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Embezzlement as a "Self-Test" for Infinite Copies of Entangled States

Li Liu

TL;DR

The paper addresses exact entanglement embezzlement in infinite-dimensional quantum systems through a C$^*$-algebraic framework, clarifying how a catalyst state $f$ must internally contain infinite, locality-respecting copies of a target state $g$ to enable exact embezzlement. It proves an equivalence between standard and no-input embezzlement, and then shows that exact embezzlement necessarily yields local containment of infinitely many copies of $g$ within $f$, formalized via the construction of $g^ extinfty$ and a hierarchy of lifted observables with certification, commutativity, and independence. This self-test-like certification of infinite copies implies that universal exact embezzlement requires catalysts with non-separable, infinite-copy structures, particularly to accommodate countable or uncountable families of target states, and raises questions about extending these results to approximate embezzlement. Overall, the work provides conceptual tools linking embezzlement, operator-algebraic locality, and state certification in infinite quantum systems, with potential implications for nonlocal games and quantum cryptography.

Abstract

We investigate the operator-algebraic structure underlying entanglement embezzlement, a phenomenon where a fixed entangled state (the catalyst) can be used to generate arbitrary target entangled states without being consumed. We show that the ability to embezzle a target state $g$ imposes strong internal constraints on the catalyst state $f$: specifically, $f$ must contain infinitely many mutually commuting, locally structured copies of $g$. This property is formalized using C*-algebraic tools and is analogous to a form of self-testing, certifying the presence of infinite copies of $g$ within $f$. Using this infinite-copy certification. Our results clarify the structural requirements for embezzlement and provide new conceptual tools for analyzing state certification in infinite-dimensional quantum systems.

Embezzlement as a "Self-Test" for Infinite Copies of Entangled States

TL;DR

The paper addresses exact entanglement embezzlement in infinite-dimensional quantum systems through a C-algebraic framework, clarifying how a catalyst state must internally contain infinite, locality-respecting copies of a target state to enable exact embezzlement. It proves an equivalence between standard and no-input embezzlement, and then shows that exact embezzlement necessarily yields local containment of infinitely many copies of within , formalized via the construction of and a hierarchy of lifted observables with certification, commutativity, and independence. This self-test-like certification of infinite copies implies that universal exact embezzlement requires catalysts with non-separable, infinite-copy structures, particularly to accommodate countable or uncountable families of target states, and raises questions about extending these results to approximate embezzlement. Overall, the work provides conceptual tools linking embezzlement, operator-algebraic locality, and state certification in infinite quantum systems, with potential implications for nonlocal games and quantum cryptography.

Abstract

We investigate the operator-algebraic structure underlying entanglement embezzlement, a phenomenon where a fixed entangled state (the catalyst) can be used to generate arbitrary target entangled states without being consumed. We show that the ability to embezzle a target state imposes strong internal constraints on the catalyst state : specifically, must contain infinitely many mutually commuting, locally structured copies of . This property is formalized using C*-algebraic tools and is analogous to a form of self-testing, certifying the presence of infinite copies of within . Using this infinite-copy certification. Our results clarify the structural requirements for embezzlement and provide new conceptual tools for analyzing state certification in infinite-dimensional quantum systems.

Paper Structure

This paper contains 13 sections, 6 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. No-Input Embezzlement
  3. Equivalence between Standard and No-input Embezzlement
  4. C*-algebraic Formulation of No-Input Embezzlement
  5. Conclusion.
  6. Embezzlement and Certification of Infinite Target States
  7. Definition of Infinite State Containment
  8. Embezzlement Implies Local Containment of Infinite Copies
  9. Implications of Certifying Infinite Copies of $g$
  10. Necessarity of Large Catalyst for Universal Embezzlement
  11. Conclusion and Discussions
  12. Acknowledgements
  13. Proof of Proposition \ref{['prop:equivalence_embezzlement']}

Key Result

Proposition 1

Let $\ket{\psi} \in \mathbb{C}^n \otimes \mathbb{C}^n$ be an entangled state. The following two notions of entanglement embezzlement are equivalent: Either form of embezzlement can be converted into the other via explicit local isometries, with the catalyst space extended by ancillary qubits in a product state $\ket{00}^{\otimes \infty}$ and local regrouping and swapping operations.

Theorems & Definitions (19)

  • Proposition 1: Equivalence of Embezzlement Models
  • Remark 1
  • Definition 1: State Containment.
  • Definition 2: Local Containment
  • Remark 2
  • Definition 3: Infinite State Containment
  • Remark 3
  • Proposition 2
  • proof
  • Definition 4: Local Containment of Infinite copies
  • ...and 9 more