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

arXiv:2609.09753

Generalized Fidelity, the Data Processing Inequality, and Convexity

Haobo Ma, Reza Rajaei, Wenlin Zhang

quant-phSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations• Revised Oct 20, 2026 (v3)
v1 · Sep 9, 2026v2 · Oct 2, 2026v3 · Oct 20, 2026
arXivHTMLPDF

Abstract

In this paper, we show that the generalized Bures--Wasserstein distance induced by the generalized fidelity fails the data processing inequality (DPI) in every dimension $n\geq2$. In dimension two, we derive three sufficient conditions guaranteeing the DPI by using an explicit formula for the real part of the generalized fidelity. Furthermore, we find a necessary and sufficient condition under which the generalized fidelity reduces to the Uhlmann fidelity. We also show that neither the generalized Bures--Wasserstein distance nor its square is convex in one state, even with the other state and the reference fixed, in every dimension $n\geq2$.

Comments
Added qubit counterexamples to both convexity and DPI

Related papers

Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance

A. Afham, Christopher Ferrie

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.

arXiv:2410.04937•Journal of Mathematical Physics•Aug 1, 2025•0 citations

Geometric conditions for saturating the data processing inequality

Sam Cree, Jonathan Sorce

Abstract The data processing inequality (DPI) is a scalar inequality satisfied by distinguishability measures on density matrices. For some distinguishability measures, saturation of the scalar DPI implies an operator equation relating the arguments of the measure. These results are typically derived using functional analytic techniques. In a complementary approach, we use geometric techniques to derive a formula that gives an operator equation from DPI saturation for any distinguishability measure; moreover, for a broad class of distinguishability measures, the derived operator equation is sufficient to imply saturation as well. Our operator equation coincides with known results for the sandwiched Rényi relative entropies, and gives new results for α - z Rényi relative entropies and a family of of quantum f -divergences, which we compute explicitly.

arXiv:2011.03473•Journal of Physics A Mathematical and Theoretical•Feb 17, 2022•2 citations

Counterexamples to the Tensorization Property of Strong Data Processing Inequalities for Quantum Divergences

Yu Cao

In information theory, the data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is given by the strong data processing inequality (SDPI) for a fixed channel and a full-rank reference state. In classical information theory, the strong data processing inequality tensorizes for a large family of $f$-divergences. However, its quantum counterpart remains largely unexplored. Tensorization of SDPI has previously been shown only for some special cases, and whether it holds more generally for quantum divergences still remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.

quant-pharXiv:2608.13204•Aug 14, 2026•0 citations

Tsirelson's bound from a generalized data processing inequality

Oscar Dahlsten, Daniel Lercher, Renato Renner

The strength of quantum correlations is bounded from above by Tsirelson's bound. We establish a connection between this bound and the fact that correlations between two systems cannot increase under local operations, a property known as the data processing inequality (DPI). More specifically, we consider arbitrary convex probabilistic theories. These can be equipped with an entropy measure that naturally generalizes the von Neumann entropy, as shown recently in Short and Wehner (2010 New J. Phys. 12 033023) and Barnum et al (2010 New J. Phys. 12 033024). We prove that if the DPI holds with respect to this generalized entropy measure then the underlying theory necessarily respects Tsirelson's bound. We, moreover, generalize this statement to any entropy measure satisfying certain minimal requirements. A consequence of our result is that not all the entropic relations used for deriving Tsirelson's bound via information causality in Pawlowski et al (2009 Nature 461 1101–4) are necessary.

arXiv:1108.4549•New Journal of Physics•Jun 18, 2012•24 citations

Saturating the Data Processing Inequality for $α-z$ Renyi Relative Entropy

Sarah Chehade

It has been shown that the $α-z$ R{é}nyi relative entropy satisfies the Data Processing Inequality (DPI) for a certain range of $α$'s and $z$'s. Moreover, the range is completely characterized by Zhang in `20. We prove necessary and algebraically sufficient conditions to saturate the DPI for the $α-z$ R{é}nyi relative entropy whenever $1

arXiv:2006.07726•Jun 13, 2020•6 citations

Equality conditions of data processing inequality forα-zRényi relative entropies

Haonan Zhang

The α–z Rényi relative entropies are a two-parameter family of Rényi relative entropies that are quantum generalizations of the classical α-Rényi relative entropies. In the work [Adv. Math. 365, 107053 (2020)], we decided the full range of (α, z) for which the data processing inequality (DPI) is valid. In this paper, we give algebraic conditions for the equality in DPI. For the full range of parameters (α, z), we give necessary conditions and sufficient conditions. For most parameters, we give equivalent conditions. This generalizes and strengthens the results of Leditzky et al. [Lett. Math. Phys. 107, 61–80 (2017)].

arXiv:2007.06644•Journal of Mathematical Physics•Oct 1, 2020•12 citations
In this paper, we show that the generalized Bures--Wasserstein distance induced by the generalized fidelity fails the data processing inequality (DPI) in every dimension n≥2n\geq2n≥2. In dimension two, we derive three sufficient conditions guaranteeing the DPI by using an explicit formula for the real part of the generalized fidelity. Furthermore, we find a necessary and sufficient condition under which the generalized fidelity reduces to the Uhlmann fidelity. We also show that neither the generalized Bures--Wasserstein distance nor its square is convex in one state, even with the other state and the reference fixed, in every dimension n≥2n\geq2n≥2.
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.
Abstract The data processing inequality (DPI) is a scalar inequality satisfied by distinguishability measures on density matrices. For some distinguishability measures, saturation of the scalar DPI implies an operator equation relating the arguments of the measure. These results are typically derived using functional analytic techniques. In a complementary approach, we use geometric techniques to derive a formula that gives an operator equation from DPI saturation for any distinguishability measure; moreover, for a broad class of distinguishability measures, the derived operator equation is sufficient to imply saturation as well. Our operator equation coincides with known results for the sandwiched Rényi relative entropies, and gives new results for α - z Rényi relative entropies and a family of of quantum f -divergences, which we compute explicitly.
In information theory, the data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is given by the strong data processing inequality (SDPI) for a fixed channel and a full-rank reference state. In classical information theory, the strong data processing inequality tensorizes for a large family of fff-divergences. However, its quantum counterpart remains largely unexplored. Tensorization of SDPI has previously been shown only for some special cases, and whether it holds more generally for quantum divergences still remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.
The strength of quantum correlations is bounded from above by Tsirelson's bound. We establish a connection between this bound and the fact that correlations between two systems cannot increase under local operations, a property known as the data processing inequality (DPI). More specifically, we consider arbitrary convex probabilistic theories. These can be equipped with an entropy measure that naturally generalizes the von Neumann entropy, as shown recently in Short and Wehner (2010 New J. Phys. 12 033023) and Barnum et al (2010 New J. Phys. 12 033024). We prove that if the DPI holds with respect to this generalized entropy measure then the underlying theory necessarily respects Tsirelson's bound. We, moreover, generalize this statement to any entropy measure satisfying certain minimal requirements. A consequence of our result is that not all the entropic relations used for deriving Tsirelson's bound via information causality in Pawlowski et al (2009 Nature 461 1101–4) are necessary.
Saturating the Data Processing Inequality for α−zα-zα−z Renyi Relative Entropy
It has been shown that the α−zα-zα−z R{é}nyi relative entropy satisfies the Data Processing Inequality (DPI) for a certain range of ααα's and zzz's. Moreover, the range is completely characterized by Zhang in `20. We prove necessary and algebraically sufficient conditions to saturate the DPI for the α−zα-zα−z R{é}nyi relative entropy whenever $1
The α–z Rényi relative entropies are a two-parameter family of Rényi relative entropies that are quantum generalizations of the classical α-Rényi relative entropies. In the work [Adv. Math. 365, 107053 (2020)], we decided the full range of (α, z) for which the data processing inequality (DPI) is valid. In this paper, we give algebraic conditions for the equality in DPI. For the full range of parameters (α, z), we give necessary conditions and sufficient conditions. For most parameters, we give equivalent conditions. This generalizes and strengthens the results of Leditzky et al. [Lett. Math. Phys. 107, 61–80 (2017)].