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:2609.05854

From Majorization to Horn and Unitary-Orbit Inequalities

Juntai Zhou

math.FAmath-phquant-phSubmitted Sep 5, 2026• Announced Sep 8, 2026
arXivHTMLPDF

Abstract

Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch--Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.

Related papers

Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits

Pablo Costa Rico, Pavel Shteyner

While many bounds have been proved for partial trace inequalities over the last decades for a large variety of quantities, recent problems in quantum information theory demand sharper bounds. In this work, we study optimal bounds for partial trace quantities in terms of the spectrum; equivalently, we determine the best bounds attainable over unitary orbits of matrices. We solve this question for Schur-convex functionals acting on a single partial trace in terms of eigenvalues for self-adjoint matrices and then we extend these results to singular values of general matrices. We subsequently extend the study to Schur-convex functionals that act on several partial traces simultaneously and present sufficient conditions for sharpness. In cases where closed-form maximizers cannot be identified, we present quadratic programs that yield new computable upper bounds for any Schur-convex functional. We additionally present examples demonstrating improvements over previously known bounds. Finally, we conclude with the study of optimal bounds for an $n$-qubit system and its subsystems of dimension $2$.

quant-pharXiv:2601.14158•Jan 21, 2026•0 citations

Poset-refined majorization relations

Alexander E. Guterman, Michael M. Wolf

Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.

math.FACross-listedarXiv:2607.28061•Jul 31, 2026

Partial trace relations beyond normal matrices

Pablo Costa Rico, Michael M. Wolf

We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations. Throughout our analysis, we pay particular attention to rank constraints. We find that every pair of matrices of equal size and trace admits dilations of any rank larger than one. We generalize Audenaert’s subadditivity inequality to encompass general matrices, multiple tensor factors, and different norms. A central ingredient for this is a novel majorization relation for Kronecker sums. As an application, we extend the interval of Werner states in which they are provably two-undistillable in any dimension d ≥ 4. We also prove new Schmidt-number witnesses and k-positive maps.

arXiv:2507.18278•Journal of Mathematical Physics•Aug 1, 2026•0 citations

Optimization of maximal quantum f-divergences between unitary orbits

Hoang Minh Nguyen, Hoang Hieu Nguyen, Công Trình Lê

Maximal quantum $f$-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum $f$-divergences along unitary orbits of two quantum states. For any operator convex function $f:(0,+\infty)\to\mathbb{R}$, we determine the exact minimum and maximum of $$ U \longmapsto \widehat S_f(ρ\|U^*σU) $$ over the unitary group, and derive explicit spectral formulas for these extremal values together with complete characterizations of the unitaries that attain them. Our approach combines the integral representation of operator convex functions with majorization theory and a unitary-orbit variational method. A key step is to show that any extremizer must commute with the reference state, which reduces the noncommutative optimization problem to a spectral permutation problem. As a consequence, the minimum is achieved by pairing the decreasing eigenvalues of $ρ$ and $σ$, while the maximum corresponds to pairing the decreasing eigenvalues of $ρ$ with the increasing eigenvalues of $σ$. Hence, the range of the maximal quantum $f$-divergence along the unitary orbit is exactly the closed interval determined by these two extremal configurations. Finally, we compare our results with recent unitary-orbit optimization results for quantum $f$-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum $f$-divergences.

arXiv:2601.08268•Jan 13, 2026•0 citations

Equality cases in monotonicity of quasi-entropies, Lieb’s concavity and Ando’s convexity

Fumio Hiai

We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let Φ:B(H)→B(K) be a trace-preserving map such that Φ* is a Schwarz map. When f is an operator monotone or operator convex function on [0, ∞), we present several equivalent conditions for the equality SfK(Φ(ρ)‖Φ(σ))=SfΦ*(K)(ρ‖σ) to hold for given positive operators ρ, σ on H and K∈B(K). The conditions include equality cases in the monotonicity versions of Lieb’s concavity and Ando’s convexity theorems. Specializing the map Φ we have equivalent conditions for equality cases in Lieb’s concavity and Ando’s convexity. Similar equality conditions are discussed also for monotone metrics and χ2-divergences. We further consider some types of linear preserver problems for those quantum information quantities.

arXiv:2304.04361•Journal of Mathematical Physics•Oct 1, 2023•0 citations

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
Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch--Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.
While many bounds have been proved for partial trace inequalities over the last decades for a large variety of quantities, recent problems in quantum information theory demand sharper bounds. In this work, we study optimal bounds for partial trace quantities in terms of the spectrum; equivalently, we determine the best bounds attainable over unitary orbits of matrices. We solve this question for Schur-convex functionals acting on a single partial trace in terms of eigenvalues for self-adjoint matrices and then we extend these results to singular values of general matrices. We subsequently extend the study to Schur-convex functionals that act on several partial traces simultaneously and present sufficient conditions for sharpness. In cases where closed-form maximizers cannot be identified, we present quadratic programs that yield new computable upper bounds for any Schur-convex functional. We additionally present examples demonstrating improvements over previously known bounds. Finally, we conclude with the study of optimal bounds for an nnn-qubit system and its subsystems of dimension 222.
Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.
We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations. Throughout our analysis, we pay particular attention to rank constraints. We find that every pair of matrices of equal size and trace admits dilations of any rank larger than one. We generalize Audenaert’s subadditivity inequality to encompass general matrices, multiple tensor factors, and different norms. A central ingredient for this is a novel majorization relation for Kronecker sums. As an application, we extend the interval of Werner states in which they are provably two-undistillable in any dimension d ≥ 4. We also prove new Schmidt-number witnesses and k-positive maps.
Maximal quantum fff-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum fff-divergences along unitary orbits of two quantum states. For any operator convex function f:(0,+∞)→Rf:(0,+\infty)\to\mathbb{R}f:(0,+∞)→R, we determine the exact minimum and maximum of U⟼S^f(ρ∥U∗σU) U \longmapsto \widehat S_f(ρ\|U^*σU) U⟼Sf​(ρ∥U∗σU) over the unitary group, and derive explicit spectral formulas for these extremal values together with complete characterizations of the unitaries that attain them. Our approach combines the integral representation of operator convex functions with majorization theory and a unitary-orbit variational method. A key step is to show that any extremizer must commute with the reference state, which reduces the noncommutative optimization problem to a spectral permutation problem. As a consequence, the minimum is achieved by pairing the decreasing eigenvalues of ρρρ and σσσ, while the maximum corresponds to pairing the decreasing eigenvalues of ρρρ with the increasing eigenvalues of σσσ. Hence, the range of the maximal quantum fff-divergence along the unitary orbit is exactly the closed interval determined by these two extremal configurations. Finally, we compare our results with recent unitary-orbit optimization results for quantum fff-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum fff-divergences.
We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let Φ:B(H)→B(K) be a trace-preserving map such that Φ* is a Schwarz map. When f is an operator monotone or operator convex function on [0, ∞), we present several equivalent conditions for the equality SfK(Φ(ρ)‖Φ(σ))=SfΦ*(K)(ρ‖σ) to hold for given positive operators ρ, σ on H and K∈B(K). The conditions include equality cases in the monotonicity versions of Lieb’s concavity and Ando’s convexity theorems. Specializing the map Φ we have equivalent conditions for equality cases in Lieb’s concavity and Ando’s convexity. Similar equality conditions are discussed also for monotone metrics and χ2-divergences. We further consider some types of linear preserver problems for those quantum information quantities.
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.