An ergodic theorem for quantum processes with applications to matrix product states

Understand

Any discrete quantum process is represented by a sequence of quantum channels.

• We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum channels.
• Under natural irreducibility conditions, we obtain a theorem showing that the state under such a process converges exponentially fast to an ergodic sequence depending on the process, but independent of the initial state.
• As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk.