Insights of quantum time into quantum evolution

If time is emergent, quantum system is entangled with quantum time as it evolves. If the system contains entanglement within itself, which we can call internal entanglement to distinguish it from the "external" time-system entanglement, the speed of evolution is enhanced. In this paper, we show the correlation between the novel time-system entanglement and the conventional internal entanglement of a system that contains two entangled qubits. We consider two cases: (1) two initially entangled qubits that evolve under local dynamics; (2) two interacting qubits such that entanglement between them is generated over time. In the first case, we obtain the main result that increasing internal entanglement speeds up the evolution and makes the system more entangled with time. For both cases, we show the dependence of time-system entanglement entropy on the distance of evolution which is characterized by fidelity. The interacting system can evolve faster than the non-interacting system if the interaction is sufficiently strong, and thus it can be entangled with time more efficiently.