Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adversarial Hypothesis Testing for Quantum Channels

Masahito Hayashi, Hao-Chung Cheng, Li Gao

TL;DR

This work develops a systematic framework for adversarial hypothesis testing across quantum-quantum and classical-quantum channels, where an adversary (Alice) selects inputs to minimize Bob's discrimination capability. It distinguishes two receiver settings (Bob informed vs non-informed) and analyzes both i.i.d. and general input regimes, deriving Stein exponents through channel divergences and their regularizations. For QQ channels, i.i.d. inputs yield a gap between informed and non-informed exponents, while general inputs collapse to a common regularized exponent; EB channels with rank-one measurements admit a single-letter characterization. For CQ channels, informing Bob leads to a single-letter exponent given by the minimal input-specific divergence, whereas non-informed settings yield a convex-hull-like infimum, with the two cases potentially diverging due to input-coherence effects. Overall, the results reveal a nuanced separation between CQ and QQ behavior in adversarial channel discrimination, and show that EB reductions do not always capture CQ phenomena.

Abstract

This paper presents a systematic study of adversarial hypothesis testing for both quantum-quantum (QQ) and classical-quantum (CQ) channels. Unlike conventional channel discrimination, we consider a framework where the sender, Alice, selects the channel input adversarially to minimize Bob's distinguishability. We analyze this problem across four settings based on whether Alice employs i.i.d. or general inputs and whether the receiver, Bob, is informed of the specific input choice (allowing his measurement to depend on the input). We characterize the Stein exponents for each setting and reveal a striking distinction in behavior: for QQ channels with i.i.d. inputs, Bob's knowledge of the input significantly enhances distinguishability, yet this advantage vanishes when general inputs are permitted. In contrast, for CQ channels, Bob being informed provides a consistent advantage over the corresponding entanglement-breaking channels for both i.i.d. and general inputs. These results demonstrate a unique phenomenon in adversarial hypothesis testing where the CQ channel does not merely behave as a special case of the QQ channel.

Adversarial Hypothesis Testing for Quantum Channels

TL;DR

This work develops a systematic framework for adversarial hypothesis testing across quantum-quantum and classical-quantum channels, where an adversary (Alice) selects inputs to minimize Bob's discrimination capability. It distinguishes two receiver settings (Bob informed vs non-informed) and analyzes both i.i.d. and general input regimes, deriving Stein exponents through channel divergences and their regularizations. For QQ channels, i.i.d. inputs yield a gap between informed and non-informed exponents, while general inputs collapse to a common regularized exponent; EB channels with rank-one measurements admit a single-letter characterization. For CQ channels, informing Bob leads to a single-letter exponent given by the minimal input-specific divergence, whereas non-informed settings yield a convex-hull-like infimum, with the two cases potentially diverging due to input-coherence effects. Overall, the results reveal a nuanced separation between CQ and QQ behavior in adversarial channel discrimination, and show that EB reductions do not always capture CQ phenomena.

Abstract

This paper presents a systematic study of adversarial hypothesis testing for both quantum-quantum (QQ) and classical-quantum (CQ) channels. Unlike conventional channel discrimination, we consider a framework where the sender, Alice, selects the channel input adversarially to minimize Bob's distinguishability. We analyze this problem across four settings based on whether Alice employs i.i.d. or general inputs and whether the receiver, Bob, is informed of the specific input choice (allowing his measurement to depend on the input). We characterize the Stein exponents for each setting and reveal a striking distinction in behavior: for QQ channels with i.i.d. inputs, Bob's knowledge of the input significantly enhances distinguishability, yet this advantage vanishes when general inputs are permitted. In contrast, for CQ channels, Bob being informed provides a consistent advantage over the corresponding entanglement-breaking channels for both i.i.d. and general inputs. These results demonstrate a unique phenomenon in adversarial hypothesis testing where the CQ channel does not merely behave as a special case of the QQ channel.
Paper Structure (10 sections, 8 theorems, 113 equations, 3 tables)

This paper contains 10 sections, 8 theorems, 113 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Adversarial Hypothesis Testing for QQ channels
  3. Preliminary and Results
  4. Proof of Theorem \ref{['thm:iid']}: i.i.d. inputs
  5. Proof of Theorem \ref{['thm:adversarial']}: general inputs
  6. EB channels with rank-one projective measurements
  7. Adversarial Hypothesis Testing for CQ Channels
  8. Setting I: Bob informed
  9. Setting II: Bob non-informed
  10. Discussions

Key Result

Theorem 1

For any two quantum channels ${\mathcal{N}}_1, {\mathcal{N}}_2$ and $\varepsilon\in(0,1)$

Theorems & Definitions (17)

  • Theorem 1: Adversarial Stein's exponent with i.i.d. inputs
  • Example 2
  • Theorem 3: Adversarial Stein's exponent with general inputs
  • Lemma 4: AM
  • proof
  • proof : Proof of Theorem \ref{['thm:iid']}
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['thm:adversarial']}
  • Theorem 6
  • ...and 7 more