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.
Highlights
Entanglement plays a key role in the study of quantum many-body systems [1, 2]
An important result coming from these studies was that the partial transpose of a bosonic Gaussian state is again a Gaussian operator; this simplifies the calculation of the negativity [13, 46]
After recalling the notion of the partial transpose for spin systems and the corresponding definition for fermions in Section 3.1, we show in Section 3.2 that the partial transpose of a Gaussian state can always be decomposed as a sum of two Gaussian operators
Summary
Entanglement plays a key role in the study of quantum many-body systems [1, 2]. Considering a pure state of a composite system, a simple measure of the entanglement between two complementary parts is given by the von Neumann (or entanglement) entropy. The partial transpose of bosonic Gaussian states remains to be Gaussian and, in turn, one has a simple formula to compute the logarithmic negativity via the covariance matrix [13]. A renewed interest in the problem was triggered recently, after a systematic approach within CFT was introduced [23] This method could be applied to calculate the entanglement negativity for various geometries in ground [24, 25] or thermal states [26, 27] of one-dimensional systems, as well as in out-of-equilibrium situations [27, 28, 29, 30]. The higher moments of the partial transpose can be exactly evaluated through simple trace formulas, providing a way to test the universal CFT predictions on fermionic Gaussian states with minimal computational costs. Various details of analytical calculations are included in three Appendices
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