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Single-shot comparison of random quantum channels and measurements

Marcin Markiewicz, Łukasz Pawela, Zbigniew Puchała

TL;DR

The paper addresses the problem of single‑shot equality testing for two unknown quantum operations drawn from Haar‑type ensembles. It shows that a universal, parameter‑free strategy—prepare the antisymmetric two‑copy input on the channel ports and measure with the symmetric projector—achieves optimal performance across all input/output dimensions, without extra ancilla. The authors derive tight Holevo–Helstrom bounds for symmetric comparison and linear‑program solutions for asymmetric tests, extended to Haar‑random POVMs via a dephasing–channel decomposition. These results provide the most general solution to single‑shot equality testing for arbitrary quantum channels and measurements, with performance governed by the output dimension and environment size, and asymptotically approaching random guessing as the environment grows.

Abstract

In this work we provide an efficiency analysis of the problem of comparison of two randomly chosen quantum operations in the single-shot regime. We provide tight bounds for the success probability of such a protocol for arbitrary quantum channels and generalized measurements.

Single-shot comparison of random quantum channels and measurements

TL;DR

The paper addresses the problem of single‑shot equality testing for two unknown quantum operations drawn from Haar‑type ensembles. It shows that a universal, parameter‑free strategy—prepare the antisymmetric two‑copy input on the channel ports and measure with the symmetric projector—achieves optimal performance across all input/output dimensions, without extra ancilla. The authors derive tight Holevo–Helstrom bounds for symmetric comparison and linear‑program solutions for asymmetric tests, extended to Haar‑random POVMs via a dephasing–channel decomposition. These results provide the most general solution to single‑shot equality testing for arbitrary quantum channels and measurements, with performance governed by the output dimension and environment size, and asymptotically approaching random guessing as the environment grows.

Abstract

In this work we provide an efficiency analysis of the problem of comparison of two randomly chosen quantum operations in the single-shot regime. We provide tight bounds for the success probability of such a protocol for arbitrary quantum channels and generalized measurements.

Paper Structure

This paper contains 17 sections, 1 theorem, 86 equations, 4 figures.

Table of Contents

  1. Introduction and motivation
  2. Comparison of unknown channels
  3. Symmetric comparison
  4. Asymmetric comparison
  5. Comparison of unknown POVMs
  6. Symmetric comparison
  7. Asymmetric comparison
  8. Discussion
  9. Random Channels
  10. Random POVMs
  11. Polar decomposition of the Choi matrix for general channels
  12. Saturation of diamond norm for general channels
  13. Solution to the linear programming problem for asymmetric comparison of random channels
  14. Reduction to two scalars.
  15. Linear-program geometry.
  16. ...and 2 more sections

Key Result

Proposition 1

A POVM channel of the form: with effects of the form Meffects, can be represented as:

Figures (4)

  • Figure 1: A schematic representation of the single-shot symmetric comparison scheme. Note that the scheme allows for usage of an additional quantum register untouched by both channels.
  • Figure 2: Plot of the optimal value of a success probability for comparison of two random channels in a symmetric scheme as a function of (continuated) output dimension $d$, presented for three different values of the environment dimension $s$. The case $s=1$ corresponds to random unitary channels.
  • Figure 3: Plot of the value of a success probability for comparison of two random channels in a symmetric scheme as a function of (continuated) environment dimension $s$, taken in the asymptotic limit of infinite output dimension.
  • Figure 4: Plot of the optimal value of a success probability for comparison of two random POVM measurements in a symmetric scheme as a function of (continuated) output dimension $d$, presented for three different values of the environment dimension $s$. The case $s=1$ corresponds to random von Neumann measurements.

Theorems & Definitions (2)

  • Proposition 1
  • proof