Distributed Quantum Hypothesis Testing under Zero-rate Communication Constraints

TL;DR

This work addresses distributed quantum hypothesis testing under zero-rate communication constraints, focusing on discriminating a bipartite state rho_AB from tilde_rho_AB with encoders at Alice and Bob and a tester Charlie. The authors derive a single-letter Stein exponent when tilde_rho_AB is a product, and a multi-letter max-min expression involving regularized measured relative entropy for the general case; they also show sufficiency of binary-outcome local measurements and prove a quantum blowing-up lemma that yields the strong converse. The results reveal a dichotomy between classical and quantum zero-rate communication: the exponent can be strictly larger (even infinite) with quantum links, and single-letterization fails in general for CQ states. The work provides a rigorous framework, extending classical distributed hypothesis testing insights into the quantum regime, and introduces techniques with potential broader applicability in quantum information theory and distributed sensing.

Abstract

The trade-offs between error probabilities in quantum hypothesis testing are by now well-understood in the centralized setting, but much less is known for distributed settings. Here, we study a distributed binary hypothesis testing problem to infer a bipartite quantum state shared between two remote parties, where one of these parties communicates to the tester at zero-rate, while the other party communicates to the tester at zero-rate or higher. As our main contribution, we derive an efficiently computable single-letter formula for the Stein's exponent of this problem, when the state under the alternative is product. For the general case, we show that the Stein's exponent when (at least) one of the parties communicates classically at zero-rate is given by a multi-letter expression involving max-min optimization of regularized measured relative entropy. While this becomes single-letter for the fully classical case, we further prove that this already does not happen in the same way for classical-quantum states in general. As a key tool for proving the converse direction of our results, we develop a quantum version of the blowing-up lemma which may be of independent interest.

Figures (1)

• Figure 1: Distributed quantum hypothesis testing under a zero-rate noiseless communication constraint. At least, one of the parties (Alice here) communicates at zero-rate to the tester (Charlie). The other party (Bob here) communicates to Charlie at zero-rate or higher.

Theorems & Definitions (16)

• Remark 1: Super-additivity of max-min measured relative entropy
• Theorem 2: Zero-rate Stein's exponent
• Remark 3: Sufficiency of binary outcome projective measures
• Lemma 4: Quantum blowing-up lemma
• Corollary 5: Stein's exponent with same marginals
• Remark 6: Stein's exponent for isotropic and Werner states
• Corollary 7: Stein's exponent for CQ states
• Proposition 8: Stein's exponent and single-letter formula
• Example 9: Violation of single-letter upper bound
• Corollary 10: Stein's exponent without support condition