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A finite sufficient set of conditions for catalytic majorization

David Elkouss, Ananda G. Maity, Aditya Nema, Sergii Strelchuk

TL;DR

This work establishes finite, computable sets of inequalities that guarantee catalytic majorization (trumping) in both LOCC-assisted state transformations and catalytic thermo-majorization under thermal operations. By leveraging symmetric polynomials F_{k,r} and norm-entropy relations, the authors derive explicit criteria that replace infinite families of conditions with a finite, checkable suite. They extend the framework to diagonal-energy states, states with coherence, and irrational thermal distributions through rational approximations and embedding maps, while connecting to generalized free energies and Rényi divergences. The contribution includes concrete numerical examples and a software toolbox to apply these finite criteria in practice, enabling practical certification of catalysis-enabled state transformations in quantum information and thermodynamics.

Abstract

The majorization relation has found numerous applications in mathematics, quantum information and resource theory, and quantum thermodynamics, where it describes the allowable transitions between two physical states. In many cases, when state vector $x$ does not majorize state vector $y$, it is nevertheless possible to find a catalyst - another vector $z$ such that $x \otimes z$ majorizes $y \otimes z$. Determining the feasibility of such catalytic transformation typically involves checking an infinite set of inequalities. Here, we derive a finite sufficient set of inequalities that imply catalysis. Extending this framework to thermodynamics, we also establish a finite set of sufficient conditions for catalytic state transformations under thermal operations. For novel examples, we provide a software toolbox implementing these conditions.

A finite sufficient set of conditions for catalytic majorization

TL;DR

This work establishes finite, computable sets of inequalities that guarantee catalytic majorization (trumping) in both LOCC-assisted state transformations and catalytic thermo-majorization under thermal operations. By leveraging symmetric polynomials F_{k,r} and norm-entropy relations, the authors derive explicit criteria that replace infinite families of conditions with a finite, checkable suite. They extend the framework to diagonal-energy states, states with coherence, and irrational thermal distributions through rational approximations and embedding maps, while connecting to generalized free energies and Rényi divergences. The contribution includes concrete numerical examples and a software toolbox to apply these finite criteria in practice, enabling practical certification of catalysis-enabled state transformations in quantum information and thermodynamics.

Abstract

The majorization relation has found numerous applications in mathematics, quantum information and resource theory, and quantum thermodynamics, where it describes the allowable transitions between two physical states. In many cases, when state vector does not majorize state vector , it is nevertheless possible to find a catalyst - another vector such that majorizes . Determining the feasibility of such catalytic transformation typically involves checking an infinite set of inequalities. Here, we derive a finite sufficient set of inequalities that imply catalysis. Extending this framework to thermodynamics, we also establish a finite set of sufficient conditions for catalytic state transformations under thermal operations. For novel examples, we provide a software toolbox implementing these conditions.

Paper Structure

This paper contains 16 sections, 27 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Majorization and Catalytic majorization
  3. A finite sufficient set of conditions for trumping of state vectors
  4. A finite sufficient set of conditions for trumping in second law of thermodynamics
  5. For states diagonal in energy eigenbasis
  6. For states having coherence
  7. Examples
  8. Discussion
  9. Proof of Theorem \ref{['thm:second_law']}: Trumping for thermal processes with thermal states having irrational entries
  10. Preliminary results
  11. Characterization in terms of relative entropy, for $p=1$:
  12. Characterization in terms of $p$ Rényi divergence, for $p\in(1,\bar{r})$:
  13. Characterization in terms of $p$ Rényi divergence, for $p \in (0,1)$:
  14. Characterization in terms of $p$ Rényi divergence, for $p \in (-\bar{s},0)$:
  15. Characterization in terms of $p$ Rényi divergence, for $p\in(\overline{r},\infty)$ and $p \in (-\infty, -\bar{s}]$:
  16. ...and 1 more sections

Key Result

Corollary 1

Let $n>1$, $\mathbf x,\mathbf y\in P_n$ and fix $r\in\mathbb N$. If Then and

Figures (1)

  • Figure 1: Checking a finite set of sufficient criteria whether there exist a catalysis $\chi$ that can alleviate restrictions on apparently prohibited transitions from state $\rho$ to state $\sigma$ having eigenvalues $\mathbf x$ and $\mathbf y$ respectively.

Theorems & Definitions (57)

  • Corollary 1
  • Lemma 1
  • proof
  • Corollary 2
  • proof
  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • ...and 47 more