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Flexible Catalysis

Máté Weisz, Sergii Strelchuk

TL;DR

The paper introduces flexible catalysis, a generalization of quantum state catalysis where a set of catalyst states can be used such that catalysts may transform among themselves while enabling the desired transformation. It formalizes transformation theories (TTs) and catalytic classes, then specializes to multisets over groups to analyze how translation, torsion, and group structure affect catalytic power. The main results show that flexible catalysis strictly extends standard catalysis and multicopy in several quantum-information channels (notably LU and LOCC for extraction) while sometimes yielding no advantage in others (e.g., PM with finite flexibility). Across LOCC, LU, and PM, flexible catalysis provides a richer framework that can enable state transformations and extractions unattainable by traditional catalytic methods, with implications for resource theories and space-bounded quantum computation. The work also outlines several open questions and potential directions, including infinite-catalyst regimes and extensions to mixed states and additional channels.

Abstract

In quantum information and computation, a central challenge is to determine which quantum states can be transformed into one another under restricted sets of free operations. While many transformations are impossible directly, catalytic processes can enable otherwise forbidden conversions: an auxiliary quantum state (the catalyst) facilitates the transformation while remaining unchanged. In this work, we introduce flexible catalysis, a generalization in which the catalyst is allowed to transform into a different auxiliary state, provided it remains a valid catalyst. We show that this framework subsumes both standard catalytic and multicopy transformations, and we analyze its advantages across several classes of free operations. In particular, we prove that when the free operations are local unitaries or permutation matrices, flexible catalysis enables state extractions that are unattainable with standard catalysis alone.

Flexible Catalysis

TL;DR

The paper introduces flexible catalysis, a generalization of quantum state catalysis where a set of catalyst states can be used such that catalysts may transform among themselves while enabling the desired transformation. It formalizes transformation theories (TTs) and catalytic classes, then specializes to multisets over groups to analyze how translation, torsion, and group structure affect catalytic power. The main results show that flexible catalysis strictly extends standard catalysis and multicopy in several quantum-information channels (notably LU and LOCC for extraction) while sometimes yielding no advantage in others (e.g., PM with finite flexibility). Across LOCC, LU, and PM, flexible catalysis provides a richer framework that can enable state transformations and extractions unattainable by traditional catalytic methods, with implications for resource theories and space-bounded quantum computation. The work also outlines several open questions and potential directions, including infinite-catalyst regimes and extensions to mixed states and additional channels.

Abstract

In quantum information and computation, a central challenge is to determine which quantum states can be transformed into one another under restricted sets of free operations. While many transformations are impossible directly, catalytic processes can enable otherwise forbidden conversions: an auxiliary quantum state (the catalyst) facilitates the transformation while remaining unchanged. In this work, we introduce flexible catalysis, a generalization in which the catalyst is allowed to transform into a different auxiliary state, provided it remains a valid catalyst. We show that this framework subsumes both standard catalytic and multicopy transformations, and we analyze its advantages across several classes of free operations. In particular, we prove that when the free operations are local unitaries or permutation matrices, flexible catalysis enables state extractions that are unattainable with standard catalysis alone.

Paper Structure

This paper contains 16 sections, 52 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Informal statements of main results
  3. Transformation Theories
  4. Basics
  5. Transformation classes
  6. The transformation theories of LOCC, LU and PM
  7. Flexible catalysis of multisets
  8. Transformation theory of multisets over a group
  9. Torsion free groups
  10. General abelian groups
  11. Adding translation as a free operation
  12. Advantages from flexibility
  13. Flexible catalysis in quantum information
  14. Open questions
  15. Acknowledgements
  16. ...and 1 more sections

Key Result

Theorem 2.1

(Informal version of Theorem LOCCpositive.) There exist bipartite quantum states $\ket{\psi},\ket{\phi}$ such that preparing $\ket{\phi}$ from $\ket{\psi}$ with certainty (i) is possible by a flexibly catalytic LOCC procedure $P$, (ii) but no single catalyst used in $P$ can do the job alone.

Theorems & Definitions (131)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 3.5
  • ...and 121 more