Entropy, purity, and fidelity in Majorana phase space

Abstract

Majorana phase-space representations for fermions map fermionic many-body physics into a distribution over one of the Cartan symmetric spaces of Lie group theory. The representation is in terms of $2M\ifmmode\times\else\texttimes\fi{}2M$ complex antisymmetric matrices, which generate the Gaussian Majorana operators. Here we show how this expansion can be utilized to calculate quantities arising in quantum thermodynamics and quantum information. Purity and the linear entropy are calculated, as well as the quantum fidelity between two general fermionic states, with numerical examples for pure states. We describe the geometrical properties of the phase space, and show that the overlap between two Gaussian Majorana states depends on the product of their antisymmetric matrices. Fermionic phase space is divided up into two orthogonal subspaces of different number parity, whose matrix representations differ by an orthogonal reflection in the phase space.