Statistical properties of Pauli matrices going through noisy channels

TL;DR

This paper analyzes how the triplet of Pauli matrices behaves statistically when passing through noisy quantum channels, revealing a non-commutative central limit theorem and a large deviation principle with explicit rate functions.

Contribution

It establishes a non-commutative central limit theorem for Pauli matrices under noise and derives a large deviation principle with an explicit rate function.

Findings

Convergence to a 3D Brownian motion in the limit

Explicit large deviation rate function depending on stationary state

Statistical characterization of Pauli matrices under noise

Abstract

We study the statistical properties of the triplet $(\sigma_x,\sigma_y,\sigma_z)$ of Pauli matrices going through a sequence of noisy channels, modeled by the repetition of a general, trace-preserving, completely positive map. We show a non-commutative central limit theorem for the distribution of this triplet, which shows up a 3-dimensional Brownian motion in the limit with a non-trivial covariance matrix. We also prove a large deviation principle associated to this convergence, with an explicit rate function depending on the stationary state of the noisy channel.