Partial time-reversal transformation and entanglement negativity in fermionic systems

TL;DR

This paper introduces a partial time-reversal transformation as a novel method to define and compute entanglement negativity in fermionic systems, effectively capturing topological features like Majorana fermions.

Contribution

It proposes the use of partial time-reversal for fermionic entanglement measures, overcoming limitations of the partial transpose in detecting topological edge states.

Findings

Partial time-reversal computes the quantum dimension of Majorana fermions.

The method is efficient and maintains Gaussianity of fermionic density matrices.

Numerical and analytical results demonstrate its effectiveness in various fermionic systems.

Abstract

The partial transpose of density matrices in many-body quantum systems, in which one takes the transpose only for a subsystem of the full Hilbert space, has been recognized as a useful tool to diagnose quantum entanglement. It can be used, for example, to define the (logarithmic) negativity. For fermionic systems, it has been known that the partial transpose of Gaussian fermionic density matrices is not Gaussian. In this work, we propose to use partial time-reversal transformation to define (an analog of) the entanglement negativity and related quantities. We demonstrate that for the symmetry-protected topological phase realized in the Kitaev chain the conventional definition of the partial transpose (and hence the entanglement negativity) fails to capture the formation of the edge Majorana fermions, while the partial time-reversal computes the quantum dimension of the Majorana fermions. Furthermore, we show that the partial time-reversal of fermionic density matrices is Gaussian and can be computed efficiently. Various results (both numerical and analytical) for the entanglement negativity using the partial-time reversal are presented for (1+1)-dimensional conformal field theories, and also for fermionic disordered systems (random single phases).