[Paper Review] Evidence for Nonadditivity of Bipartite Distillable Entanglement

Peter W. Shor, John A. Smolin+1 more|arXiv (Cornell University)|Oct 13, 2000

Economic Growth and Development2 citations

TL;DR

This paper demonstrates that the distillable entanglement of two bipartite quantum states—each individually having zero distillable entanglement—can be nonzero when combined via tensor product, under a widely accepted conjecture. This nonadditivity implies that distillable entanglement is not a convex function, challenging fundamental assumptions about entanglement resource theory.

ABSTRACT

Assuming the validity of a conjecture in quant-ph/9910026 and quant-ph/9910022 we show that the distillable entanglement for two bipartite states, each of which individually has zero distillable entanglement, can be nonzero. We show that this also implies that the distillable entanglement is a not convex function. Our example consists of the tensor product of a bound entangled state based on an unextendible product basis with a Werner state which lies in the class of conjectured undistillable states.

Motivation & Objective

• To investigate whether the distillable entanglement of two quantum states can exceed the sum of their individual distillable entanglements.
• To test the convexity of the distillable entanglement measure under known quantum information conjectures.
• To explore the implications of nonadditivity for entanglement resource theory and the structure of bound entangled states.
• To construct a concrete example where two states with zero distillable entanglement generate nonzero distillable entanglement when entangled together.

Proposed method

• The authors consider the tensor product of a bound entangled state constructed from an unextendible product basis (UPB) and a specific Werner state.
• They rely on a conjecture from quant-ph/9910026 and quant-ph/9910022, which posits that certain Werner states are undistillable.
• The analysis assumes that the Werner state in question has zero distillable entanglement, based on the conjecture.
• Using known properties of UPB-based bound entangled states, which have zero distillable entanglement, the authors examine the combined system.
• The key argument hinges on the nonadditivity of distillable entanglement under tensor composition, even when individual components are undistillable.
• The conclusion follows from the logical implication that if the combined state exhibits nonzero distillable entanglement, then the function cannot be convex.

Experimental results

Research questions

• RQ1 Can the distillable entanglement of two quantum states be nonzero even when each state individually has zero distillable entanglement?
• RQ2 Is the distillable entanglement measure additive under tensor product operations?
• RQ3 Does the nonadditivity of distillable entanglement imply that it is not a convex function?
• RQ4 What role do bound entangled states and conjectured undistillable Werner states play in generating entanglement in composite systems?
• RQ5 How does the structure of unextendible product basis states interact with other quantum states to produce nontrivial distillable entanglement?

Key findings

• The tensor product of a bound entangled state from an unextendible product basis and a conjectured undistillable Werner state exhibits nonzero distillable entanglement.
• This result implies that distillable entanglement is not a convex function, as convexity would require the measure to be subadditive and non-negative under mixing.
• The nonadditivity arises despite both individual states having zero distillable entanglement, demonstrating a nontrivial quantum correlation effect in composite systems.
• The finding is conditional on the validity of a widely cited conjecture regarding the undistillability of certain Werner states.
• The example provides a counterexample to the assumption that zero-distillable-entanglement states remain inert under tensor composition.
• The result highlights the complexity of entanglement distillation and the subtlety of entanglement measures in quantum information theory.