High-Dimensional Entanglement in States with Positive Partial Transposition

TL;DR

This paper constructs explicit high-dimensional PPT quantum states with linearly scaling Schmidt numbers, demonstrating that PPT states can possess significant high-dimensional entanglement and answering open questions in entanglement theory.

Contribution

It provides the first explicit construction of PPT states with linear Schmidt number scaling and links Schmidt number to entangled sub-block matrices, advancing understanding of PPT entanglement.

Findings

Constructed PPT states with linear Schmidt number scaling.

Proved random PPT states typically have high Schmidt numbers.

Linked Schmidt number to entangled sub-block matrices, generalizing known results.

Abstract

Genuine high-dimensional entanglement, i.e. the property of having a high Schmidt number, constitutes a resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT), on the other hand, are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question, whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e. linear, scaling in local dimension should be possible, albeit without providing an insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to answer a recent question by Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states invariant under partial transposition on the smaller of their two subsystems cannot have maximal Schmidt number. This generalizes a well-known result by Kraus et al. We also show that the Schmidt number of absolutely PPT states cannot be maximal, contributing to an open problem in entanglement theory.