[Paper Review] No-go theorem for entanglement distillation using catalysis

Ludovico Lami, Bartosz Regula+1 more|arXiv (Cornell University)|May 5, 2023

Quantum Information and Cryptography92 references4 citations

TL;DR

This paper establishes a fundamental no-go theorem: entanglement catalysis cannot overcome bound entanglement, meaning no catalytic transformation—even with correlated catalysts or generalized operations—can distill entanglement from a bound entangled state. The key result is that the catalytically distillable entanglement of any positive-partial-transpose (PPT) state is zero, proving catalysis cannot restore asymptotic reversibility in entanglement theory.

ABSTRACT

The use of ancillary quantum systems known as catalysts is known to be able to enhance the capabilities of entanglement transformations under local operations and classical communication. However, the limits of these advantages have not been determined, and in particular it is not known if such assistance can overcome the known restrictions on asymptotic transformation rates -- notably the existence of bound entangled (undistillable) states. Here we establish a general limitation of entanglement catalysis: we show that catalytic transformations can never allow for the distillation of entanglement from a bound entangled state with positive partial transpose, even if the catalyst may become correlated with the system of interest, and even under permissive choices of free operations. This precludes the possibility that catalysis can make entanglement theory asymptotically reversible. Our methods are based on new asymptotic bounds for the distillable entanglement and entanglement cost assisted by correlated catalysts.

Motivation & Objective

• To determine whether entanglement catalysis can overcome the fundamental limitation imposed by bound entangled states, which cannot be distilled into maximally entangled states under standard LOCC operations.
• To investigate whether the use of catalysts—ancillary systems that remain unchanged after transformation—can enhance asymptotic entanglement conversion rates beyond the known bounds of entanglement theory.
• To examine whether catalytic transformations, even when allowing for correlations between the catalyst and the system, can enable distillation from bound entangled states such as PPT states.
• To extend the analysis beyond entanglement to other quantum resource theories, particularly quantum coherence, to assess the generality of the catalytic limitations.
• To establish new asymptotic bounds on distillable entanglement and entanglement cost under catalytic operations, particularly using the relative entropy of PPT entanglement as an upper bound.

Proposed method

• Derives a general upper bound on catalytically assisted distillable entanglement using the relative entropy of PPT entanglement, extending a known bound from non-catalytic protocols to catalytic scenarios.
• Applies the data processing inequality for relative entropy to measurement channels that project onto classical-quantum states, enabling the decomposition of composite systems into subsystems.
• Introduces a compatibility condition between the set of free operations (e.g., PPT-preserving maps) and the catalyst's measurement, ensuring that the reduced states remain within the allowed set of free states.
• Uses convexity and trace-preserving properties of the catalyst's measurement to derive lower bounds on the relative entropy distance between states, which are then used to establish monotonicity under catalytic operations.
• Applies the framework to both entanglement and coherence resource theories, showing that the same no-go result holds for incoherent operations in the coherence theory context.
• Employs a duality between entanglement cost and distillable entanglement under catalytic operations, proving that the catalytic entanglement cost of a PPT state exceeds its distillable entanglement, thus proving irreversibility.

Experimental results

Research questions

• RQ1 Can catalytic transformations enable the distillation of entanglement from a bound entangled state, specifically a PPT state?
• RQ2 Is there any form of catalytic assistance—allowing for catalyst correlation or generalized operations—that can overcome the bound entanglement barrier?
• RQ3 Does the relative entropy of PPT entanglement remain an upper bound on distillable entanglement under catalytic LOCC operations?
• RQ4 Can catalysis restore asymptotic reversibility in entanglement theory, given that standard protocols exhibit irreversibility due to bound entanglement?
• RQ5 Do similar limitations apply to other quantum resource theories, such as quantum coherence, under catalytic transformations?

Key findings

• The catalytically distillable entanglement of any positive-partial-transpose (PPT) state is zero, proving that catalysis cannot overcome bound entanglement.
• The relative entropy of PPT entanglement serves as a valid upper bound on distillable entanglement even under catalytic LOCC operations, extending its known validity from non-catalytic settings.
• Even when the catalyst becomes correlated with the system, or when using PPT-preserving operations (a broader class than LOCC), entanglement distillation from bound entangled states remains impossible.
• The entanglement cost of a PPT state under catalytic operations strictly exceeds its distillable entanglement, confirming the irreversibility of the transformation and precluding asymptotic reversibility.
• The same no-go result extends to the resource theory of quantum coherence: catalysis cannot enable full reversibility in coherence manipulation under incoherent operations.
• The framework yields new asymptotic bounds on entanglement cost and distillable entanglement under catalytic assistance, which are of independent interest beyond the main result.