Hypercontractivity for Quantum Erasure Channels via Variable Multipartite Log-Sobolev Inequality

• URL: http://arxiv.org/abs/2311.14321v2
• Date: Wed, 30 Apr 2025 10:41:02 GMT
• Title: Hypercontractivity for Quantum Erasure Channels via Variable Multipartite Log-Sobolev Inequality
• Authors: Zongbo Bao, Yangjing Dong, Fengning Ou, Penghui Yao,
• Abstract summary: We prove an almost optimal hypercontractive inequality for products of quantum erasure channels.<n>This is the first tensorization-type hypercontractivity bound for quantum channels with no fixed states.
• Score: 3.6623643759959914
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove an almost optimal hypercontractive inequality for products of quantum erasure channels, generalizing the hypercontractivity for classical binary erasure channels. To our knowledge, this is the first tensorization-type hypercontractivity bound for quantum channels with no fixed states. The traditional inductive arguments for classical hypercontractivity cannot be generalized to the quantum setting due to the nature of the non-commutativity of matrices. To overcome the difficulty, we establish a novel quantum log-Sobolev inequality for Bernoulli entropy, which includes the classical log-Sobolev inequality and the quantum log-Sobolev inequality as one-partite cases. To our knowledge, its classical counterpart is also unknown prior to this work. We establish a connection between our quantum log-Sobolev inequality and the hypercontractivity bound for quantum erasure channels via a refined quantum Gross' lemma, extending the analogous connection between the quantum log-Sobolev inequality and the hypercontractivity for qubit unital channels. As an application, we prove an almost tight bound (up to a constant factor) on the classical communication complexity of two-party common randomness generation assisted with erased-noisy EPR states, generalizing the tight bound on the same task assisted with erased-noisy random strings due to Guruswami and Radhakrishnan.

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