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

Pablo Costa Rico, Pavel Shteyner

TL;DR

The paper develops a unified majorization-based framework to obtain sharp spectral bounds for Schur-convex functionals of partial traces over unitary orbits. It proves exact maximizers for a single partial trace (diagonal in the eigenbasis for the second trace and flipped diagonal for the first), extends these results to singular values, and analyzes the more intricate case of jointly bounding both partial traces, offering sufficient spectral conditions for diagonal optimality and practical quadratic-programming relaxations when closed-form maximizers fail. The approach yields improved norm and entropy bounds and extends to multipartite $n$-qubit systems, enabling sharp spectral bounds for quantumMarginal-type quantities. This has direct implications for quantum information tasks involving entanglement measures and information-theoretic quantities, by providing computable, spectrally sharp bounds. Overall, the framework balances the breadth of covering sets (starting with diagonals) against the tractability of spectral conditions, with clear pathways for refinement via SDP/QP methods.

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 $2$.

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

TL;DR

The paper develops a unified majorization-based framework to obtain sharp spectral bounds for Schur-convex functionals of partial traces over unitary orbits. It proves exact maximizers for a single partial trace (diagonal in the eigenbasis for the second trace and flipped diagonal for the first), extends these results to singular values, and analyzes the more intricate case of jointly bounding both partial traces, offering sufficient spectral conditions for diagonal optimality and practical quadratic-programming relaxations when closed-form maximizers fail. The approach yields improved norm and entropy bounds and extends to multipartite -qubit systems, enabling sharp spectral bounds for quantumMarginal-type quantities. This has direct implications for quantum information tasks involving entanglement measures and information-theoretic quantities, by providing computable, spectrally sharp bounds. Overall, the framework balances the breadth of covering sets (starting with diagonals) against the tractability of spectral conditions, with clear pathways for refinement via SDP/QP methods.

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 -qubit system and its subsystems of dimension .
Paper Structure (13 sections, 19 theorems, 102 equations, 2 tables)

This paper contains 13 sections, 19 theorems, 102 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation and Preliminaries
  4. Local diagonalization of partial traces and majorization
  5. Optimal bounds for a single partial trace
  6. Maxima
  7. Minima
  8. Optimal bounds for both partial traces on two copies of a system
  9. Sufficient conditions for majorization with diagonal matrix
  10. Necessary ranks for majorization with diagonal matrix
  11. Conclusion
  12. Counterexamples for majorization by diagonal matrices
  13. Technical results for Theorem \ref{['theo:2xd']}

Key Result

Lemma 3.1

Let $C\in \operatorname{L}\!\bigl( \mathbb{C}^{d} \IfNoValueTF{_} {} {\otimes\mathbb{C}^{_}} \bigr) 1,d_2$ be a self-adjoint matrix. Then there exists a unitarily similar matrix $\hat{C}$ of $C$ such that:

Theorems & Definitions (40)

  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1: Local Diagonalization Lemma
  • proof
  • Remark 3.2
  • Proposition 3.3
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • proof
  • ...and 30 more