[Paper Review] Catalytic and asymptotic equivalence for quantum entanglement

Ray Ganardi, Tulja Varun Kondra+1 more|arXiv (Cornell University)|May 5, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper establishes a fundamental equivalence between catalytic and asymptotic entanglement transformations for distillable quantum states, demonstrating that using an entangled catalyst cannot increase the asymptotic singlet distillation rate. The authors prove that both frameworks are fully equivalent under conditions allowing correlations across multiple copies, resolving a long-standing question in quantum resource theory and showing that catalysis offers no advantage in asymptotic entanglement distillation beyond what is already achievable via standard asymptotic protocols.

ABSTRACT

Entanglement is a fundamental resource in quantum information processing, yet understanding its manipulation and transformation remains a challenge. Many tasks rely on highly entangled pure states, but obtaining such states is often challenging due to the presence of noise. Typically, entanglement manipulation procedures involving asymptotically many copies of a state are considered to overcome this problem. These procedures allow for distilling highly entangled pure states from noisy states, which enables a wide range of applications, such as quantum teleportation and quantum cryptography. When it comes to manipulating entangled quantum systems on a single copy level, using entangled states as catalysts can significantly broaden the range of achievable transformations. Similar to the concept of catalysis in chemistry, the entangled catalyst is returned unchanged at the end of the state manipulation procedure. Our results demonstrate that despite the apparent conceptual differences between the asymptotic and catalytic settings, they are actually strongly connected and fully equivalent for all distillable states. Our methods rely on the analysis of many-copy entanglement manipulation procedures which may establish correlations between different copies. As an important consequence, we demonstrate that using an entangled catalyst cannot enhance the asymptotic singlet distillation rate of a distillable quantum state. Our findings provide a comprehensive understanding of the capabilities and limitations of both catalytic and asymptotic state transformations of entangled states, and highlight the importance of correlations in these processes.

Motivation & Objective

• To resolve the conceptual and operational relationship between catalytic and asymptotic entanglement transformations in quantum information theory.
• To determine whether entanglement catalysis can enhance the asymptotic distillation rate of quantum states.
• To investigate the role of correlations in multi-copy entanglement manipulation protocols.
• To establish a general equivalence between marginal asymptotic rates and standard asymptotic rates in quantum resource theories.
• To clarify the limitations of catalysis in the context of entanglement distillation and resource theory.

Proposed method

• The authors analyze many-copy entanglement manipulation protocols that allow for correlations between different copies of a quantum state.
• They employ the squashed entanglement measure to bound transformation rates and prove inequalities involving entanglement entropy and free operations.
• A key technique involves using the super-additivity and lower semi-continuity of resource measures like squashed entanglement to relate marginal and standard asymptotic rates.
• The proof relies on constructing LOCC protocols that achieve approximate transformations with vanishing error in the limit of large n.
• The authors generalize their results to arbitrary quantum resource theories by assuming super-additivity and lower semi-continuity of the resource measure.
• They use continuity and monotonicity properties of entanglement measures to derive bounds on achievable transformation rates.

Figure 1: Equivalence between catalysis and reducibility. The left part of the figure shows a state $\rho$ being converted into another state $\sigma$ through a catalytic transformation by using a catalyst in the state $\tau$ . The right part of the figure shows that $\rho$ is asymptotically reducib

Experimental results

Research questions

• RQ1 Is there a fundamental equivalence between catalytic and asymptotic entanglement transformations for distillable states?
• RQ2 Can the use of an entangled catalyst increase the asymptotic singlet distillation rate of a quantum state?
• RQ3 What is the role of correlations between multiple copies in entanglement manipulation protocols?
• RQ4 Under what conditions do marginal asymptotic rates coincide with standard asymptotic rates in quantum resource theories?
• RQ5 How do the properties of entanglement measures like squashed entanglement constrain the achievable transformation rates?

Key findings

• Catalytic and asymptotic entanglement transformations are fully equivalent for all distillable quantum states, despite their apparent conceptual differences.
• The addition of an entangled catalyst cannot increase the asymptotic singlet distillation rate of a distillable state, proving a fundamental limitation of catalysis in this context.
• The transformation rate from a state ρ to a target state σ is bounded by the ratio of their squashed entanglement: R_sq(ρ→σ) ≤ E_sq(ρ)/E_sq(σ).
• For general quantum resource theories, marginal asymptotic rates equal standard asymptotic rates if the resource measure is super-additive and lower semi-continuous at the target state.
• The proof establishes that the asymptotic rate r satisfies r < E_sq(ρ)/E_sq(σ) − ε + δ for arbitrarily small ε, δ > 0, leading to the tight bound R_sq(ρ→σ) ≤ E_sq(ρ)/E_sq(σ).
• The results confirm that catalysis does not provide an advantage in asymptotic entanglement distillation, highlighting the sufficiency of standard asymptotic protocols for distillable states.

Figure 2: Distillation of multipartite entanglement. We call a multipartite state distillable if it can be asymptotically converted into states comprising a singlet between each of the parties. The figure shows the desired final state for $3$ parties. The multipartite distillable entanglement is the