Abstract
We introduce a subsystem generalization of the spectral form factor via pseudoentropy, the von-Neumann entropy for the reduced transition matrix. We consider a transition matrix between the thermofield double state and its time-evolved state in two-dimensional conformal field theories, and study the time dependence of the pseudoentropy for a single interval. We show that the real part of the pseudoentropy behaves similarly to the spectral form factor; it starts from the thermal entropy, initially drops to the minimum, then it starts increasing, and then finally it approaches the vacuum entanglement entropy. We also study the theory dependence of its behavior by considering theories on a compact space.
Highlights
We introduce a subsystem generalization of the spectral form factor via pseudoentropy, the vonNeumann entropy for the reduced transition matrix
We show that the real part of the pseudoentropy behaves to the spectral form factor; it starts from the thermal entropy, initially drops to the minimum, it starts increasing, and it approaches the vacuum entanglement entropy
The above expression (8) is already suggestive of the spectral form factor if we focus on the real part
Summary
Quantum entanglement is being studied extensively in several fields such as high-energy physics, condensed matter physics, and quantum information theory. Zðβ þ itÞ ≡ Trðe−βHþitHÞ; ð2Þ where H is the Hamiltonian, β is the inverse temperature of the system, En and Em are energy eigenvalues, and ZðβÞ is the partition function of the system at β:Zðβ þ itÞ is the partition function analytically continued to the real time as β → β þ it It characterizes the discreteness of the energy spectrum. ZðβþitÞ ZðβÞ j2 is diagnostic of the pair correlation of energy eigenvalues This squared quantity is called the spectrum form factor (see [14], for example). Note that (3) and (4) are two-state generalizations of the density matrix and the reduced density matrix These transition matrices can appear naturally in the postselected process, where jψi is the initial state and jφi is the final state. Throughout this paper, we normalize all dimensionful parameters by a lattice spacing
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