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arXiv:2605.17167

A Weighted Spectral Quantum Fidelity

Công Trình Lê, Vu The Khoi, Ho Minh Toan, Trung Hoa Dinh

Published May 16, 2026• 0 citations
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Abstract

We introduce and study a one-parameter family of fidelity-type quantities based on the weighted spectral geometric mean, which we call the \emph{weighted spectral fidelity} \( \mathsf{F}_t^{\mathrm{spec}}(ρ,σ):=\Tr\!\big[ρ(ρ^{-1}\sharpσ)^{2t}\big],\ t\in[0,1]. \) This family interpolates smoothly between the trivial overlap ($t=0,1$) and the Uhlmann (root) fidelity at $t=\tfrac12$, and it is distinct from the sandwiched Rényi family except at this midpoint. We establish core structural features-unitary invariance, tensor stabilization and multiplicativity, flip symmetry, endpoint behavior, and a orthogonality criterion. We further show explicit \emph{violations of DPI} for generic $t\neq\tfrac12$. For concavity in the state variables we obtain concavity in each variable separately. Closed forms are obtained for pure states and for qubits in Bloch coordinates. We also extend the first Fuchs--van de Graaf inequality to $\mathsf{F}_t^{\mathrm{spec}}$ for all $t\in[0,1]$, while the second inequality fails away from the midpoint.

DOI
10.48550/arxiv.2605.17167

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Uhlmann's fidelity function is one of the most widely used similarity measures in quantum theory. One definition of this function is that it is the minimum classical fidelity associated with a quantum-to-classical measurement procedure of two quantum states. In 2010, Matsumoto introduced another fidelity function which is dual to Uhlmann's in the sense that it is the maximimum classical fidelity associated with a classical-to-quantum preparation procedure for two quantum states. Matsumoto's fidelity can also be defined using the well-established notion definition of the matrix geometric mean. In this work, we examine Matsumoto's fidelity through the lens of semidefinite programming to give simple proofs that it possesses many desirable properties for a similarity measure, including monotonicity under quantum channels, joint concavity, and unitary invariance. Finally, we provide a geometric interpretation of this fidelity in terms of the Riemannian space of positive definite matrices, and show how this picture can be useful in understanding some of its peculiar properties.
We use nearly parallel pure states to characterize positive linear functionals φφφ on Mn\mathbb{M}_nMn​ as positive multiples of the trace if and only if φ(A♮B)≤φ(A)φ(B)φ(A \natural B) \leq \sqrt{φ(A) φ(B)}φ(A♮B)≤φ(A)φ(B)​ for all positive definite matrices AAA and BBB. Here A♮B=(A−1#B)1/2A(A−1#B)1/2A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}A♮B=(A−1#B)1/2A(A−1#B)1/2 represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality φ(A♮B)≤φ((A+B)/2)φ(A \natural B) \leq φ((A+B)/2)φ(A♮B)≤φ((A+B)/2) for all positive definite matrices AAA and BBB. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.
We introduce a family of fidelities, termed generalized fidelity, which are based on the Riemannian geometry of the Bures-Wasserstein manifold. We show that this family of fidelities generalizes standard quantum fidelities such as Uhlmann-, Holevo-, and Matsumoto-fidelity and demonstrate that it satisfies analogous celebrated properties. The generalized fidelity naturally arises from a generalized Bures distance, the natural distance obtained by linearizing the Bures-Wasserstein manifold. We prove various invariance and covariance properties of generalized fidelity as the point of linearization moves along geodesic-related paths. We also provide a Block-matrix characterization and prove an Uhlmann-like theorem, as well as provide further extensions to the multivariate setting and to quantum Rényi divergences, generalizing Petz-, Sandwiched-, Reverse sandwiched-, and Geometric-Rényi divergences of order α.
The fidelity-based smooth min-relative entropy is a distinguishability measure that has appeared in a variety of contexts in prior work on quantum information, including resource theories like thermodynamics and coherence. Here we provide a comprehensive study of this quantity. First we prove that it satisfies several basic properties, including the dataprocessing inequality. We also establish connections between the fidelity-based smooth min-relative entropy and other widely used information-theoretic quantities, including smooth min-relative entropy and smooth sandwiched Rényi relative entropy, of which the sandwiched Rényi relative entropy and smooth max-relative entropy are special cases. After that, we use these connections to establish the second-order asymptotics of the fidelity-based smooth min-relative entropy and all smooth sandwiched Rényi relative entropies, finding that the first-order term is the quantum relative entropy and the second-order term involves the quantum relative entropy variance. Utilizing the properties derived, we also show how the fidelity-based smooth min-relative entropy provides one-shot bounds for operational tasks in general resource theories in which the target state is mixed, with a particular example being randomness distillation. The above observations then lead to second-order expansions of the upper bounds on distillable randomness, as well as the precise second-order asymptotics of the distillable randomness of particular classical–quantum states. Finally, we establish semi-definite programs for smooth maxrelative entropy and smooth conditional min-entropy, as well as a bilinear program for the fidelity-based smooth min-relative entropy, which we subsequently use to explore the tightness of a bound relating the last to the first.
The notion of fidelity for quantum states is a measure of how much two states overlap. In the matrix formalism of quantum mechanics, states are represented by density operators, i.e., positive semi-definite matrices with trace equal to 1 in a complex Euclidean space Mn(C)M_n(\mathbb{C})Mn​(C). Felipe-Sosa and Felipe (2022) introduced the notion of quantum states on certain Krein spaces with indefinite metric induced by a fundamental symmetry JJJ, calling these JJJ-states. We define an analogous notion of measurement for JJJ-states to the regular quantum theory and use it to show that a notion of fidelity holds in the Krein setting. We also show that an analogous result to the Fuchs-Caves measurement holds in this setting. Following the developments of Bag, Rohilla, and Trivedi (2024), we then extend this definition of fidelity to UUU-quantum states on SSS-spaces. We demonstrate that the analogous geometric motivation holds in the Krein and SSS-space setting, as holds for quantum fidelity and geometric means of operators.