Inequivalence of Pure State Ensembles for Open Quantum Systems: The Preferred Ensembles Are Those That Are Physically Realizable

Abstract

An open quantum system in steady state ${\hat{\rho }}_{\mathrm{ss}}$ can be represented by a weighted ensemble of pure states ${\hat{\rho }}_{\mathrm{ss}}=\Sigma k^{}{\wp }_k|{\psi }_k〉〈{\psi }_k|$ in infinitely many ways. A physically realizable (PR) ensemble is one for which some continuous measurement of the environment will collapse the system into a pure state $|\psi \left(t\right)〉$, stochastically evolving such that the proportion of time for which $|\psi \left(t\right)〉=|{\psi }_k〉$ equals ${\wp }_k$. Some, but not all, ensembles are PR. This constitutes the preferred ensemble fact. We present the necessary and sufficient conditions for a given ensemble to be PR, and illustrate the method by showing that the coherent state ensemble is not PR for an atom laser.