Abstract
Motivated by the holographic complexity proposals, in this paper, we investigate the time dependence of the complexity for the Fermionic thermofield double state (TFD) using the Nielsen approach and Fubini-Study (FS) approach separately. In both two approaches, we discuss the results for different reference states: the Dirac vacuum state and the Gaussian state which has no spatial entanglement (NSE). For Dirac vacuum reference state, we find that the complexity by both two approaches is time independent and the circuit complexity shares the same expression for both methods with the $L^2$ norm. For the NSE reference state, the complexity by the Nielsen approach is time-dependent while it by the FS approach is time-independent. Further, we find our dynamical results are in good agreement with the bosonic case, where the complexity evolves in time and saturates after a time at the order of the inverse temperature. And we show that the complexity of formation is also shared same similar behaviors with the holographic complexity.
Highlights
Applications of quantum information concepts to high energy physics and gravity have led to many promising results
We review some key elements of Nielsen’s approach to evaluate circuit complexity, and demonstrate how both the timeindependent and time-dependent fermionic thermofield double state (TFD) states of 2N decoupled harmonic oscillators can be generated by quadratic operators, for later use, we write down the covariance matrices, which serves as an alternate–and more natural–characterization of Gaussian states in the last part of this section
In this paper, motivated by recent development of circuit complexity for TFD states in the free scalar theory, we have generalized some of those results to the time-dependent TFD states in free, noninteracting fermionic system
Summary
Applications of quantum information concepts to high energy physics and gravity have led to many promising results. In the context of holographic gravity, the TFD states (1.1) dual to the left and right sides of the geometry which is connected by a wormhole, or ER bridge, and the volume of this geometry increases for a time which is exponential in the number of degrees of freedom (d.o.f.) of the boundary theory This time is much larger than other characteristic times in holographic gravity which indicates that entanglement alone is not enough to capture the dynamics behind the horizon and there must exist some other quantity we yet do not know which have the information about this late-time evolution of the wormhole interior. We review some key elements of Nielsen’s approach to evaluate circuit complexity, and demonstrate how both the timeindependent and time-dependent fermionic TFD states of 2N decoupled harmonic oscillators can be generated by quadratic operators, for later use, we write down the covariance matrices, which serves as an alternate–and more natural–characterization of Gaussian states in the last part of this section.
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