Is there a probability distribution associated with fermionic Gaussian states

Question

I am writing this as a mathematician trying to understand fermionic Gaussian states.

Up to global phase, a quantum state can be faithfully represented in terms of a quasi-probability distribution on its phase space by its Wigner function.

In particular, if this quantum state is a (bosonic) Gaussian state, then this restricts to a multivariate Gaussian probability distribution on the phase space. In other words, the state is described by a (positive definite) matrix determining the covariance between the positions and momentums between modes, as well as a vector determining the mean of this probability distribution. The evolution of these states is closed under the action of Gaussian unitaries, which can be regarded as the probabilistic evolution of the phase space.

However, in the literature, there is a notion of fermionic Gaussian states. See the article of Hackl and Bianchi for reference. However, the covariance matrix is given by a skew-symmetric matrix. Is there any way in which these fermionic Gaussian states can be represented by probability distributions in such a way that their evolution by fermionic unitaries corresponds to the probabilistic evolution of the phase space? Or does the name Gaussian in this case only come from the relation to bosonic fermions, and not Gaussian probability theory?

Answer 1

This is probably only a partial answer. Firstly, although there are exceptions, most fermion phase space representations are defined with the aid of Grassmann functions (or more precisely, linear combinations of functions with coefficients given by Grassmann numbers).

Secondly, the Gaussian fermion state is a little misleading. Although it is expressed as a Gaussian function in terms of Grassmann variables, there is usually only one term in the Taylor series expansion of such a fermion Gaussian function that survives. The other terms become zero due to the Grassmann property. Nevertheless, thanks to the calculation technology that has been develop for Gaussian functions, one can do sophisticated calculation using such fermion Gaussian functions on phase space.

Hope this helps.