Distributed Quantum Hypothesis Testing under Zero-rate Communication Constraints

TL;DR

This paper investigates distributed quantum hypothesis testing with zero-rate communication constraints, deriving new formulas for error exponents and introducing novel quantum information techniques.

Contribution

It provides the first single-letter formula for the Stein's exponent in distributed quantum hypothesis testing under zero-rate communication.

Findings

Derived a single-letter formula for the Stein's exponent with product alternative states.

Established a multi-letter expression involving regularized measured relative entropy for general cases.

Proved a max-min characterization of the exponent for states with specific commutation and support properties.

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 (asymptotic) 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 the product of their marginals. For proving the converse direction of our result, we utilize a novel technique based on reverse hypercontractivity of a quantum markov semigroup combined with the pinching method. For the general case with vanishing type I error probability, 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 regularized measured relative entropy maximized over a sub-class of binary outcome separable measurements. When the state under the alternative commutes with the product of marginals under the null and has a larger support, we show that the exponent is characterized as a max-min optimization of regularized measured relative entropy over a class of local binary outcome projective measurements. While this expression 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. The converse proof of the max-min characterization relies on an extension of the classical blowing-up lemma to bipartite quantum states whose marginals commute, which could be of independent interest.