In spite of their simple description in terms of rotations or symplectic transformations in phase space, quadratic Hamiltonians such as those modeling the most common Gaussian operations on bosonic modes remain poorly understood in terms of entropy p
roduction. For instance, determining the von Neumann entropy produced by a Bogoliubov transformation is notably a hard problem, with generally no known analytical solution. Here, we overcome this difficulty by using the replica method, a tool borrowed from statistical physics and quantum field theory. We exhibit a first application of this method to the field of quantum optics, where it enables accessing entropies in a two-mode squeezer or optical parametric amplifier. As an illustration, we determine the entropy generated by amplifying a binary superposition of the vacuum and an arbitrary Fock state, which yields a surprisingly simple, yet unknown analytical expression.
We discuss the possibility of non-minimal gauge invariance of transverse-momentum-dependent parton densities (TMDs) that allows direct access to the spin degrees of freedom of fermion fields entering the operator definition of (quark) TMDs. This is a
chieved via enhanced Wilson lines that are supplied with the spin-dependent Pauli term $sim F^{mu u}[gamma_mu, gamma_ u]$, thus providing an appropriate tool for the microscopic investigation of the spin and color structure of TMDs. We show that this generalization leaves the leading-twist TMD properties unchanged but modifies those of twist three by contributing to their anomalous dimensions. We also comment on Collins recent criticism of our approach.
