Exploring Quantum Average-Case Distances: proofs, properties, and examples

TL;DR

This paper introduces and analyzes average-case quantum distances based on random circuits, proving their properties, operational interpretations, and providing examples, thereby advancing the understanding of quantum statistical distinguishability.

Contribution

It establishes the properties and approximations of average-case quantum distances using random circuits and provides analytical examples demonstrating their practical relevance.

Findings

Average-case distances can be approximated by degree two polynomials for certain random circuits.

These distances have properties like subadditivity, convexity, and data-processing inequalities.

The maximal ratio between worst-case and average-case distances is bounded by dimension-dependent powers.

Abstract

In this work, we perform an in-depth study of recently introduced average-case quantum distances. The average-case distances approximate the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantify $\textit{average-case}$ statistical distinguishability via random circuits. We prove that once a family of random circuits forms an $\delta$-approximate $4$-design, with $\delta=o(d^{-8})$, then the average-case distances can be approximated by simple explicit functions that can be expressed via degree two polynomials in objects of interest. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity, joint convexity, and (restricted) data-processing inequalities. Notably, all of the distances utilize the Hilbert-Schmidt norm which provides an operational interpretation it did not possess before. We also derive upper bounds on the maximal ratio between worst-case and average-case distances. For each dimension $d$ this ratio is at most $d^{\frac{1}{2}},\ d, \ d^{\frac{3}{2}}$ for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.