Core Concepts
The authors introduce several quantum generalizations of multivariate classical fidelities, such as the average pairwise z-fidelity, multivariate semi-definite programming fidelity, secrecy-based multivariate fidelity, and multivariate log-Euclidean fidelity. These multivariate quantum fidelities satisfy desirable properties like reduction to classical multivariate fidelities for commuting states, data processing inequality, symmetry, faithfulness, orthogonality, direct-sum property, and joint concavity.
Abstract
The paper begins by reviewing classical and quantum bivariate fidelities, including the Uhlmann and Holevo fidelities. It then introduces several multivariate classical fidelities, such as the Matusita multivariate fidelity and the average pairwise fidelity, and establishes relationships between them.
The main contribution is the introduction of quantum generalizations of these multivariate classical fidelities. The authors propose three main variants that reduce to the average pairwise fidelity for commuting states:
Average pairwise z-fidelities: These generalize the classical average pairwise fidelity using the z-fidelity for z ≥ 1/2. The Uhlmann and Holevo average pairwise fidelities are special cases.
Multivariate semi-definite programming (SDP) fidelity: This is obtained by extending the SDP formulation of the Uhlmann fidelity to multiple states.
Secrecy-based multivariate fidelity: This is inspired by an existing secrecy measure and also reduces to the average pairwise fidelity for commuting states.
The authors show that all three of these variants satisfy the desired properties for a multivariate fidelity, including reduction to classical multivariate fidelities, data processing inequality, symmetry, faithfulness, orthogonality, direct-sum property, and joint concavity. They also establish uniform continuity bounds for some of these fidelities.
Additionally, the authors introduce the multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. They show that it satisfies most of the desired properties and has an operational interpretation in terms of quantum hypothesis testing.
Finally, the authors define maximal and minimal extensions of multivariate classical fidelities and analyze their properties.