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

We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited states. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.Let us also note that there is no need to consider a tolerance with this continuous description, since the Y (s) can always be adjusted to produce exactly the desired transformation (2.1). 4 Note that our notation here differs slightly from that in [32] where the overall factor of −i was absorbed in the O I , which were then anti-Hermitian operators. 5 We define the boundary conditions more precisely below in the discussion around eq. (2.11).8 To correctly account for the dimensions of q and p, these expressions should include a specific scale, e.g., a = 1 √ 2 (ω 1 q + i p/ω 1 ) yields a properly dimensionless annihilation operator. One effect of the Bogoliubov transformations (3.5) is then to scale this scale, e.g., ω 1 → e r ω 1 with ϕ = ϑ = 0 in eq. (3.7). See [38] for further discussion.9 Note that we can change a toã = e iϕ a without changing the vacuum, which corresponds to a U(1) subgroup of Bogoliubov transformations that do not change the vacuum. For N bosonic degrees of freedom, there is the freedom of unitarily mixing all N annihilation operators (and creation operators respectively) among themselves, leading to a U(N ) subgroup of different choices of a i that all define the same vacuum. 20 Of course, the symmetric combinations are trivial, since {ξ a , ξ b } = G ab ≡ δ ab .

In a seminal paper [Phys. Rev. Lett. 71, 1291], Page proved that the entanglement entropy of typical pure states is Styp ln DA − (1/2)D 2 A /D, for 1 DA ≤ √ D, where DA and D are the Hilbert space dimensions of the subsystem and the system, respectively. Typical pure states are hence (nearly) maximally entangled. We develop tools to compute the average entanglement entropy S of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models, ln DA − (ln DA)2 / ln D. Consequently we prove that: (i) if the subsystem size is a finite fraction of the system size then S < ln DA in the thermodynamic limit, i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal, i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.Introduction. The concept of entanglement is a cornerstone in modern quantum physics. Different measures of entanglement have been extensively used to probe the structure of pure quantum states [1], and they have started to be measured in experiments with ultracold atoms in optical lattices [2,3]. Here, we are interested in the bipartite entanglement entropy (referred to as the entanglement entropy) in fermionic lattice systems. In such systems, an upper bound for the entanglement entropy of a subsystem A (smaller than its complement) is S max = ln D A , where D and D A are the dimensions of the Hilbert space of the system and of the subsystem, with D A ≤ √ D (see Fig. 1 for an example for spinless fermions). Note that ln D A ∝ V A , where V A is the number of sites in A, i.e., this upper bound scales with the "volume" of A. (When A is larger than its complement, the Hilbert space of the complement is the one that determines S.) Almost twenty-four years ago, motivated by the puzzle of information in black hole radiation [4], Page proved [5] that typical (with respect to the Haar measure) pure states nearly saturate that bound (the correction is exponentially small) [6][7][8][9][10]. Their reduced density matrices are thermal at infinite temperature [11][12][13].In stark contrast with typical pure states, ground states and low-lying excited states of local Hamiltonians are known to exhibit an area-law entanglement [1]. Namely, their entanglement entropy scales with the area of the boundary of the subsystem. On the other hand, most eigenstates of local Hamiltonians at nonzero energy densities above the ground state are expected to have a volume-law entanglement entropy (with the exception of many-body localized systems [14,15]). Within the eigenstate thermalization hypothesis (ETH) [16][17][18], one expects volume-law entanglement in all eigenstates (excluding those at the edges of the spectrum) of quantum chaotic Hamiltonians [19][20][21][22][23], with those in the center of the spectrum exhibiting maximal entanglement [...

Abstract:The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate h KS given by the sum of all positive Lyapunov exponents of the system. We prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian. The derivation takes into account the case of time-dependent Hamiltonians with Floquet instabilities. We show that the entanglement entropy S A of a Gaussian state grows linearly for large times in unstable systems, with a rate Λ A ≤ h KS determined by the Lyapunov exponents and the choice of the subsystem A. We apply our results to the analysis of entanglement production in unstable quadratic potentials and due to periodic quantum quenches in many-body quantum systems. Our results are relevant for quantum field theory, for which we present three applications: a scalar field in a symmetry-breaking potential, parametric resonance during post-inflationary reheating and cosmological perturbations during inflation. Finally, we conjecture that the same rate Λ A appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.

We study the time evolution of the entanglement entropy in bosonic systems with timeindependent, or time-periodic, Hamiltonians. In the first part, we focus on quadratic Hamiltonians and Gaussian initial states. We show that all quadratic Hamiltonians can be decomposed into three parts: (a) unstable, (b) stable, and (c) metastable. If present, each part contributes in a characteristic way to the time-dependence of the entanglement entropy: (a) linear production, (b) bounded oscillations, and (c) logarithmic production. In the second part, we use numerical calculations to go beyond Gaussian states and quadratic Hamiltonians. We provide numerical evidence for the conjecture that entanglement production through quadratic Hamiltonians has the same asymptotic behavior for non-Gaussian initial states as for Gaussian ones. Moreover, even for non-quadratic Hamiltonians, we find a similar behavior at intermediate times. Our results are of relevance to understanding entanglement production for quantum fields in dynamical backgrounds and ultracold atoms in optical lattices. arXiv:1710.04279v2 [hep-th]

Much has been learned about universal properties of entanglement entropies in ground states of quantum many-body lattice systems. Here we unveil universal properties of the average bipartite entanglement entropy of eigenstates of the paradigmatic quantum Ising model in one dimension. The leading term exhibits a volume-law scaling that we argue is universal for translationally invariant quadratic models. The subleading term is constant at the critical field for the quantum phase transition and vanishes otherwise (in the thermodynamic limit), i.e., the critical field can be identified from subleading corrections to the average (over all eigenstates) entanglement entropy.

We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-Kähler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

We derive a formula for the entanglement entropy of squeezed states on a lattice in terms of the complex structure J. The analysis involves the identification of squeezed states with group-theoretical coherent states of the symplectic group and the relation between the coset Sp(2N, R)/Isot(J 0 ) and the space of complex structures. We present two applications of the new formula: (i) we derive the area law for the ground state of a scalar field on a generic lattice in the limit of small speed of sound, (ii) we compute the rate of growth of the entanglement entropy in the presence of an instability and show that it is asymptotically bounded from above by the Kolmogorov-Sinai rate. *