Abstract:We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. For either conjecture, we find that the late time limit for the rate of change of complexity is saturated at times of the order of the inverse temperature. Adding a charge to the eternal black holes washes out the early time behaviour, i.e. complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.

We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a statedependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form suð1; 1Þ algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals. DOI: 10.1103/PhysRevLett.120.121602 Introduction.-Applications of quantum information concepts to high-energy physics and gravity have recently led to many far-reaching developments. In particular, it has become apparent that special properties of entanglement in holographic [1] quantum field theory (QFT) states are crucial for the emergence of smooth higher-dimensional (bulk) geometries in the gauge-gravity duality [2]. Much of the progress in this direction was achieved by building on the holographic entanglement entropy proposal by Ryu and Takayanagi [3], which geometrizes the von Neumann entropy of a reduced density matrix of a QFT in a subregion in terms of the area of codimension-2 bulk minimal surfaces anchored at the boundary of this subregion (see, e.g., Ref. [4] for a recent overview). However, Ryu-Takayanagi surfaces are often unable to access the whole holographic geometry [5][6][7]. This observation has led to significant interest in novel, from the point of view of quantum gravity, codimension-1 (volume) and codimension-0 (action) bulk quantities, whose behavior suggests conjecturing a link with the information theory notion of quantum state complexity [8][9][10][11][12][13][14]. In fact, a certain identification between complexity and action was originally suggested by Toffoli [15,16] outside the context of holography.

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

It was recently conjectured that the quantum complexity of a holographic boundary state can be computed by evaluating the gravitational action on a bulk region known as the Wheeler-DeWitt patch. We apply this complexity=action duality to evaluate the 'complexity of formation' [1, 2], i.e., the additional complexity arising in preparing the entangled thermofield double state with two copies of the boundary CFT compared to preparing the individual vacuum states of the two copies. We find that for boundary dimensions d > 2, the difference in the complexities grows linearly with the thermal entropy at high temperatures. For the special case d = 2, the complexity of formation is a fixed constant, independent of the temperature. We compare these results to those found using the complexity=volume duality. arXiv:1610.08063v2 [hep-th]

We examine holographic complexity in time-dependent Vaidya spacetimes with both the complexity=volume (CV) and complexity=action (CA) proposals. We focus on the evolution of the holographic complexity for a thin shell of null fluid, which collapses into empty AdS space and forms a (one-sided) black hole. In order to apply the CA approach, we introduce an action principle for the null fluid which sources the Vaidya geometries, and we carefully examine the contribution of the null shell to the action. Further, we find that adding a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is essential if the gravitational action is to properly describe the complexity of the boundary state. For both the CV proposal and the CA proposal (with the extra boundary counterterm), the late time limit of the growth rate of the holographic complexity for the one-sided black hole is precisely the same as that found for an eternal black hole.

In this second part of the study initiated in [1], we investigate holographic complexity for eternal black hole backgrounds perturbed by shock waves, with both the complexity=action (CA) and complexity=volume (CV) proposals. In particular, we consider Vaidya geometries describing a thin shell of null fluid with arbitrary energy falling in from one of the boundaries of a two-sided AdS-Schwarzschild spacetime. We demonstrate how known properties of complexity, such as the switchback effect for light shocks, as well as analogous properties for heavy ones, are imprinted in the complexity of formation and in the full time evolution of complexity. Following our discussion in [1], we find that in order to obtain the expected properties of the complexity, the inclusion of a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is required for the CA proposal. arXiv:1805.07262v1 [hep-th] 18 May 2018 1 Our geometries are more properly interpreted in terms of a thermal quench, e.g., [68,69], where some boundary coupling is rapidly varied at t R = −t w . Instead, eq. (2.4) corresponds to an excited state in which the excitation becomes coherent at t R = −t w (but with no variations of the couplings). The corresponding bulk geometry involves a null shell which emerges from the white hole singularity and reflects off of the right asymptotic boundary at t R = −t w to become a collapsing shell, e.g., see [15,47,49]. Since our evaluations of the holographic complexity always involve t R > −t w , our results would be the same for either geometry.2 The details of the operator will not be important for our analysis, however, for the special case of d = 2, [50] provides a detailed description of the dual of the Vaidya-AdS 3 geometry.14 Again, d = 2 is a special case with r * 1 (0) = 0 -see eq. (3.56) below.

In this paper we obtain holographic formulas for the transport coefficients κ and τ π present in the second-order derivative expansion of relativistic hydrodynamics in curved spacetime associated with a non-conformal strongly coupled plasma described holographically by an Einstein+Scalar action in the bulk. We compute these coefficients as functions of the temperature in a bottom-up non-conformal model that is tuned to reproduce lattice QCD thermodynamics at zero baryon chemical potential. We directly compute, besides the speed of sound, 6 other transport coefficients that appear at second-order in the derivative expansion. We also give an estimate for the temperature dependence of 11 other transport coefficients taking into account the simplest contribution from non-conformal effects that appear near the QCD crossover phase transition. Using these results, we construct an Israel-Stewart-like theory in flat spacetime containing 13 of these 17 transport coefficients that should be suitable for phenomenological applications in the context of numerical hydrodynamic simulations of the strongly-coupled, non-conformal quark-gluon plasma. Using several different approximations, we give parametrizations for the temperature dependence of all the second-order transport coefficients that appear in this theory in a format that can be easily implemented in existing numerical hydrodynamic codes.

We use symmetry arguments developed by Gubser to construct the first radially-expanding explicit solutions of the Israel-Stewart formulation of hydrodynamics. Along with a general semi-analytical solution, an exact analytical solution is given which is valid in the cold plasma limit where viscous effects from shear viscosity and the relaxation time coefficient are important. The radially expanding solutions presented in this paper can be used as nontrivial checks of numerical algorithms employed in hydrodynamic simulations of the quark-gluon plasma formed in ultra-relativistic heavy ion collisions. We show this explicitly by comparing such analytic and semi-analytic solutions with the corresponding numerical solutions obtained using the music viscous hydrodynamics simulation code.