Entanglement echo and dynamical entanglement transitions

We formulate dynamical phase transitions in subsystems embedded in larger quantum systems. Introducing the entanglement echo as an overlap of the initial and instantaneous entanglement ground states, we show its analytic structure after a quench provides natural definition of dynamical phase transitions in the subsystem. These transitions come in two varieties, the entanglement-type transitions and the bulk-type Loschmidt transitions. The entanglement-type transitions arise from periodic reorganization of quantum correlations between the subsystem and its environment, manifesting in instantaneous entanglement ground state degeneracies. Furthermore, the entanglement echo distinguishes the direction of the quench, resolves spatially distinct dynamical phase transitions for non-uniform quenches and give rise to sharply-defined transitions for mixed initial states. We propose an experimental probe to identify entanglement-type transitions through temporal changes in subsystem fluctuations.