Typicality of pure states randomly sampled according to the Gaussian adjusted projected measure

TL;DR

This paper demonstrates that for mixed quantum states generated by a specific random sampling measure, most pure states exhibit expectation values close to the ensemble average, highlighting a typicality property in high-dimensional Hilbert spaces.

Contribution

It establishes a typicality result for pure states sampled according to the Gaussian adjusted projected measure, connecting ensemble averages with most individual pure states in high-dimensional spaces.

Findings

Most pure states have expectation values close to the ensemble average.

The typicality holds for high-dimensional subspaces with uniform sampling.

The result applies to experimentally realistic observables.

Abstract

Consider a mixed quantum mechanical state, describing a statistical ensemble in terms of an arbitrary density operator $\rho$ of low purity, $\tr\rho^2\ll 1$, and yielding the ensemble averaged expectation value $\tr(\rho A)$ for any observable $A$. Assuming that the given statistical ensemble $\rho$ is generated by randomly sampling pure states $|\psi>$ according to the corresponding so-called Gaussian adjusted projected measure $[$Goldstein et al., J. Stat. Phys. 125, 1197 (2006)$]$, the expectation value $<\psi|A|\psi>$ is shown to be extremely close to the ensemble average $\tr(\rho A)$ for the overwhelming majority of pure states $|\psi>$ and any experimentally realistic observable $A$. In particular, such a `typicality' property holds whenever the Hilbert space $\hr$ of the system contains a high dimensional subspace $\hr_+\subset\hr$ with the property that all $|\psi>\in\hr_+$ are realized with equal probability and all other $|\psi> \in\hr$ are excluded.