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Mapping the Space of Quantum Expectation Values by Jarov, Seraphim; Van Raamsdonk, Mark is a scholarly article available to read on EtoBox.

What is Mapping the Space of Quantum Expectation Values about?

For a quantum system with Hilbert space ${\cal H}$ of dimension $N$ and a set $S$ of $n$ Hermitian operators ${\cal O}_i$, a basic question is to understand the set $E_S \subset \mathbb{R}^n$ of points $\vec{e}$ where $e_i = {\rm tr}(\rho {\cal O}_i)$ 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 $\rho({\vec{\beta}}) = e^{-\sum_i \beta_i {\cal O}_i}/{\rm tr}(e^{-\sum_i \beta_i {\cal O}_i})$ for ${\cal O}_i \in S$. In fact, the map $\vec{\beta} \to \vec{E}(\vec{\beta}) = {\rm tr}(\vec{\cal O} \rho({\vec{\beta}}))$ is a diffeomorphism from $\mathbb{R}^n$ to the interior of $E_S$ with symmetric, positive Jacobian; using this fact, we provide an algorithm

Author

Jarov, Seraphim; Van Raamsdonk, Mark

Published

2023

Language

EN