Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Modern Optics

Fidelity for Mixed Quantum States

Richard Jozsa

Published Dec 1, 1994• 1,898 citations
Publisher

Abstract

We propose a definition of fidelity for mixed quantum states in terms of Uhlmann's ‘transition probability’ formula F(ϱ1, ϱ2) = {trace [(√ϱ1ϱ2 × √ϱ1)1/2]}2 and give new elementary proofs of its essential properties.
DOI
10.1080/09500349414552171

Related papers

Efficient fidelity estimation: alternative derivation and related applications

Diego S Starke, Marcos L. W. Basso, Jonas Maziero

Abstract In [Phys. Rev. A 107 012427 (2023)], Baldwin and Jones prove that Uhlmann–Jozsa’s fidelity between two quantum states ρ and σ, i.e., F ( ρ , σ ) ≔ Tr ρ σ ρ 2 , can be written in a simplified form as F ( ρ , σ ) = Tr ρ σ 2 . In this article, we give an alternative proof of this result, using a function power series expansion and the properties of the trace function. Our approach not only reinforces the validity of the simplified expression but also facilitates the exploration of novel dissimilarity functions for quantum states and more complex trace functions of density operators.

arXiv:2312.12438•Communications in Theoretical Physics•Jun 10, 2024•3 citations

The fidelity of density operators in an operator-algebraic framework

Douglas Farenick, Samuel Jaques, Mizanur Rahaman

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.

arXiv:1602.08177•Journal of Mathematical Physics•Oct 1, 2016•6 citations

Operational distance and fidelity for quantum channels

V. P. Belavkin, Giacomo Mauro D’Ariano, Maxim Raginsky

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.

arXiv:quant-ph/0408159•Journal of Mathematical Physics•May 20, 2005•50 citations

Uhlmann’s Theorem for Relative Entropies

Giulia Mazzola, David Sutter, Renato Renner

Uhlmann’s theorem states that, for any two quantum states ρABand σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their reduced states ρAand σA. In this work, we generalize Uhlmann’s theorem to α-R´enyi relative entropies for α ∈ [1/2, ∞], a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to α = 1/2, α = 1, and α = ∞, respectively.

arXiv:2502.01749•IEEE Transactions on Information Theory•Jul 23, 2025•2 citations

Uhlmann’s Theorem for Measured Divergences

Kun Fang, Hamza Fawzi, Omar Fawzi

Uhlmann’s theorem is a cornerstone of quantum information theory, stating that for any quantum state ρABand any state σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their marginals ρAand σA. This property underpins many results and applications in quantum information science. In this work, we generalize Uhlmann’s theorem to a broad class of measuredf-divergences, including the measured α-Rényi divergences for all α ≥ 0. The well-known Uhlmann’s theorem for the fidelity corresponds to the special case α = 1/2. Since most commonly used quantum Rényi divergences, including the Petz and sandwiched Rényi divergences, cannot satisfy this property (except for degenerate cases), this fundamentally distinguishes measuredf-divergences from other quantum divergences and highlights their unique mathematical structure.

arXiv:2502.07745•IEEE Transactions on Information Theory•Dec 29, 2025•2 citations

A limit formula for the quantum fidelity

Gaetana Spedalieri, Christian Weedbrook, Stefano Pirandola

Quantum fidelity is a central tool in quantum information, quantifying how much two quantum states are similar. Here we propose a limit formula for the quantum fidelity between a mixed state and a pure state. As an example of an application, we apply this formula to the case of multimode Gaussian states, achieving a simple expression in terms of their first and second-order statistical moments.

arXiv:1204.2473•Journal of Physics A Mathematical and Theoretical•Dec 12, 2012•30 citations
Abstract In [Phys. Rev. A 107 012427 (2023)], Baldwin and Jones prove that Uhlmann–Jozsa’s fidelity between two quantum states ρ and σ, i.e., F ( ρ , σ ) ≔ Tr ρ σ ρ 2 , can be written in a simplified form as F ( ρ , σ ) = Tr ρ σ 2 . In this article, we give an alternative proof of this result, using a function power series expansion and the properties of the trace function. Our approach not only reinforces the validity of the simplified expression but also facilitates the exploration of novel dissimilarity functions for quantum states and more complex trace functions of density 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.
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.
Uhlmann’s theorem is a cornerstone of quantum information theory, stating that for any quantum state ρABand any state σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their marginals ρAand σA. This property underpins many results and applications in quantum information science. In this work, we generalize Uhlmann’s theorem to a broad class of measuredf-divergences, including the measured α-Rényi divergences for all α ≥ 0. The well-known Uhlmann’s theorem for the fidelity corresponds to the special case α = 1/2. Since most commonly used quantum Rényi divergences, including the Petz and sandwiched Rényi divergences, cannot satisfy this property (except for degenerate cases), this fundamentally distinguishes measuredf-divergences from other quantum divergences and highlights their unique mathematical structure.