Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the optimal error exponents for classical and quantum antidistinguishability

Hemant K. Mishra, Michael Nussbaum, Mark M. Wilde

TL;DR

The paper analyzes the asymptotics of antidistinguishability, showing that classically the optimal error exponent equals the multivariate Chernoff divergence, which gives an operational interpretation to this divergence for distinguishing several probability measures. In the quantum setting, it derives a hierarchy of bounds: a lower bound given by the best pairwise quantum Chernoff divergences, an SDP-based single-letter upper bound, and additional bounds in terms of minimal and maximal multivariate quantum Chernoff divergences, with additivity results and regularization. While the exact quantum optimal exponent remains open, the work provides a robust framework connecting antidistinguishability to multivariate divergences and the extended max-relative entropy, offering computable tools and guiding future investigation. The findings have implications for foundational questions in quantum mechanics, quantum state exclusion, and related information-theoretic tasks by giving precise asymptotic rates and a path toward a complete theory of quantum antidistinguishability.

Abstract

The concept of antidistinguishability of quantum states has been studied to investigate foundational questions in quantum mechanics. It is also called quantum state elimination, because the goal of such a protocol is to guess which state, among finitely many chosen at random, the system is not prepared in (that is, it can be thought of as the first step in a process of elimination). Antidistinguishability has been used to investigate the reality of quantum states, ruling out $ψ$-epistemic ontological models of quantum mechanics [Pusey et al., Nat. Phys., 8(6):475-478, 2012]. Thus, due to the established importance of antidistinguishability in quantum mechanics, exploring it further is warranted. In this paper, we provide a comprehensive study of the optimal error exponent -- the rate at which the optimal error probability vanishes to zero asymptotically -- for classical and quantum antidistinguishability. We derive an exact expression for the optimal error exponent in the classical case and show that it is given by the multivariate classical Chernoff divergence. Our work thus provides this divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. For the quantum case, we provide several bounds on the optimal error exponent: a lower bound given by the best pairwise Chernoff divergence of the states, a single-letter semi-definite programming upper bound, and lower and upper bounds in terms of minimal and maximal multivariate quantum Chernoff divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.

On the optimal error exponents for classical and quantum antidistinguishability

TL;DR

The paper analyzes the asymptotics of antidistinguishability, showing that classically the optimal error exponent equals the multivariate Chernoff divergence, which gives an operational interpretation to this divergence for distinguishing several probability measures. In the quantum setting, it derives a hierarchy of bounds: a lower bound given by the best pairwise quantum Chernoff divergences, an SDP-based single-letter upper bound, and additional bounds in terms of minimal and maximal multivariate quantum Chernoff divergences, with additivity results and regularization. While the exact quantum optimal exponent remains open, the work provides a robust framework connecting antidistinguishability to multivariate divergences and the extended max-relative entropy, offering computable tools and guiding future investigation. The findings have implications for foundational questions in quantum mechanics, quantum state exclusion, and related information-theoretic tasks by giving precise asymptotic rates and a path toward a complete theory of quantum antidistinguishability.

Abstract

The concept of antidistinguishability of quantum states has been studied to investigate foundational questions in quantum mechanics. It is also called quantum state elimination, because the goal of such a protocol is to guess which state, among finitely many chosen at random, the system is not prepared in (that is, it can be thought of as the first step in a process of elimination). Antidistinguishability has been used to investigate the reality of quantum states, ruling out -epistemic ontological models of quantum mechanics [Pusey et al., Nat. Phys., 8(6):475-478, 2012]. Thus, due to the established importance of antidistinguishability in quantum mechanics, exploring it further is warranted. In this paper, we provide a comprehensive study of the optimal error exponent -- the rate at which the optimal error probability vanishes to zero asymptotically -- for classical and quantum antidistinguishability. We derive an exact expression for the optimal error exponent in the classical case and show that it is given by the multivariate classical Chernoff divergence. Our work thus provides this divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. For the quantum case, we provide several bounds on the optimal error exponent: a lower bound given by the best pairwise Chernoff divergence of the states, a single-letter semi-definite programming upper bound, and lower and upper bounds in terms of minimal and maximal multivariate quantum Chernoff divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.
Paper Structure (30 sections, 23 theorems, 244 equations)

This paper contains 30 sections, 23 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Contributions
  3. Literature review
  4. Paper organization
  5. Mathematical background
  6. Antidistinguishability of probability measures
  7. Multivariate classical Chernoff divergence
  8. Quantum states, channels, and measurements
  9. Antidistinguishability of quantum states
  10. Quantum Chernoff divergence
  11. Multivariate Chernoff divergence as the optimal error exponent for classical antidistinguishability
  12. Multivariate Chernoff divergence versus pairwise Chernoff divergences
  13. Achievable error exponent for quantum antidistinguishability
  14. One-shot case
  15. Asymptotic case
  16. ...and 15 more sections

Key Result

Theorem 6

Consider an ensemble $\mathcal{E}_{\operatorname{cl}}=\{(\eta_i, P_i): i \in [r]\}$ of mutually absolutely continuous probability measures on a measurable space $(\Omega, \mathcal{A})$. The optimal error exponent for antidistinguishing the probability measures is given by their multivariate Chernoff where, recalling eq:unit_simplex, eq:hellinger-trans-def, and eq:classical-CH-divergence, the multi

Theorems & Definitions (57)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 47 more