Semi-definite optimization of the measured relative entropies of quantum states and channels

2024-06-27

arXiv.org

AI Summary

This work addresses the efficient computation of the measured relative entropy—a quantity with operational significance in quantum hypothesis testing and recent relevance to hybrid quantum-classical hardware. We establish, for the first time, exact semidefinite programming (SDP) characterizations of the measured relative entropy for both quantum states and quantum channels, unifying optimal value computation with explicit construction of optimal measurement strategies. Methodologically, we combine SDP representations of weighted geometric means of operators, convex optimization formulations of logarithmic operator connections, and variational principles—yielding polynomial-time exact computability. Our framework delivers not only tight theoretical bounds but also directly implementable optimal measurements, thereby significantly enhancing both the design efficiency and experimental feasibility of quantum hypothesis testing protocols.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum-classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semi-definite programming, by making use of variational formulas for the measured relative entropies of states and semi-definite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semi-definite programs output the optimal values of the measured relative entropies of states and channels, but they also provide numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.

Problem

• Efficient calculation of measured relative entropies via semi-definite programming
• Hybrid quantum-classical strategies for hypothesis testing tasks
• Optimal strategies characterization for quantum hypothesis testing

Innovation

• Semi-definite programming for quantum entropies
• Hybrid quantum-classical strategies optimization
• Optimal hypothesis testing protocols design