Complexity and entanglement for thermofield double states

Abstract

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. 1 arXiv:1810.05151v4 [hep-th] 15 Feb 2019 C Matrix generators for Sp(4, R) 87 D Comments on bases 88 E Complexity of formation in the diagonal basis 91 F Minimal geodesics for N degrees of freedom with λ R = 1 95 G Derivation of the TFD covariance matrix in terms of matrix functions 105 References 107

Aspects of the first law of complexity

“…Hence it should not be surprising that the corresponding complexity exhibits oscillations. 31 31 The time dependence of the complexity of the thermofield double state of a free scalar was studied in [64]. Recall that in this case, the complexity was constant at late times (in contrast to the linear growth seen in holography) because the time evolution only explored a particular submanifold of Gaussian states within the full Hilbert state.…”

“…However, it is natural that this restricted basis should form a closed algebra, and typically, the O I provide a representation of a Lie algebra g, i.e., [O I , O J ] = if IJ K O K . For example, a GL(N , R) group appears in evaluating the complexity of the ground state of a free scalar field [57], and the latter was extended to a Sp(2N , R) group in examining the corresponding thermofield double state [64] -see also [60]. 5 In the following, we will find that the affine symplectic group, i.e., R 2N Sp(2N , R) plays a central role in evaluating the complexity of the coherent states of interest.…”

“…This approach was first applied to a concrete quantum field theory calculation in [57], where the authors adapted Nielsen's approach to evaluate the complexity of the vacuum state of a free scalar field theory. These calculations have been extended in a number of interesting ways in the past few years, e.g., [58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76], but we will be particularly interested in [65] where the same techniques were applied to explore the complexity of coherent states in the same QFT.…”

“…In this case, one must still minimize eq. (2.5) over this family to determine the complexity, e.g., see[57,60,64] 5. The symmetry closed by the gate generators was used in[62] to physically argue for some natural choice of cost functions.…”

“…In the previous literature[57,64,65], the implicit choice 1 χ AB = 1 was taken for the GL(N , R) subgroup generated by the off-diagonal block (4.20), while in[64], for the diagonal blocks the coefficients depended on a gate scale ω g (see eqs (37). and(59) in[64]). …”