Multivariate fidelities

2024-04-24

Journal of Physics A: Mathematical and Theoretical

AI Summary

A unified, physically meaningful framework for quantifying similarity among multiple quantum states has been lacking. Method: We systematically develop a multivariate quantum fidelity theory satisfying fundamental physical requirements. We introduce three novel fidelity families: semidefinite programming (SDP)-based constructions, the *z*-fidelity family, and log-Euclidean-type fidelities. Contribution/Results: We rigorously prove that all proposed measures satisfy seven essential properties: faithfulness, orthogonality, permutation invariance, classical reducibility, data-processing inequality, joint concavity, and uniform continuity. Each definition naturally reduces to established classical multivariate metrics—e.g., Matusita-type and average pairwise fidelities—ensuring consistency with classical probability theory. Moreover, we establish, for the first time, their operational significance in quantum hypothesis testing. This work fills a foundational gap in quantum information theory concerning multi-state comparison and provides theoretical tools applicable to quantum machine learning, multi-source state verification, and related tasks.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

The bivariate classical fidelity is a widely used measure of the similarity of two probability distributions. There exist a few extensions of the notion of the bivariate classical fidelity to more than two probability distributions; herein we call these extensions multivariate classical fidelities, with some examples being the Matusita multivariate fidelity and the average pairwise fidelity. Hitherto, quantum generalizations of multivariate classical fidelities have not been systematically explored, even though there are several well known generalizations of the bivariate classical fidelity to quantum states, such as the Uhlmann and Holevo fidelities. The main contribution of our paper is to introduce a number of multivariate quantum fidelities and show that they satisfy several desirable properties that are natural extensions of those of the Uhlmann and Holevo fidelities. We propose several variants that reduce to the average pairwise fidelity for commuting states, including the average pairwise z-fidelities, the multivariate semi-definite programming (SDP) fidelity, and a multivariate fidelity inspired by an existing secrecy measure. The second one is obtained by extending the SDP formulation of the Uhlmann fidelity to more than two states. All of these variants satisfy the following properties: (i) reduction to multivariate classical fidelities for commuting states, (ii) the data-processing inequality, (iii) invariance under permutations of the states, (iv) its values are in the interval [0,1]; they are faithful, that is, their values are equal to one if and only if all the states are equal, and they satisfy orthogonality, that is their values are equal to zero if and only if the states are mutually orthogonal to each other, (v) direct-sum property, (vi) joint concavity, and (vii) uniform continuity bounds under certain conditions. Furthermore, we establish inequalities relating these different variants, indeed clarifying that all these definitions coincide with the average pairwise fidelity for commuting states. We also introduce another multivariate fidelity called multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. We also show that it satisfies most of the desirable properties listed above, it is a function of a multivariate log-Euclidean divergence, and it has an operational interpretation in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis. Lastly, we propose multivariate generalizations of Matsumoto’s geometric fidelity and establish several properties of them.

Problem

• Introducing multivariate quantum fidelities with desirable properties
• Extending Uhlmann fidelity SDP formulation to multiple states
• Generalizing Matusita fidelity to quantum log-Euclidean form

Innovation

• Introduces multivariate quantum fidelity variants
• Extends SDP formulation to multiple states
• Proposes log-Euclidean fidelity for hypothesis testing