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, $\trρ^2\ll 1$, and yielding the ensemble averaged expectation value $\tr(ρA)$ 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 $\tr(ρA)$ 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 $|ψ>\in\hr_+$ are realized with equal probability and all other $|ψ> \in\hr$ are excluded.

arXiv commentaccepted for publication in J. Stat. Phys

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Cited by in corpus (21)

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• Origin of the Canonical Ensemble: Thermalization with Decoherence
• Thermal Pure Quantum States of Many-Particle Systems
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• Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
• Generalized quantum microcanonical ensemble from random matrix product states
• Eigenstate thermalization scaling in approaching the classical limit
• Deep thermalization in Gaussian continuous-variable quantum systems
• Maximum Geometric Quantum Entropy
• Canonical Typicality For Other Ensembles Than Micro-Canonical
• Typical pure nonequilibrium steady states and irreversibility for quantum transports
• Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces
• Spin and the Thermal Equilibrium Distribution of Wave Functions
• Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles
• A classical interpretation of the Scrooge distribution