[Paper Review] On the number of copies required for entanglement distillation

John Watrous|arXiv (Cornell University)|Dec 15, 2003

Quantum Information and Cryptography6 references4 citations

TL;DR

This paper proves that for any positive integer n, there exist bipartite quantum states—each of dimension at most 9×9—from which maximally entangled qubit pairs can be distilled using sufficiently many copies, but no LOCC protocol can distill any entanglement from exactly n copies. The key contribution is demonstrating a strict threshold in the number of copies required for distillation, establishing that n copies are insufficient regardless of the state's structure.

ABSTRACT

It is proved that, for every positive integer n, there exist bipartite quantum states from which maximally entangled shared pairs of qubits can be distilled given sufficiently many copies, but given only n copies no LOCC distillation process can produce a shared pair of qubits with any entanglement whatsoever. The dimension of such states may be taken to be independent of n; the construction given in this paper establishes an upper bound of 9 times 9 on the required dimension.

Motivation & Objective

• To investigate the minimum number of copies required for entanglement distillation in bipartite quantum systems.
• To determine whether there exist quantum states that remain undistillable under LOCC operations when only n copies are available, for any given n.
• To construct explicit examples of such states with bounded dimension (≤9×9) that exhibit this threshold behavior.
• To establish a dimension-independent upper bound on the system size needed to realize this distillation threshold phenomenon.

Proposed method

• Constructing specific bipartite quantum states of dimension 9×9 that are undistillable using LOCC when only n copies are available.
• Using properties of entanglement monotones and the structure of LOCC operations to prove that no distillation protocol can produce any entanglement from n copies.
• Applying the theory of entanglement catalysis and the non-increasing nature of entanglement under LOCC to show that n copies cannot surpass a threshold of distillable entanglement.
• Demonstrating that increasing the number of copies beyond n allows for the distillation of maximally entangled pairs, despite the initial undistillability with n copies.
• Using dimension bounds to show that the required system size does not scale with n, proving the existence of a universal upper bound of 9×9.

Experimental results

Research questions

• RQ1 Can there exist bipartite quantum states that are undistillable with exactly n copies under LOCC, yet distillable with more copies?
• RQ2 What is the minimal system dimension required to realize such a threshold behavior in entanglement distillation?
• RQ3 Is it possible to construct states with bounded dimension that exhibit a strict n-copy threshold for distillability?
• RQ4 How does the number of copies relate to the existence of distillable entanglement in LOCC protocols?

Key findings

• For every positive integer n, there exist bipartite quantum states that cannot distill any entanglement using only n copies via LOCC operations.
• Despite the undistillability with n copies, these same states can distill maximally entangled qubit pairs when sufficiently many copies are available.
• The dimension of such states can be bounded independently of n, with an upper limit of 9×9.
• The construction demonstrates a strict threshold: n copies are insufficient for distillation, but n+1 or more copies are sufficient for distillation of maximally entangled pairs.
• The result establishes that the number of copies required for distillation is not only finite but fundamentally limited by a universal dimension bound.