On the distribution of the mean energy in the unitary orbit of quantum states

TL;DR

This paper analyzes the distribution of mean energy in quantum states reachable via cyclic processes, showing it closely follows a Gaussian distribution in high-dimensional systems under general conditions.

Contribution

It proves that the mean energy distribution in the unitary orbit of quantum states is approximately Gaussian, providing bounds on moments and characteristic functions.

Findings

Distribution is close to Gaussian for large system dimensions

Bounds established for moments and characteristic functions

Discrepancy with normal distribution diminishes as dimension increases

Abstract

Given a closed quantum system, the states that can be reached with a cyclic process are those with the same spectrum as the initial state. Here we prove that, under a very general assumption on the Hamiltonian, the distribution of the mean extractable work is very close to a gaussian with respect to the Haar measure. We derive bounds for both the moments of the distribution of the mean energy of the state and for its characteristic function, showing that the discrepancy with the normal distribution is increasingly suppressed for large dimensions of the system Hilbert space.