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Catalytic Activation of Bell Nonlocality

Jessica Bavaresco, Nicolas Brunner, Antoine Girardin, Patryk Lipka-Bartosik, Pavel Sekatski

TL;DR

This work addresses whether Bell nonlocality can be activated starting from Bell-local entangled states by introducing catalytic activation: a Bell-local $\rho_{AB}$ combined with a catalyst $\omega_{C_A C_B}$ is transformed via local operations into a Bell-nonlocal $\tau_{A'B'}$, while the catalyst is returned unchanged. A central result links catalytic activation to many-copy activation, showing that if $\rho_{AB}$ becomes Bell-nonlocal when replicated $n$ times, then one-copy catalytic activation exists with a specifically constructed catalyst $\omega_{C_A C_B}$; the output $\tau_{A'B'}$ is Bell-nonlocal and the catalyst is preserved. The paper identifies large classes of activatable states, including those with singlet fraction $F(\rho_{AB}) > 1/d$, and demonstrates activation even with a Bell-local catalyst, with extensions to CHSH and other activation schemes. These findings reveal a new form of quantum catalysis connecting entanglement and Bell nonlocality, with potential implications for device-independent tasks and metrological advantages.

Abstract

The correlations of certain entangled states can be perfectly simulated classically via a local model. Hence such states are termed Bell local, as they cannot lead to Bell inequality violation. Here, we show that Bell nonlocality can nevertheless be activated for certain Bell-local states via a catalytic process. Specifically, we present a protocol where a Bell-local state, combined with a catalyst, is transformed into a Bell-nonlocal state while the catalyst is returned exactly in its initial state. Importantly, this transformation is deterministic and based only on local operations. Moreover, this procedure is possible even when the state of the catalyst is itself Bell local, demonstrating a new form of superactivation of Bell nonlocality, as well as an interesting form of quantum catalysis. On the technical level, our main tool is a formal connection between catalytic activation and many-copy activation, which is of independent interest.

Catalytic Activation of Bell Nonlocality

TL;DR

This work addresses whether Bell nonlocality can be activated starting from Bell-local entangled states by introducing catalytic activation: a Bell-local combined with a catalyst is transformed via local operations into a Bell-nonlocal , while the catalyst is returned unchanged. A central result links catalytic activation to many-copy activation, showing that if becomes Bell-nonlocal when replicated times, then one-copy catalytic activation exists with a specifically constructed catalyst ; the output is Bell-nonlocal and the catalyst is preserved. The paper identifies large classes of activatable states, including those with singlet fraction , and demonstrates activation even with a Bell-local catalyst, with extensions to CHSH and other activation schemes. These findings reveal a new form of quantum catalysis connecting entanglement and Bell nonlocality, with potential implications for device-independent tasks and metrological advantages.

Abstract

The correlations of certain entangled states can be perfectly simulated classically via a local model. Hence such states are termed Bell local, as they cannot lead to Bell inequality violation. Here, we show that Bell nonlocality can nevertheless be activated for certain Bell-local states via a catalytic process. Specifically, we present a protocol where a Bell-local state, combined with a catalyst, is transformed into a Bell-nonlocal state while the catalyst is returned exactly in its initial state. Importantly, this transformation is deterministic and based only on local operations. Moreover, this procedure is possible even when the state of the catalyst is itself Bell local, demonstrating a new form of superactivation of Bell nonlocality, as well as an interesting form of quantum catalysis. On the technical level, our main tool is a formal connection between catalytic activation and many-copy activation, which is of independent interest.

Paper Structure

This paper contains 9 sections, 5 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bell nonlocality
  3. Catalytic activation of Bell nonlocality
  4. Activation with a Bell-local catalyst
  5. Catalytic activation of CHSH
  6. Other forms of catalytic activation of Bell nonlocality
  7. Discussion and Outlook
  8. Proof of Lemma \ref{['lem: cata']}
  9. Instrument-based catalytic activation of Bell nonlocality

Key Result

Theorem 1

Let $\rho_{AB}$ be a Bell-local entangled state, such that $n$ copies exhibit Bell nonlocality, i.e., such that $\rho_{AB}^{\otimes n}$ violates a Bell inequality. Then, one copy of $\rho_{AB}$ exhibits catalytic Bell nonlocality activation, with the state of the catalyst being given by where $\otimes[ii]_{\widetilde{R}_A \widetilde{R}_B} \equiv \dyad{i}_{\widetilde{R}_A} \otimes \dyad{i}_{\widet

Figures (1)

  • Figure 1: (a) The standard Bell nonlocality scenario, where an entangled state $\rho_{AB}$ is shared between two parties, $A$ and $B$, who perform local measurements to obtain the distribution $p(ab|xy)$. (b) In this work, we discuss a scenario for catalytic activation of Bell nonlocality. Before being measured, the state $\rho_{AB}$ is catalytically transformed into $\tau_{A' B'}$ by means of deterministic local operations. That is, the joint state of the systems and the bipartite catalyst $\rho_{AB}\otimes \omega_{C_A C_B}$ is mapped to a global output state $\tau_{A' B' C_A C_B}$ (by joint operations on $AC_A$ and $BC_B$), in such a way that the marginal state of the catalyst is returned unchanged $\tau_{C_A C_B}=\omega_{C_A C_B}$. We show that the output state of the measured systems $\tau_{A' B'}$ can be Bell nonlocal, even when starting from a state $\rho_{AB}$ that is Bell local, i.e., it cannot lead to nonlocality in scenario (a). (c) A different variant of catalytic activation of Bell nonlocality, where the catalyst returned only after the local measurements.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Lemma 1
  • proof