Optimal Entanglement Certification from Moments of the Partial Transpose

TL;DR

This paper introduces two systematic methods for entanglement detection using moments of the partially transposed quantum state, improving certification efficiency for medium-sized quantum devices.

Contribution

It proposes two new entanglement detection techniques based on moments of the partial transpose, including an optimal method with necessary and sufficient conditions.

Findings

Hankel matrix-based entanglement criteria

Reduction to known $p_3$-PPT criterion

An optimal method with complete entanglement characterization

Abstract

For the certification and benchmarking of medium-size quantum devices efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known $p_3$-PPT criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.