Constructing entanglement witnesses for infinite-dimensional systems

TL;DR

This paper demonstrates that all entangled states in infinite-dimensional systems can be detected using simple entanglement witnesses of a specific form, and introduces methods to construct such witnesses.

Contribution

It provides a universal form for entanglement witnesses in infinite-dimensional systems and new methods for their construction, surpassing traditional criteria like PPT and realignment.

Findings

Every entangled state has a simple entanglement witness of the form αI+T.

Constructed witnesses detect entanglement not identified by PPT or realignment.

Methods enable detection of a broader class of entangled states.

Abstract

It is shown that, every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $\alpha I+T$ with $\alpha$ a nonnegative number and $T$ a finite rank self-adjoint operator. We also provide two methods of constructing entanglement witness and apply them to obtain some entangled states that cannot be detected by the PPT criterion and the realignment criterion.