Joint probability distributions for projection probabilities of random orthonormal states

TL;DR

This paper derives analytic expressions for joint probability densities of measurement outcomes in quantum systems modeled by random orthogonal or unitary matrices, linking quantum chaos, random matrix theory, and mesoscopic conductance phenomena.

Contribution

It provides new analytic formulas for joint probabilities of measurement outcomes in quantum states modeled by random matrices, connecting quantum chaos and conductance fluctuations.

Findings

Derived explicit joint probability density functions for measurement outcomes.

Connected random matrix theory with universal conductance fluctuations.

Applicable to quantum chaos and mesoscopic physics scenarios.

Abstract

A finite dimensional quantum system for which the quantum chaos conjecture applies has eigenstates, which show the same statistical properties than the column vectors of random orthogonal or unitary matrices. Here, we consider the different probabilities for obtaining a specific outcome in a projective measurement, provided the system is in one of its eigenstates. We then give analytic expressions for the joint probability density for these probabilities, with respect to the ensemble of random matrices. In the case of the unitary group, our results can be applied, also, to the phenomenon of universal conductance fluctuations, where the same mathematical quantities describe partial conductances in a two-terminal mesoscopic scattering problem with a finite number of modes in each terminal.