Symmetries in physical dilations of open quantum systems

Symmetry plays a fundamental role in Physics and quantum mechanics. More specifically, it is a key concept in the theory of open quantum systems, as the symmetries of a master equation set some constraints on the open system dynamics and on the structure of the steady states. However, a well-known and remarkable result states that Noether's theorem is not valid for open quantum systems. That is, a symmetry does not always correspond to a conserved quantity, and vice versa. Then, a natural question arises: given a symmetry in the dynamics of an open quantum system, can we always find an analogous symmetry and a conserved quantity in the physical dilations of this open dynamics? In other words, is there always an observable living in the extended Hilbert space that is conserved under the time evolution driven by the unitary operator given by the Stinespring-like dilation? In this talk, we address this question and show some results and conjectures about it.