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.

Consistent formulations of relativistic viscous hydrodynamics involve short-lived modes, leading to asymptotic rather than convergent gradient expansions. In this Letter we consider the Müller-Israel-Stewart theory applied to a longitudinally expanding quark-gluon plasma system and identify hydrodynamics as a universal attractor without invoking the gradient expansion. We give strong evidence for the existence of this attractor and then show that it can be recovered from the divergent gradient expansion by Borel summation. This requires careful accounting for the short-lived modes which leads to an intricate mathematical structure known from the theory of resurgence.

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 report on the approach toward the hydrodynamic regime of boost-invariant N=4 super Yang-Mills plasma at strong coupling starting from various far-from-equilibrium states at τ=0. The results are obtained through a numerical solution of Einstein's equations for the dual geometries, as described in detail in the companion article [M. P. Heller, R. A. Janik, and P. Witaszczyk, arXiv:1203.0755]. Despite the very rich far-from-equilibrium evolution, we find surprising regularities in the form of clear correlations between initial entropy and total produced entropy, as well as between initial entropy and the temperature at thermalization, understood as the transition to a hydrodynamic description. For 29 different initial conditions that we consider, hydrodynamics turns out to be definitely applicable for proper times larger than 0.7 in units of inverse temperature at thermalization. We observe a sizable anisotropy in the energy-momentum tensor at thermalization, which is nevertheless entirely due to hydrodynamic effects. This suggests that effective thermalization in heavy-ion collisions may occur significantly earlier than true thermalization.