Quantum ensembles and the statistical operator: a tutorial

TL;DR

This tutorial clarifies the concept of quantum ensembles, explores their relationship with statistical operators, and discusses methods for discriminating between different ensembles, including practical applications and subtleties involved.

Contribution

It provides a detailed explanation of quantum ensembles, addresses assumptions and paradoxes, and reviews techniques for ensemble discrimination using multiparticle correlations.

Findings

Different ensembles with the same statistical operator can be distinguished using correlations.

Approximate quantum cloning plays a role in ensemble discrimination.

Practical examples demonstrate ensemble discrimination methods.

Abstract

The main purpose of this tutorial is to elucidate in details what should be meant by ensemble of states in quantum mechanics, and to properly address the problem of discriminating, exactly or approximately, two different ensembles. To this aim, we review the notion and the definition of quantum ensemble as well as its relationships with the concept of statistical operator in quantum mechanics. We point out the implicit assumptions contained in introducing a correspondence between quantum ensembles and the corresponding single-particle statistical operator, and discuss some issues arising when these assumptions are not satisfied. We review some subtleties leading to apparent paradoxes, and illustrate the role of approximate quantum cloning. In particular, we review some examples of practical interest where different (but equivalent) preparations of a quantum system, i.e. different ensembles corresponding to the same single-particle statistical operator, may be successfully discriminated exploiting multiparticle correlations, or some a priori knowledge about the number of particles in the ensemble.