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 ,...,, 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.