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Intersection and union of subspaces with applications to communication over authenticated classical-quantum channels and composite hypothesis testing

Naqueeb Ahmad Warsi, Ayanava Dasgupta

TL;DR

This work introduces a quantum analogue of intersection/union for events via a projection-based construction grounded in Jordan's lemma, enabling simultaneous handling of multiple quantum hypotheses without commutativity assumptions. The authors apply this framework to two principal problems: (i) communication over authenticated classical-quantum channels and the resulting authentication capacity C_auth = max_{P_X} I[X; B_{(s0)}], and (ii) asymmetric composite hypothesis testing across several quantum state collections, providing single-letter exponents and finite-set trade-offs through generalized intersection projectors. A key methodological contribution is the construction of Pi^\star to approximate set operations with controlled trace bounds, enabling channel discrimination with minimal state disturbance and robust hypothesis testing bounds. The results advance quantum joint-typicality-style reasoning in one-shot and finite-blocklength regimes and highlight a trade-off between exact single-letter exponents and additive correction terms arising from noncommutativity. Overall, the paper offers a unified projection-based toolkit for quantum analogues of classical set operations with concrete operational implications in authentication and hypothesis testing.

Abstract

In information theory, we often use intersection and union of the typical sets to analyze various communication problems. However, in the quantum setting it is not very clear how to construct a measurement which behaves analogously to intersection and union of the typical sets. In this work, we construct a projection operator which behaves very similarly to intersection and union of the typical sets. Our construction relies on the Jordan's lemma. Using this construction we study the problem of communication over authenticated classical-quantum channels and derive its capacity. As another application of our construction, we also study the problem of quantum asymmetric composite hypothesis testing.

Intersection and union of subspaces with applications to communication over authenticated classical-quantum channels and composite hypothesis testing

TL;DR

This work introduces a quantum analogue of intersection/union for events via a projection-based construction grounded in Jordan's lemma, enabling simultaneous handling of multiple quantum hypotheses without commutativity assumptions. The authors apply this framework to two principal problems: (i) communication over authenticated classical-quantum channels and the resulting authentication capacity C_auth = max_{P_X} I[X; B_{(s0)}], and (ii) asymmetric composite hypothesis testing across several quantum state collections, providing single-letter exponents and finite-set trade-offs through generalized intersection projectors. A key methodological contribution is the construction of Pi^\star to approximate set operations with controlled trace bounds, enabling channel discrimination with minimal state disturbance and robust hypothesis testing bounds. The results advance quantum joint-typicality-style reasoning in one-shot and finite-blocklength regimes and highlight a trade-off between exact single-letter exponents and additive correction terms arising from noncommutativity. Overall, the paper offers a unified projection-based toolkit for quantum analogues of classical set operations with concrete operational implications in authentication and hypothesis testing.

Abstract

In information theory, we often use intersection and union of the typical sets to analyze various communication problems. However, in the quantum setting it is not very clear how to construct a measurement which behaves analogously to intersection and union of the typical sets. In this work, we construct a projection operator which behaves very similarly to intersection and union of the typical sets. Our construction relies on the Jordan's lemma. Using this construction we study the problem of communication over authenticated classical-quantum channels and derive its capacity. As another application of our construction, we also study the problem of quantum asymmetric composite hypothesis testing.
Paper Structure (22 sections, 14 theorems, 136 equations, 2 figures)

This paper contains 22 sections, 14 theorems, 136 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Comparison with previous works
  3. Notations and Facts
  4. Proof of Lemma \ref{['lbound']}
  5. Construction of the Intersection Projector $\Pi^{\star}$
  6. Proof for the properties of $\Pi^{\star}$
  7. Communication Over Authenticated Classical-Quantum channel
  8. Proof of Lemma \ref{['mr']}
  9. Proof for the properties of $\Pi^\star$
  10. Asymmetric Composite Hypothesis Testing
  11. Asymmetric Composite Hypothesis Testing $($The case of $\{\rho_s\}_{s=1}^{|\mathcal{S}|}$ and $\sigma )$
  12. Asymmetric Composite Hypothesis Testing $($The case of $\rho$ and $\{\sigma_s\}_{s=1}^{|\mathcal{S}|})$
  13. Asymmetric Composite Hypothesis Testing $($The case of $\{\rho_s\}_{s=1}^{|\mathcal{S}_1|}$ and $\{\sigma_t\}_{t=1}^{|\mathcal{S}_2|})$
  14. Comparison with Berta2017OnCQ
  15. Conclusion and Acknowledgements
  16. ...and 7 more sections

Key Result

Lemma 1

(Upper-bound) Let $\rho, \sigma_1$ and $\sigma_2 \in \mathcal{D}(\mathcal{H})$ such that $\hbox{supp}(\rho) \subseteq \hbox{supp}(\sigma_1)$ and $\hbox{supp}(\rho) \subseteq \hbox{supp}(\sigma_2)$. Further, for ${k_1}, k_2 >0,$ let $\Pi_1, \Pi_2$ be such that $\Pi_1:=\{\sigma_1 \preceq 2^{-{k_1}}\rh

Figures (2)

  • Figure 1: A parallel construction of the intersection $\Pi^{\star}$ of the collection $\left\{\Pi_1,\Pi_2,\cdots,\Pi_{2^t}\right\}$ of projectors mentioned in the statement of Lemma $4$.
  • Figure 2: A sequential construction of the intersection $\Pi^{\star}$ of the collection $\left\{\Pi_1,\Pi_2,\cdots,\Pi_{2^t}\right\}$ of projectors mentioned in the statement of Lemma $4$.

Theorems & Definitions (30)

  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • proof
  • Lemma 4
  • Remark 1
  • Lemma 5: The case of more than two projectors
  • proof
  • Lemma 6
  • ...and 20 more