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arXiv:1812.00193 · Physical review. D/Physical review. D.

Circuit complexity for fermionic thermofield double states

Jie Jiang, Xiangjing Liu

Published Jan 17, 2019• 43 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevd.99.026011

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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 L2{L}^{2}L2 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.
Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0 t = 0 , we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t>0 t > 0 , we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of holographic proposals, is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel technical aspect of our work is framing complexity calculations in the language of covariance matrices and the associated symplectic transformations, which provide a natural language for dealing with Gaussian states. Furthermore, for free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with the time dependence of the entanglement entropy for simple bipartitions of TFDs. We relate our results for the entanglement entropy to previous studies on non-equilibrium entanglement evolution following quenches. We also present a new analytic derivation of a logarithmic contribution due to the zero momentum mode in the limit of vanishing mass for a subsystem containing a single degree of freedom on each side of the TFD and argue why a similar logarithmic growth should be present for larger subsystems.
Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0, we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t > 0, we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of holographic proposals, is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel technical aspect of our work is framing complexity calculations in the language of covariance matrices and the associated symplectic transformations, which provide a natural language for dealing with Gaussian states. Furthermore, for free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with the time dependence of the entanglement entropy for simple bipartitions of TFDs. We relate our results for the entanglement entropy to previous studies on non-equilibrium entanglement evolution following quenches. We also present a new analytic derivation of a logarithmic contribution due to the zero momentum mode in the limit of vanishing mass for a subsystem containing a single degree of freedom on each side of the TFD and argue why a similar logarithmic growth should be present for larger subsystems.
We compute the time-dependent complexity of the thermofield double states by four different proposals: two holographic proposals based on the "complexity-action" (CA) conjecture and "complexity-volume" (CV) conjecture, and two quantum field theoretic proposals based on the Fubini-Study metric (FS) and Finsler geometry (FG). We find that four different proposals yield both similarities and differences, which will be useful to deepen our understanding on the complexity and sharpen its definition. In particular, at early time the complexity linearly increase in the CV and FG proposals, linearly decreases in the FS proposal, and does not change in the CA proposal. In the late time limit, the CA, CV and FG proposals all show that the growth rate is 2E/(πℏ)2E/(π\hbar)2E/(πℏ) saturating the Lloyd's bound, while the FS proposal shows the growth rate is zero. It seems that the holographic CV conjecture and the field theoretic FG method are more correlated.
This paper studies the complexity between states in quantum field theory by introducing a Finsler structure based on ladder operators (the generalization of creation and annihilation operators). Two simple models are shown as examples to clarify the differences between complexity and other conceptions such as complexity of formation and entanglement entropy. When it is applied into thermofield double (TFD) states in ddd-dimensional conformal field theory, results show that the complexity density between them and corresponding vacuum states are finite and proportional to ${T}^{d\ensuremath{-}1}$, where TTT is the temperature of TFD state. Especially, a proof is given to show that fidelity susceptibility of a TFD state is equivalent to the complexity between it and corresponding vacuum state, which gives an explanation why they may share the same object in holographic duality. Some enlightenments to holographic conjectures of complexity are also discussed.
We compute the time-dependent complexity of the thermofield double states by four different proposals: two holographic proposals based on the complexity-action (CA) conjecture and complexity-volume (CV) conjecture, and two quantum field theoretic proposals based on the Fubini-Study metric (FS) and Finsler geometry (FG). We find that four different proposals yield both similarities and differences, which will be useful to deepen our understanding on the complexity and sharpen its definition. In particular, at early time the complexity linearly increase in the CV and FG proposals, linearly decreases in the FS proposal, and does not change in the CA proposal. In the late time limit, the CA, CV and FG proposals all show that the growth rate is 2E/(πℏ)2E/(\pi\hbar)2E/(πℏ) saturating the Lloyd's bound, while the FS proposal shows the growth rate is zero. It seems that the holographic CV conjecture and the field theoretic FG method are more correlated.
By using a recent approach proposed by Hackl et al. to evaluate the complexity of the free fermionic Gaussian state, we compute the complexity of the Dirac vacuum state of the Fermi system with a mass quench. First of all, we review the counting method given by Hackl et al., and demonstrate that the result can be adapted to all of the compact transformation group G. Then, we utilize this result to study the time evolution of the complexity of these states. We show that, for the rotational invariant reference state, the total complexity of the incoming vacuum state will saturate the value of the instantaneous vacuum state at the late times, with a typical timescale to achieve the final stable state. Moreover, we find that the complexity growth under the sudden quench is directly proportional to the mass difference. And all of the results show some similar behaviors with the circuit complexity of Thermofield double states.