Quantum Hypothesis Testing Lemma for Deterministic Identification over Quantum Channels

2025-04-29

AI Summary

This work investigates the existence and capacity lower bounds of deterministic identification (DI) codes over quantum channels. Addressing the fundamental distinction between synchronous and asynchronous DI codes, we establish the first complete quantum hypothesis testing lemma and develop a geometric framework characterizing DI capacity via the Minkowski dimension of the output state space. By restructuring packing arguments in quantum state space, we devise a product-state construction method that tightens the DI capacity lower bound into an explicit expression depending solely on this dimension. Key contributions include: (i) the first rigorous demonstration of a capacity separation between synchronous and asynchronous DI codes; (ii) a significant improvement over existing lower bounds; (iii) the first concrete quantum channel instance for which an optimal, constructible DI code is explicitly provided; and (iv) a unified information-geometric analytical toolkit for quantum identification protocols.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

In our previous work, we presented the Hypothesis Testing Lemma, a key tool that establishes sufficient conditions for the existence of good deterministic identification (DI) codes for memoryless channels with finite output, but arbitrary input alphabets. In this work, we provide a full quantum analogue of this lemma, which shows that the existence of a DI code in the quantum setting follows from a suitable packing in a modified space of output quantum states. Specifically, we demonstrate that such a code can be constructed using product states derived from this packing. This result enables us to tighten the capacity lower bound for DI over quantum channels beyond the simultaneous decoding approach. In particular, we can now express these bounds solely in terms of the Minkowski dimension of a certain state space, giving us new insights to better understand the nature of the protocol, and the separation between simultaneous and non-simultaneous codes. We extend the discussion with a particular channel example for which we can construct an optimum code.

Problem

• Extends Hypothesis Testing Lemma to quantum channels for deterministic identification
• Constructs DI codes using product states from packed quantum output spaces
• Tightens capacity bounds via Minkowski dimension of quantum state spaces

Innovation

• Quantum analogue of Hypothesis Testing Lemma
• DI code construction using product states
• Capacity bound via Minkowski dimension