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:2609.12547

Relative entropy representations via tracial joint spectral measures

Lukas Schmitt

quant-phmath-phSubmitted Sep 11, 2026• Announced Sep 14, 2026• 0 citations
arXivHTMLPDF

Abstract

We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}μ_{A,B}(a,b)$, where $μ_{A,B}$ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve $t\mapsto \mathrm{tr}[A-tB]_+$ is piecewise affine exactly when $A$ and $B$ commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.

Comments
13+5 pages

Related papers

A UNIFIED TREATMENT OF CONVEXITY OF RELATIVE ENTROPY AND RELATED TRACE FUNCTIONS, WITH CONDITIONS FOR EQUALITY

Anna Jenčová, Mary Beth Ruskai

We consider a generalization of relative entropy derived from the Wigner–Yanase–Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K* ApK B1-pLieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for [Formula: see text]. In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.

arXiv:0903.2895•Reviews in Mathematical Physics•Oct 1, 2010•61 citations

Fundamental properties of Tsallis relative entropy

Shigeru Furuichi, 研二郎 柳, 憲 栗山

Fundamental propenies for the Tsallis relative entropy in both classical and quantum systems are studied.As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz.The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible.The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity.Moreover, the generalized Peierls-Bogoliubov inequality is also proven.

•Dec 1, 2004•0 citations

Fundamental properties of Tsallis relative entropy

Shigeru Furuichi, Kenjiro Yanagi, Ken Kuriyama

Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls–Bogoliubov inequality is also proven.

arXiv:cond-mat/0406178•Journal of Mathematical Physics•Nov 29, 2004•178 citations

Integral formula for quantum relative entropy implies data processing inequality

Péter E. Frenkel

Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.

arXiv:2208.12194•Quantum•Sep 7, 2023•15 citations

Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings

Marius Junge, Nicholas D. LaRacuente

Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.

arXiv:2009.11866•Journal of Mathematical Physics•Dec 1, 2022•9 citations

Reverse relative entropy and finite-time contraction for conditional expectations

Christoph Hartmann

This preprint proves reverse relative entropy and finite-time contraction inequalities for trace-preserving conditional expectations on finite-dimensional matrix algebras. A spectral-block reduction for an algebra-valued multiplier and a scalar boundary reduction yield explicit constants gamma_C and Lambda_p(C). Complete bounds follow from complete order bounds. For qubit depolarization, the results give 0.940046944825 <= alpha_c < 0.954706390024 in the convention exp(-2 alpha t), and 0.265671889879 < eta_c < 0.270961852526 at retained weight p=1/2. The contraction coefficient compares each state with its own conditional-expectation equilibrium. The paper also proves a stronger bound for commuting states, with sharpness in specified classes, and derives consequences for products of local channels. The exact complete qubit constants and the unrestricted sharp ensemble inequality are not determined here. The accompanying archive contains LaTeX sources, exact inputs, symbolic and interval verifiers, numerical regression checks, and reproduced outputs. Developed with AI assistance; not peer reviewed or formally verified. Companion paper: Christoph Hartmann, Sharp reverse entropy inequalities for uniform qubit ensembles (2026).

•Sep 29, 2026•0 citations
We show that for positive semidefinite AAA and BBB the relative entropy can be written as D(A∥B)=2∫alog⁡(a/b) dμA,B(a,b)D(A\|B)=2\int a\log(a/b)\,\mathrm{d}μ_{A,B}(a,b)D(A∥B)=2∫alog(a/b)dμA,B​(a,b), where μA,Bμ_{A,B}μA,B​ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve t↦tr[A−tB]+t\mapsto \mathrm{tr}[A-tB]_+t↦tr[A−tB]+​ is piecewise affine exactly when AAA and BBB commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.
We consider a generalization of relative entropy derived from the Wigner–Yanase–Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K* ApK B1-pLieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for [Formula: see text]. In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.
Fundamental propenies for the Tsallis relative entropy in both classical and quantum systems are studied.As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz.The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible.The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity.Moreover, the generalized Peierls-Bogoliubov inequality is also proven.
Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls–Bogoliubov inequality is also proven.
Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.
Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.
This preprint proves reverse relative entropy and finite-time contraction inequalities for trace-preserving conditional expectations on finite-dimensional matrix algebras. A spectral-block reduction for an algebra-valued multiplier and a scalar boundary reduction yield explicit constants gamma_C and Lambda_p(C). Complete bounds follow from complete order bounds. For qubit depolarization, the results give 0.940046944825 <= alpha_c < 0.954706390024 in the convention exp(-2 alpha t), and 0.265671889879 < eta_c < 0.270961852526 at retained weight p=1/2. The contraction coefficient compares each state with its own conditional-expectation equilibrium. The paper also proves a stronger bound for commuting states, with sharpness in specified classes, and derives consequences for products of local channels. The exact complete qubit constants and the unrestricted sharp ensemble inequality are not determined here. The accompanying archive contains LaTeX sources, exact inputs, symbolic and interval verifiers, numerical regression checks, and reproduced outputs. Developed with AI assistance; not peer reviewed or formally verified. Companion paper: Christoph Hartmann, Sharp reverse entropy inequalities for uniform qubit ensembles (2026).