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Semi-definite optimization of the measured relative entropies of quantum states and channels

Zixin Huang, Mark M. Wilde

TL;DR

This work delivers efficient semidefinite-programming methods for computing measured quantum relative entropies, both Rényi and standard, for states and channels. By leveraging variational formulas together with hypographs/epigraphs of the weighted geometric mean and the operator connection of the logarithm, the authors obtain SDPs that output not only optimal entropy values but also optimal measurements and input states, enabling practical hybrid quantum–classical hypothesis testing strategies. The results scale favorably with system dimensions (e.g., $2d\times2d$ for states and $2d_A d_B\times2d_A d_B$ for channels) and accommodate energy constraints, making them applicable to near-term experimental tasks. This bridges theoretical entropy measures with implementable discrimination protocols, offering concrete numerical tools for designing quantum hypothesis tests under realistic constraints.

Abstract

The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum--classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semi-definite programming, by making use of variational formulas for the measured relative entropies of states and semi-definite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semi-definite programs output the optimal values of the measured relative entropies of states and channels, but they also provide numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.

Semi-definite optimization of the measured relative entropies of quantum states and channels

TL;DR

This work delivers efficient semidefinite-programming methods for computing measured quantum relative entropies, both Rényi and standard, for states and channels. By leveraging variational formulas together with hypographs/epigraphs of the weighted geometric mean and the operator connection of the logarithm, the authors obtain SDPs that output not only optimal entropy values but also optimal measurements and input states, enabling practical hybrid quantum–classical hypothesis testing strategies. The results scale favorably with system dimensions (e.g., for states and for channels) and accommodate energy constraints, making them applicable to near-term experimental tasks. This bridges theoretical entropy measures with implementable discrimination protocols, offering concrete numerical tools for designing quantum hypothesis tests under realistic constraints.

Abstract

The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum--classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semi-definite programming, by making use of variational formulas for the measured relative entropies of states and semi-definite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semi-definite programs output the optimal values of the measured relative entropies of states and channels, but they also provide numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.
Paper Structure (25 sections, 13 theorems, 109 equations)

This paper contains 25 sections, 13 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of results
  4. Organization of the paper
  5. Notation and Preliminaries
  6. Weighted geometric mean and its properties
  7. Hypograph and epigraph of the weighted geometric mean
  8. Operator connections
  9. Measured Rényi relative entropy of states
  10. Definition and basic properties
  11. Variational formulas for measured Rényi relative entropy of states
  12. Optimizing the measured Rényi relative entropy of states
  13. Measured relative entropy of states
  14. Definition and basic properties
  15. Variational formulas for the measured relative entropy of states
  16. ...and 10 more sections

Key Result

Proposition 1

It suffices to optimize $D_{\alpha}^{M}(\rho \Vert\sigma)$ over rank-one POVMs; i.e., where each $\varphi_{x}$ is a rank-one operator such that $\sum_{x\in \mathcal{X}}\varphi_{x}=I$.

Theorems & Definitions (18)

  • Definition 1: Measured Rényi relative entropy
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Definition 2: Measured relative entropy
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Lemma 8
  • ...and 8 more