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.

翻译：在集中式设置中，量子假设检验中错误概率间的权衡现已得到充分理解，但分布式设置下的情况尚不明确。本文研究一个分布式二元假设检验问题，旨在推断两个远程方共享的双向量子态，其中一方以（渐近）零速率向检验者通信，而另一方则以零速率或更高速率通信。我们的主要贡献是，当备择假设下的态为其边缘态的乘积时，推导出该问题斯坦因指数的可高效计算的单字符公式。在证明结果的对偶方向时，我们利用了一种基于量子马尔可夫半群逆超收缩性与夹逼法相结合的新技术。对于第一类错误概率趋于零的一般情形，我们证明当至少一方以经典零速率通信时，斯坦因指数由涉及正则化测量相对熵的多字符表达式给出，该表达式在一类二元结果可分测量子类上最大化。当备择假设下的态与零假设下边缘乘积态对易且具有更大支撑时，我们证明该指数可表征为对一类局部二元结果投影测量上正则化测量相对熵的极大极小优化。尽管该表达式在完全经典情形下退化为单字符形式，但我们进一步证明，对于经典-量子态，通常不会以相同方式退化。极大极小表征的对偶证明依赖于经典爆破引理到边缘可对易双向量子态的推广，这一结果本身可能具有独立价值。