On the partial transpose of fermionic Gaussian states

TL;DR

This paper analyzes how the partial transpose acts on fermionic Gaussian states, providing a method to compute entanglement measures and traces, with applications to quantum chains and conformal field theory.

Contribution

It introduces a basis choice that transforms the partial transpose into a combination of Gaussian operators, enabling efficient entanglement calculations for fermionic systems.

Findings

Derived exact formulas for traces of partial transposes.

Applied method to quantum Ising chain with results matching conformal field theory.

Provided a lower bound for logarithmic negativity in symmetric geometries.

Abstract

We consider Gaussian states of fermionic systems and study the action of the partial transposition on the density matrix. It is shown that, with a suitable choice of basis, these states are transformed into a linear combination of two Gaussian operators that are uniquely defined in terms of the covariance matrix of the original state. In case of a reflection symmetric geometry, this result can be used to efficiently calculate a lower bound for a well-known entanglement measure, the logarithmic negativity. Furthermore, exact expressions can be derived for traces involving integer powers of the partial transpose. The method can also be applied to the quantum Ising chain and the results show perfect agreement with the predictions of conformal field theory.