从幺正等价中涌现的时间
Emergence of Time from Unitary Equivalence
AI总结：
本文通过模哈密顿量与子系统哈密顿量之间的幺正等价关系揭示时间涌现机制，并利用费米子热场双态中的量子混沌诊断量展示模流与时间演化之间的对偶性。
摘要：
我们讨论了模哈密顿量 $\hat{H}_{\mathrm{mod}}$ 与子系统哈密顿量 $\hat{H}$ 之间通过幺正算符 $\hat{U}$ 实现的幺正等价概念 $\hat{H}\sim\hat{U}^{\dagger}\hat{H}_{\mathrm{mod}}\hat{U}$，其背景是通过幺正算符实现时间的涌现。这一概念暗示了模流与时间演化之间的对偶性。此外，要求幺正等价意味着“模混沌界限”与“混沌界限”之间存在联系。进一步地，我们利用费米子系统的热场双态中的量子混沌诊断量来展示这种对偶性。量子混沌诊断量是刻画量子系统中混沌行为的数学度量。通过考察热场双态中的这些量，我们阐明了它们与模哈密顿量之间的对偶性。我们展示了关联函数、谱形状因子和 Loschmidt 回波与模哈密顿量之间的具体对偶性。谱形状因子是提供量子系统能谱信息的量，而 Loschmidt 回波刻画了系统模时间演化对微扰的敏感性。最后，我们证明不同的纠缠谱不会对子系统哈密顿量施加相同的约束。纠缠谱与纠缠熵相关，并提供与子系统相关的约化密度矩阵本征值的信息。我们讨论了涉及量子混沌、时间涌现以及模哈密顿量与子系统哈密顿量之间关系的复杂概念。这些思想是量子信息理论及相关领域中正在进行的研究的一部分。
英文摘要：
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.
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