Gibbs States and the Consistency of Local Density Matrices

Suppose we have an n-qubit system, and we are given a collection of local density matrices ρ1, . . . , ρm, where each ρi describes some subset of the qubits. We say that ρ1, . . . , ρm are “consistent” if there exists a global state σ (on all n qubits) whose reduced density matrices match ρ1, . . . , ρm. We prove the following result: if ρ1, . . . , ρm are consistent with some state σ ≻ 0, then they are also consistent with a state σ of the form σ = (1/Z) exp(M1+· · ·+Mm), where each Mi is a Hermitian matrix acting on the same qubits as ρi, 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.