Gibbs States and the Consistency of Local Density Matrices

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Suppose we have an n-qubit system, and we are given a collection of local density matrices rho₁,...,rhoₘ, where each rhoᵢ describes some subset of the qubits. We say that rho₁,...,rhoₘ are "consistent" if there exists a global state sigma (on all n qubits) whose reduced density matrices match rho₁,...,rhoₘ. We prove the following result: if rho₁,...,rhoₘ are consistent with some state sigma > 0, then they are also consistent with a state sigma' of the form sigma' = (1/Z) exp(M₁+...+Mₘ), where each Mᵢ is a Hermitian matrix acting on the same qubits as rhoᵢ, 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 <T₁>,...,<Tᵣ>, where the observables T₁,...,Tᵣ 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.