Mapping the space of quantum expectation values

Abstract

For a quantum system with Hilbert space $\mathcal{H}$ of dimension $N$ and a set $S$ of $n$ Hermitian operators ${\mathcal{O}}_i$, a basic question is to understand the set $E_S\subset {\mathbb{R}}^n$ of points $\vec{e}$ where $e_i=\mathrm{tr}(\rho {\mathcal{O}}_i)$ for an allowed state $\rho$. A related question is to determine whether a given set of expectation values $\vec{e}$ 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 $\vec{e}$ in the interior of $E_S$ is achieved uniquely by a state of the form $\rho (\vec{\beta })=e^{-{\sum }_i{\beta }_i{\mathcal{O}}_i}/\mathrm{tr}(e^{-{\sum }_i{\beta }_i{\mathcal{O}}_i})$ for ${\mathcal{O}}_i\in S$. In fact, the map $\vec{\beta }\to \vec{E}(\vec{\beta })=\mathrm{tr}(\vec{\mathcal{O}}\rho (\vec{\beta }))$ is a diffeomorphism from ${\mathbb{R}}^n$ to the interior of $E_S$ with symmetric, positive Jacobian; using this fact, we provide an algorithm to invert $\vec{E}(\vec{\beta })$ and thus determine a state $\rho (\vec{\beta }(\vec{e}))$ with specified expectation values $\vec{e}$ provided that these lie in $E_S$. The algorithm is based on defining a first order differential equation in the space of parameters $\vec{\beta }$ that is guaranteed to converge to $\vec{\beta }(\vec{e})$ in a precise way, with $|\vec{E}(\vec{\beta }(t))-\vec{e}|=C{e}^{-t}$.

35 pages, LaTeX