Adversarial quantum channel discrimination

TL;DR

This work advances quantum hypothesis testing by formulating adversarial quantum channel discrimination, where an adversary controls inputs and potentially adapts during testing. It establishes a quantum Stein's lemma analogue: the optimal type-II error exponent under a fixed type-I constraint equals the regularized minimum output channel divergence $D^{\inf,\infty}({\cal N}\|{\cal M})$, with non-adaptive strategies already achieving the optimum and a strong converse holding in general. The authors develop chain rules for quantum divergences in this adversarial setting, prove a tightness result via amortized divergences, and provide a relative entropy accumulation theorem that extends entropy-accumulation techniques to sequences of channels, linking quantum information theory with cryptography. Computationally, the exponent is efficiently computable via semidefinite programming despite regularization, enabling practical evaluation of ultimate limits in channel discrimination and informing device-verification and cryptographic contexts.

Abstract

We introduce a new framework for quantum channel discrimination in an adversarial setting, where the tester plays against an adversary. We show that in asymmetric hypothesis testing, the optimal type-II error exponent is precisely characterized by a new notion of quantum channel divergence (termed the minimum output channel divergence). This serves as a direct analog of the quantum Stein's lemma in this new framework, and complements previous studies on ``best-case'' channel discrimination, thereby providing a complete understanding of the ultimate limits of quantum channel discrimination. Notably, the optimal error exponent can be achieved by simple non-adaptive adversarial strategies, and despite the need for regularization, it remains efficiently computable and satisfies the strong converse property in general. Furthermore, we show that entropy accumulation, a powerful tool in quantum cryptography, can be reframed as an adversarial channel discrimination problem, establishing a new connection between quantum information theory and quantum cryptography.

Figures (6)

• Figure 1: Adaptive and non-adaptive strategies for adversarial quantum channel discrimination. Here, ${\cal U}$ and ${\cal V}$ (in gray) are the Stinespring dilations of the quantum channels ${\cal N}$ and ${\cal M}$, respectively; ${\cal P}^i$ and ${\cal Q}^i$ (in red) denote the adversary's internal operations; ${\operatorname{id}}$ is the identity map; and $\{M_2, I-M_2\}$ (in blue) represents the tester's quantum measurement. (a) Adaptive strategies: the adversary accesses the environmental systems $E_i$ and uses quantum memory $R_i$ to implement adaptive strategies. (b) Non-adaptive strategies: the adversary ignores the environmental systems $E_i$ and performs no updates (i.e., identity map) between rounds.
• Figure 2: (a) Subadditivity for the channel divergence $D^{{ \rm M},\inf'}$ where ${\cal A}_{0.5, 0}$ and ${\cal A}_{p, 0.9}$ are the GAD channels and $p$ ranges from $0$ to $1$; (b) Random test for the chain rule property, where the quantum channels are chosen as ${\cal A}_{0.5, 0}$ and ${\cal A}_{0.5, 0.9}$, the quantum states are 500 randomly generated quantum states with real entries, $x_1 = D_{{ \rm M}}(\rho_R\|\sigma_R) + D_{{ \rm M}}^{\inf}({\cal A}_{0.5, 0}\|{\cal A}_{0.5, 0.9})$ and $x_2 = D_{{ \rm M}}(\rho_R\|\sigma_R) + D_{{ \rm M}}^{\inf'}({\cal A}_{0.5, 0}\|{\cal A}_{0.5, 0.9})$.
• Figure 3: Illustration of the setting of the relative entropy accumulation.
• Figure 4: Illustration of the setting of the relative entropy accumulation with Stinespring dilation.
• Figure 5: Illustration of an extended setting of adversarial quantum channel discrimination.

Theorems & Definitions (22)

• Theorem 1
• Lemma 2
• Theorem 3
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7