Concentration of measure for quantum states with a fixed expectation value

TL;DR

This paper studies the typical properties of quantum states with a fixed expectation value of an observable, demonstrating measure concentration phenomena and providing analytical tools for expectation estimation and sampling.

Contribution

It introduces a new measure concentration result for quantum states with fixed expectation values and develops methods for expectation estimation and state sampling.

Findings

Quantum states with fixed expectation values exhibit measure concentration.

A new analytical method for estimating expectation values is proposed.

A generalized sampling method for these quantum states is developed.

Abstract

Given some observable H of a finite-dimensional quantum system, we investigate the typical properties of random quantum state vectors that have a fixed expectation value with respect to H. Under some some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.