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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 ρ\rho of low purity, \tr\rho^2\ll 1, and yielding the ensemble averaged expectation value \tr(\rho A) for any observable AA. Assuming that the given statistical ensemble ρ\rho is generated by randomly sampling pure states ψ>|\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 ψ>|\psi> and any experimentally realistic observable AA. 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

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