Relative entropy of fermionic states from Wigner function characteristics

The study of entanglement and its various measures has ubiquitous interest in quantum physics. Particularly in quantum condensed matter, novel states both at the bottom and the middle of the spectrum have been shown to exhibit universal scaling of entanglement entropy (EE) both in static and dynamical situations. Using recent Wigner-characteristic based techniques to calculate EE, we give a microscopic understanding of the universal scaling of EE with subsystem fraction from the structure of eigenstates. We will also show how to adapt the technique to compute the relative entropy between two fermionic density matrices. Of particular interest is the relative entropy between the reduced density matrix of a many body quantum state and a thermal ensemble defined on the same degrees of freedom. This provides an information theoretic measure of how close reduced density matrices obtained by partial tracing an arbitrary Fock state are close to a thermal state. We contrast this with the usual Eigenstate Thermalisation Hypothesis prediction based on expectation values of local observables to adjudge thermalisation, or the lack thereof.