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arXiv:2606.09939

Quantum Fidelity on Krein and S-spaces

Morgan Jones

quant-phmath.FASubmitted Jun 7, 2026• Announced Jun 9, 2026• 0 citations
v1 · Jun 7, 2026v2 · Jun 30, 2026
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Abstract

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 $M_n(\mathbb{C})$. The notion of quantum states in this setting has already started to be considered. We will define an analogous notion of measurement for so-called $J$-states and use it to show that a notion of fidelity holds in the Krein setting. We will also show that there exists an analogous result to the Fuchs-Caves measurement holds in the Krein setting. We will then will extend this definition of fidelity to so-called $U$-quantum states on $S$-spaces. We will demonstrate that the analogous geometric motivation holds in the Krein and $S$-space setting, as holds for quantum fidelity and geometric means of operators.

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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). The notion of quantum states in this setting has already started to be considered. We will define an analogous notion of measurement for so-called JJJ-states and use it to show that a notion of fidelity holds in the Krein setting. We will also show that there exists an analogous result to the Fuchs-Caves measurement holds in the Krein setting. We will then will extend this definition of fidelity to so-called UUU-quantum states on SSS-spaces. We will demonstrate that the analogous geometric motivation holds in the Krein and SSS-space setting, as holds for quantum fidelity and geometric means of operators.
Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.
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.
Fidelity is the standard measure for quantifying the similarity between two quantum states. It is equal to the square of the minimum Bhattacharyya coefficient between the probability distributions induced by quantum measurements on the two states. Though established for over thirty years, the structure of fidelity-optimal quantum measurements remains unclear when the two density operators are singular (not invertible). Here we address this gap, with a focus on minimal optimal measurements, which admit no nontrivial coarse graining that is still optimal. We show that there exists either a unique minimal optimal measurement or infinitely many inequivalent choices. Moreover, the first case holds if and only if the two density operators satisfy a weak commutativity condition. In addition, we provide a complete characterization of all minimal optimal measurements when one state is pure, leveraging geometric insights from the Bloch-sphere representation. The connections with quantum incompatibility, operator pencils, and geometric means are highlighted.
We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (“generalized transition probability”) of Uhlmann, is topologically equivalent to the trace-norm distance.
This dissertation undertakes the study of quantum delity, a distinguishability measure in the context of quantum mechanics, from the operator algebraic viewpoint. The notion of delity provides a quantitative measure of how close one state of a quantum system is to another state. High delity occurs when the two states are very close to each other. Evidently, this concept is closely related to a metric on the quantum states which is known as the Bures metric. In this thesis, delity and the Bures metric have been studied in the context of (i) unital C -algebras that possess a faithful positive trace functional and (ii) semi nite von Neumann algebras. In addition, these notions have been analysed in the matrix algebras in an e ort to relate to the quantum information theory literature.
For Hilbert spaces H⊆L2(R)\mathcal H\subseteq L^2(\mathbb R)H⊆L2(R) we consider the convex sets D+(H)\mathcal D_+(\mathcal H)D+​(H) of Wigner-positive states (WPS), i.e.~density matrices over H\mathcal HH with non-negative Wigner function. We investigate the topological structure of these sets, namely concerning closure, compactness, interior and boundary (in a relative topology induced by the trace norm). We also study their geometric structure and construct minimal sets of states that generate D+(H)\mathcal D_+(\mathcal H)D+​(H) through convex combinations. If H\mathcal HH is finite-dimensional, the existence of such sets follows from a central result in convex analysis, namely the Krein-Milman theorem. In the infinite-dimensional case H=L2(R)\mathcal H=L^2(\mathbb R)H=L2(R) this is not so, due to lack of compactness of the set D+(H)\mathcal D_+(\mathcal H)D+​(H). Nevertheless, we prove that a Krein-Milman theorem holds in this case, which allows us to extend most of the results concerning the sets of generators to the infinite-dimensional setting. Finally, we study the relation between the finite and infinite-dimensional sets of WPS, and prove that the former provide a hierarchy of closed subsets, which are also proper faces of the latter. These results provide a basis for an operational characterisation of the extreme points of the sets of WPS, which we undertake in a companion paper. Our work offers a unified perspective on the topological and geometric properties of the sets of WPS in finite and infinite dimensions, along with explicit constructions of minimal sets of generators.