Abstract
For a quantum system with Hilbert space H of dimension N and a set S of n Hermitian operators O_i, a basic question is to understand the set E_S ⊂ Rⁿ of points e where e_i = tr(ρ O_i) for an allowed state ρ. A related question is to determine whether a given set of expectation values e lies in E_S and in this case to describe the most general state with these expectation values. In this paper, we describe various ways to characterize E_S, reviewing basic results that are perhaps not widely known and adding new ones. One important result (originally due to E. Wichmann) is that for a set S of linearly independent traceless operators, every set of expectation values e in the interior of E_S is achieved uniquely by a state of the form ρ(β) = e^-Σ_i β_i O_i/ tr(e^-Σ_i β_i O_i) for O_i ∈ S. In fact, the map β → E(β) = tr( O ρ(β)) is a diffeomorphism from Rⁿ to the interior of E_S with symmetric, positive Jacobian; using this fact, we provide an algorithm to invert E(β) and thus determine a state ρ(β(e)) with specified expectation values e provided that these lie in E_S. The algorithm is based on defining a first order differential equation in the space of parameters β that is guaranteed to converge to β(e) in a precise way, with |E(β(t)) - e| = C e^-t.