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

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 $$ 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.Comment: accepted for publication in J. Stat. Phy