Density Matrices

2006/03/02 by Yi-Kai Liu, Liu, Yi-Kai · 1 citation
Computer Science · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Physical sciences #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.quant-ph/0603012

openalex publication_date 2006/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose we have an n-qubit system, and we are given a collection of local density matrices rho₁,...,rho_m, where each rho_i describes some subset of the qubits. We say that rho₁,...,rho_m are "consistent" if there exists a global state sigma (on all n qubits) whose reduced density matrices match rho₁,...,rho_m. We prove the following result: if rho₁,...,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(M₁+...+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 ,...,, where the observables T₁,...,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.

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