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

• URL: http://arxiv.org/abs/2601.14158v1
• Date: Tue, 20 Jan 2026 17:05:45 GMT
• Title: Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits
• Authors: Pablo Costa Rico, Pavel Shteyner,
• Abstract summary: We show that we can find the equivalent quantities of the spectrum for partial trace information theory.<n>We also show that we can maximize the bounds for any Schuform upper bounds for any dimension functional.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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 $2$.

Related papers

• Constructive Approximation under Carleman's Condition, with Applications to Smoothed Analysis [13.02728413691724]
  We develop a fairly tight analogue of the underlying DenjoyCarleman via complex analysis.<n>We establish $L2$ approximation-theoretic results for functions over general classes of distributions.<n>As another application, we show that the Paley--Wiener class of functions band to $[-,]$ admits superexponential rates of approximation over all strictly sub-exponential distributions.
  arXiv  Detail & Related papers  (2025-12-04T01:40:05Z)
• Conditional Distribution Quantization in Machine Learning [83.54039134248231]
  Conditional expectation mathbbE(Y mid X) often fails to capture the complexity of multimodal conditional distributions mathcalL(Y mid X)<n>We propose using n-point conditional quantizations--functional mappings of X that are learnable via gradient descent--to approximate mathcalL(Y mid X)
  arXiv  Detail & Related papers  (2025-02-11T00:28:24Z)
• The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank [0.6963971634605796]
  We construct a convergent family of outer approximations for the problem of optimizing functions over convex subject bodies to constraints. A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.
  arXiv  Detail & Related papers  (2024-06-13T18:00:09Z)
• Entrywise error bounds for low-rank approximations of kernel matrices [55.524284152242096]
  We derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues. We validate our theory with an empirical study of a collection of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2024-05-23T12:26:25Z)
• Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications [8.438718130535296]
  We prove upper bounds on the covering number of numerous sets in Euclidean space.<n>We illustrate the power of the result on three computational applications.
  arXiv  Detail & Related papers  (2023-11-09T03:06:59Z)
• Approximation of optimization problems with constraints through kernel Sum-Of-Squares [77.27820145069515]
  We show that pointwise inequalities are turned into equalities within a class of nonnegative kSoS functions. We also show that focusing on pointwise equality constraints enables the use of scattering inequalities to mitigate the curse of dimensionality in sampling the constraints.
  arXiv  Detail & Related papers  (2023-01-16T10:30:04Z)
• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields [53.31927549039624]
  We show that a piecewise discretization preserves better contrast from existing discretization problems. We apply this theory to the problem of matching two images.
  arXiv  Detail & Related papers  (2021-07-13T12:31:06Z)
• Maximal violation of steering inequalities and the matrix cube [1.5229257192293197]
  We show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal.
  arXiv  Detail & Related papers  (2021-05-24T14:36:24Z)
• Inexact and Stochastic Generalized Conditional Gradient with Augmented Lagrangian and Proximal Step [2.0196229393131726]
  We analyze inexact and versions of the CGALP algorithm developed in the authors' previous paper. This allows one to compute some gradients, terms, and/or linear minimization oracles in an inexact fashion. We show convergence of the Lagrangian to an optimum and feasibility of the affine constraint.
  arXiv  Detail & Related papers  (2020-05-11T14:52:16Z)
• On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
  We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
  arXiv  Detail & Related papers  (2020-04-09T17:54:18Z)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.