Gibbs States and the Consistency of Local Density Matrices

TitleGibbs States and the Consistency of Local Density Matrices
Publication TypeJournal Article
Year of Publication2006
AuthorsLiu, Y-K
Date Published2006/03/02

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
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 ,...,, 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.