Emergence of Time from Unitary Equivalence

We discuss the concept of unitary equivalence $\hat{H}\sim\hat{U}^{\dagger}\hat{H}_{\mathrm{mod}}\hat{U}$ between the modular Hamiltonian $\hat{H}_{\mathrm{mod}}$ and the subsystem Hamiltonian $\hat{H}$ in the context of realizing the emergence of time through a unitary operator $\hat{U}$. This concept suggests a duality between the modular flow and time evolution. Additionally, requiring unitary equivalence implies a connection between the "Modular Chaos Bound" and the "Chaos Bound". Furthermore, we demonstrate this duality using quantum chaos diagnostic quantities in the thermofield double state of a fermionic system. Quantum chaos diagnostic quantities are mathematical measures that characterize chaotic behavior in quantum systems. By examining these quantities in the thermofield double state, we illustrate the duality between them and the modular Hamiltonian. We show a specific duality between correlators, the spectral form factor, and the Loschmidt echo with the modular Hamiltonian. The spectral form factor is a quantity that provides information about the energy spectrum of a quantum system, while the Loschmidt echo characterizes the sensitivity of a system's modular time evolution to perturbations. Finally, we demonstrate that a different entanglement spectrum does not impose the same constraint on the subsystem Hamiltonian. The entanglement spectrum is related to entanglement entropy and provides information about the eigenvalues of the reduced density matrix associated with a subsystem. We discuss complex concepts related to the interplay between quantum chaos, time emergence, and the relationship between modular and subsystem Hamiltonians. These ideas are part of ongoing research in quantum information theory and related fields.