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Entanglement gain in supercatalytic state transformations

Guillermo Díez-Pastor, Julio I. de Vicente

TL;DR

This work introduces the entanglement gain $G\in[0,1]$ as a unifying figure of merit for supercatalytic state transformations, where an auxiliary catalyst is returned in an enhanced form. It proves that any catalytic transformation can be realized in a supercatalytic fashion with maximal gain $G=1$ if one tolerates arbitrarily large or cleverly chosen catalysts, but shows that many transformations are not fully supercatalyzable under common Schmidt-rank constraints. The study of minimal supercatalysis reveals that, even with the borrowed state restricted to minimal Schmidt rank, $G$ cannot reach 1 in general, though there exist families of transformations for which $G$ can be made arbitrarily close to 1; the optimal strategy (miserly vs intermediate lending) depends intricately on the input/output states. The results provide both state-dependent upper bounds and constructive examples, offering insights into the resource-theoretic potential of supercatalysis and raising open questions about universal bounds and practical implementations.

Abstract

Catalysis refers to the possibility of performing an otherwise impossible local state transformation by sharing an additional state, i.e. a catalyst, which is returned at the end of the protocol. There is a stronger version, known as supercatalysis, in which the borrowed catalyst is returned in an enhanced form, i.e. more entangled. However, this phenomenon has remained little explored. In this work we introduce the supercatalytic entanglement gain as a figure of merit taking values in [0,1] that quantifies the performance of the protocol (with 0 corresponding to the standard case of catalysis and 1 representing the maximal possible gain) and we study in which cases it can be greater than zero and which strategies can maximize it. While it turns out that every catalytic transformation can be implemented in a supercatalytic fashion with entanglement gain equal to 1 if the state that is borrowed is chosen appropriately, other choices can make the gain strictly less than 1 and even 0. In fact, we prove that a large class of catalytic transformations are not fully supercatalyzable, i.e. there is at least one choice of catalyst for which the entanglement gain vanishes. On the other hand, the construction that shows that supercatalysis is always possible with maximal gain uses artificially highly entangled catalysts. For this reason, we also study minimal supercatalysis, where the entanglement content of the borrowed state is constrained in a precise and natural way. While we consider a scenario where we prove it is impossible to have entanglement gain equal to 1 in this case, we show that there exist minimal supercatalytic transformations with gain as close to 1 as desired. We also explore several examples and observe that, although choosing a catalyst with the least possible entanglement is often an optimal strategy for minimal supercatalysis, this is not necessarily always the case.

Entanglement gain in supercatalytic state transformations

TL;DR

This work introduces the entanglement gain as a unifying figure of merit for supercatalytic state transformations, where an auxiliary catalyst is returned in an enhanced form. It proves that any catalytic transformation can be realized in a supercatalytic fashion with maximal gain if one tolerates arbitrarily large or cleverly chosen catalysts, but shows that many transformations are not fully supercatalyzable under common Schmidt-rank constraints. The study of minimal supercatalysis reveals that, even with the borrowed state restricted to minimal Schmidt rank, cannot reach 1 in general, though there exist families of transformations for which can be made arbitrarily close to 1; the optimal strategy (miserly vs intermediate lending) depends intricately on the input/output states. The results provide both state-dependent upper bounds and constructive examples, offering insights into the resource-theoretic potential of supercatalysis and raising open questions about universal bounds and practical implementations.

Abstract

Catalysis refers to the possibility of performing an otherwise impossible local state transformation by sharing an additional state, i.e. a catalyst, which is returned at the end of the protocol. There is a stronger version, known as supercatalysis, in which the borrowed catalyst is returned in an enhanced form, i.e. more entangled. However, this phenomenon has remained little explored. In this work we introduce the supercatalytic entanglement gain as a figure of merit taking values in [0,1] that quantifies the performance of the protocol (with 0 corresponding to the standard case of catalysis and 1 representing the maximal possible gain) and we study in which cases it can be greater than zero and which strategies can maximize it. While it turns out that every catalytic transformation can be implemented in a supercatalytic fashion with entanglement gain equal to 1 if the state that is borrowed is chosen appropriately, other choices can make the gain strictly less than 1 and even 0. In fact, we prove that a large class of catalytic transformations are not fully supercatalyzable, i.e. there is at least one choice of catalyst for which the entanglement gain vanishes. On the other hand, the construction that shows that supercatalysis is always possible with maximal gain uses artificially highly entangled catalysts. For this reason, we also study minimal supercatalysis, where the entanglement content of the borrowed state is constrained in a precise and natural way. While we consider a scenario where we prove it is impossible to have entanglement gain equal to 1 in this case, we show that there exist minimal supercatalytic transformations with gain as close to 1 as desired. We also explore several examples and observe that, although choosing a catalyst with the least possible entanglement is often an optimal strategy for minimal supercatalysis, this is not necessarily always the case.

Paper Structure

This paper contains 13 sections, 15 theorems, 50 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background
  4. Supercatalysis
  5. Existence of supercatalyzable and fully supercatalyzable transformations
  6. Supercatalyzable transformations
  7. Fully supercatalyzable transformations
  8. Proof of Theorem \ref{['th:nofull1']}
  9. Proof of Theorem \ref{['th:nofull2']}
  10. Supercatalytic protocols with constrained auxiliary states: minimal scenario
  11. Exploring optimal protocols in the minimal scenario
  12. Minimal supercatalysis with entanglement gain arbitrarily close to 1
  13. Conclusions

Key Result

Theorem 1

Let $\left| a\right>,\left| b\right>\in S(\mathcal{H}_1)$ be such that $SR(|a\rangle),SR(|b\rangle)\leq4$ and $\left| a\right>\nrightarrow\left| b\right>$. Then, there exists a catalyst $\left| c\right>\in C_2(\left| a\right>,\left| b\right>)$ with Schmidt vector $c = (x, 1-x)$ for $x \in [0.5, 1]$

Figures (5)

  • Figure 1: Alice and Bob share two entangled states: the main system, on which they want to perform a certain transformation, and an auxiliary system that acts as a catalyst for the transformation.
  • Figure 2: $G_{\max}(\left| a\right>,\left| b\right>,\left| c\right>)$ (together with its upper bound given in Eq. (\ref{['minbound']})) vs. the entanglement entropy of the borrowed state, $E(\left| c\right>)$, for Example 1.
  • Figure 3: $G_{\max}(\left| a\right>,\left| b\right>,\left| c\right>)$ (together with its upper bound given in Eq. (\ref{['minbound']})) vs. the entanglement entropy of the borrowed state, $E(\left| c\right>)$, for Example 2.
  • Figure 4: $G_{\max}(\left| a\right>,\left| b\right>,\left| c\right>)$ (together with its upper bound given in Eq. (\ref{['minbound']})) vs. the entanglement entropy of the borrowed state, $E(\left| c\right>)$, for Example 3.
  • Figure 5: $G_{\max}(\left| a\right>,\left| b\right>,\left| c\right>)$ (together with its upper bound given in Eq. (\ref{['minbound']})) vs. the entanglement entropy of the borrowed state, $E(\left| c\right>)$, for Example 4.

Theorems & Definitions (39)

  • Theorem 1: XiaoRunMin
  • Definition 2
  • Proposition 3: BandRoy
  • proof
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 9
  • ...and 29 more