Thermofield Double State Research Articles

Subsystem complexity in warped AdS

We compute the ultraviolet divergences of holographic subregion complexity for the left and right factors of the thermofield double state in warped AdS3 black holes, both for the action and the volume conjectures. Besides the linear divergences, which are also present in the BTZ black hole, additional logarithmic divergences appear. For the action conjecture, these log divergences are not affected by the arbitrarity in the length scale associated with the counterterm needed to ensure reparameterization invariance. We find that the subregion action complexity obeys the superadditivity property for the thermofield double in warped AdS3, independently from the action counterterm coefficient. We study the temperature dependence of subregion complexity at constant angular momentum and we find that it is correlated with the sign of the specific heat.

Stringy effects and the role of the singularity in holographic complexity

There has been considerable recent interest in holographic complexity. The two leading conjectures on this subject hold that the quantum complexity of the boundary thermofield double state should be dual to either the volume of the Einstein-Rosen bridge connecting the two sides (CV conjecture) or to the action of the Wheeler-de-Witt patch of the bulk spacetime (CA conjecture). Although these conjectures are frequently studied in the context of pure Einstein gravity, from the perspective of string theory it is also natural to consider models of gravity in which general relativity is perturbed by higher powers of the Riemann tensor, suppressed by powers of the string length; in a holographic context, these corrections are dual to corrections in inverse powers of the ’t Hooft coupling. In this paper, we investigate the CV and CA conjectures in two stringy models of higher-curvature gravity. We find that the CV complexification rate remains well-behaved, but conversely that these corrections induce new divergences in the CA complexification rate that are absent in pure Einstein gravity. These divergences are intrinsically linked to the singularity, and appear to be generic in higher curvature theories. To the best of our knowledge, infinities originating at the singularity have not yet been observed elsewhere in the literature. We argue that these divergences imply that, in the CA picture, the complexification rate of the boundary theory is a nonanalytic function of the ’t Hooft coupling.

Holographic CFT states for localized perturbations to AdS black holes

In this note, we describe a holographic CFT construction of states dual to scalar perturbations of the maximally extended three-dimensional AdS-Schwarzschild black hole. The states are constructed by adding sources for a scalar operator to the path integral that constructs the thermofield double state. For a scalar field of arbitrary mass, we provide the general map between sources and scalar field perturbations at linear order. With this, we investigate to what extent it is possible using this construction to produce perturbations localized to one side of the black hole horizon. Our results suggest that it is possible to produce arbitrarily localized perturbations (and thus, plausibly, general linear perturbations to the black hole initial data slice), but that the amplitude of the perturbation must be taken small as the perturbation becomes more localized in order that the sources do not diverge, as found for the pure AdS case considered in arXiv:1709.10101.

Quantum chaos transition in a two-site Sachdev-Ye-Kitaev model dual to an eternal traversable wormhole

It has been recently proposed by Maldacena and Qi that an eternal traversable wormhole in a two-dimensional anti--de Sitter space is the gravity dual of the low temperature limit of two Sachdev-Ye-Kitaev (SYK) models coupled by a relevant interaction (which we will refer to as spin operator). We study spectral and eigenstate properties of this coupled SYK model. We find that level statistics in the tail of the spectrum, and for a sufficiently weak coupling, show substantial deviations from random matrix theory, which suggests that traversable wormholes are not quantum chaotic. By contrast, for sufficiently strong coupling, corresponding to the black hole phase, level statistics are well described by random matrix theory. This transition in level statistics coincides approximately with a previously reported Hawking-Page transition for weak coupling. We show explicitly that this thermodynamic transition turns into a sharp crossover as the coupling increases. Likewise, this critical coupling also corresponds to the one at which the overlap between the ground state and the thermofield double state (TFD) is smallest. In the range of sizes we can reach by exact diagonalization, the ground state is well approximated by the TFD only in the strong coupling limit. This is due to the fact that the ground state is close to the eigenstate of the spin operator corresponding to the lowest eigenvalue which is an exact TFD at infinite temperature. In this region, the spectral density is separated into blobs centered around the eigenvalues of the spin operator. For weaker couplings, the exponential decay of coefficients in a tensor product basis, typical of the TFD, becomes power law. Finally, we also find that the total Hamiltonian has an additional discrete symmetry which has not been reported previously.

Entanglement entropy of two coupled SYK models and eternal traversable wormhole

In this paper, we study the entanglement entropy between two SYK systems with bilinear coupling. We use the replica trick to calculate the entanglement entropy in the ground state. In parallel, we calculate the entanglement entropy through the Ryu-Takayanagi formula in gravity. For the ground state that is dual to an eternal traversable wormhole in AdS2, the bulk quantum correction to the entanglement entropy is of the same order as the minimal surface area. The ground state of the coupled system is close to a thermofield double state with particular temperature and they have the same entanglement entropy. From the gravity point of view, we explain why the two states have the same entanglement entropy. We also study a case with time-dependent coupling, which involves finding the quantum extremal surface in the bulk.

Entanglement negativity and minimal entanglement wedge cross sections in holographic theories

We calculate logarithmic negativity, a quantum entanglement measure for mixed quantum states, in quantum error-correcting codes and find it to equal the minimal cross sectional area of the entanglement wedge in holographic codes with a quantum correction term equal to the logarithmic negativity between the bulk degrees of freedom on either side of the entanglement wedge cross section. This leads us to conjecture a holographic dual for logarithmic negativity that is related to the area of a cosmic brane with tension in the entanglement wedge plus a quantum correction term. This is closely related to (though distinct from) the holographic proposal for entanglement of purification. We check this relation for various configurations of subregions in AdS${}_3$/CFT${}_2$. These are disjoint intervals at zero temperature, as well as a single interval and adjacent intervals at finite temperature. We also find this prescription to effectively characterize the thermofield double state. We discuss how a deformation of a spherical entangling region complicates calculations and speculate how to generalize to a covariant description.

Comments on black hole interiors and modular inclusions

We show how the traversable wormhole induced by a double-trace deformation of the thermofield double state can be understood as a modular inclusion of the algebras of exterior operators. The effect of this deformation is the creation of a new region of spacetime deep in the bulk, corresponding to a non-trivial center between the left and right algebras. This set-up provides a precise framework for investigating how black hole interiors are encoded in the CFT. In particular, we use modular theory to demonstrate that state dependence is an inevitable feature of any attempt to represent operators behind the horizon. Building on this geometrical structure, we propose that modular inclusions may provide a more precise means of investigating the nascent relationship between entanglement and geometry in the context of the emergent spacetime paradigm.

Complexity and entanglement for thermofield double states

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 = 0t=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>0t>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.

Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic Quantum Systems

Characterizing the work statistics of driven complex quantum systems is generally challenging because of the exponential growth with the system size of the number of transitions involved between different energy levels. We consider the quantum work distribution associated with the driving of chaotic quantum systems described by random matrix Hamiltonians and characterize exactly the work statistics associated with a sudden quench for arbitrary temperature and system size. Knowledge of the work statistics yields the Loschmidt echo dynamics of an entangled state between two copies of the system of interest, the thermofield double state. This echo dynamics is dictated by the spectral form factor. We discuss its relation to frame potentials and its use to assess information scrambling.

Complexity and the bulk volume, a new York time story

We study the boundary description of the volume of maximal Cauchy slices using the recently derived equivalence between bulk and boundary symplectic forms. The volume of constant mean curvature slices is known to be canonically conjugate to “York time”. We use this to construct the boundary deformation that is conjugate to the volume in a handful of examples, such as empty AdS, a backreacting scalar condensate, or the thermofield double at infinite time. We propose a possible natural boundary interpretation for this deformation and use it to motivate a concrete version of the complexity=volume conjecture, where the boundary complexity is defined as the energy of geodesics in the Kähler geometry of half sided sources. We check this conjecture for Bañados geometries and a mini-superspace version of the thermofield double state. Finally, we show that the precise dual of the quantum information metric for marginal scalars is given by a particularly simple symplectic flux, instead of the volume as previously conjectured.

How to build the thermofield double state

Given two copies of any quantum mechanical system, one may want to prepare them in the thermofield double state for the purpose of studying thermal physics or black holes. However, the thermofield double is a unique entangled pure state and may be difficult to prepare. We propose a local interacting Hamiltonian for the combined system whose ground state is approximately the thermofield double. The energy gap for this Hamiltonian is of order the temperature. Our construction works for any quantum system satisfying the Eigenvalue Thermalization Hypothesis.

Expanding the black hole interior: partially entangled thermal states in SYK

We introduce a family of partially entangled thermal states in the SYK model that interpolates between the thermo-field double state and a pure (product) state. The states are prepared by a euclidean path integral describing the evolution over two euclidean time segments separated by a local scaling operator mathcal{O} . We argue that the holographic dual of this class of states consists of two black holes with their interior regions connected via a domain wall, described by the worldline of a massive particle. We compute the size of the interior region and the entanglement entropy as a function of the scale dimension of mathcal{O} and the temperature of each black hole. We argue that the one-sided bulk reconstruction can access the interior region of the black hole.

Scrambling time from local perturbations of the rotating BTZ black hole

In this paper, we investigate the entanglement entropy of the rotating BTZ black hole perturbed by a massive back-reacting free falling particle. Then, mutual information between two finite intervals in two asymptotic regions of rotating BTZ is derived. It allows us to find the scrambling time, the time scale in which mutual information vanishes. We give a dual large c CFT description in terms of a thermofield double state with different temperatures for left and right moving modes that is perturbed by a local operator. Exact matching between gravity and CFT results is obtained.

Circuit complexity for fermionic thermofield double states

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.

Holographic entanglement of purification for thermofield double states and thermal quench

We explore the properties of holographic entanglement of purification (EoP) for two disjoint strips in the Schwarzschild-AdS black brane and the Vaidya-AdS black brane spacetimes. For two given strips on the same boundary of Schwarzschild-AdS spacetime, there is an upper bound of the separation beyond which the holographic EoP will always vanish no matter how wide the strips are. In the case that two strips are in the two boundaries of the spacetime respectively, we find that the holographic EoP exists only when the strips are wide enough. If the width is finite, the EoP can be nonzero in a finite time region. For thermal quench case, we find that the equilibrium time of holographic EoP is only sensitive to the width of strips, while that of the holographic mutual information is sensitive not only to the width of strips but also to their separation.

Chaos and entanglement spreading in a non-commutative gauge theory

Holographic theories with classical gravity duals are maximally chaotic: they saturate a set of bounds on the spread of quantum information. In this paper we question whether non-locality can affect such bounds. Specifically, we consider the gravity dual of a prototypical theory with non-local interactions, namely, mathcal{N}=4 non-commutative super Yang Mills. We construct shock waves geometries that correspond to perturbations of the thermofield double state with definite momentum and study several chaos related properties of the theory, including the butterfly velocity, the entanglement velocity, the scrambling time and the maximal Lyapunov exponent. The latter two are unaffected by the non-commutative parameter θ, however, both the butterfly and entanglement velocities increase with the strength of the non-commutativity. This implies that non-local interactions can enhance the effective light-cone for the transfer of quantum information, eluding previously conjectured bounds encountered in the context of local quantum field theory. We comment on a possible limitation on the retrieval of quantum information imposed by non-locality.

Bulk metric reconstruction from boundary entanglement

Most of the literature in the \emph{bulk reconstruction program} in holography focuses on recovering local bulk operators propagating on a quasilocal bulk geometry and the knowledge of the bulk geometry is always assumed or guessed. The fundamental problem of the bulk reconstruction program, which is \emph{recovering the bulk background geometry (metric)} from the boundary CFT state is still outstanding. In this work, we formulate a recipe to extract the bulk metric itself from the boundary state, specifically, the modular Hamiltonian information of spherical subregions in the boundary. Our recipe exploits the recent construction of Kabat and Lifschytz \cite{Kabat:2017mun} to first compute the bulk two point function of scalar fields directly in the CFT without knowledge of the bulk metric or the equations of motion, and then to take a large scaling dimension limit (WKB) to extract the geodesic distance between two close points in the bulk i.e. the metric. As a proof of principle, we consider three dimensional bulk and selected CFT states such as the vacuum and the thermofield double states. We show that they indeed reproduce the pure AdS and the regions outside the Rindler wedge and the BTZ black hole \emph{up to a rigid conformal factor}. Since our approach does not rely on symmetry properties of the CFT state, it can be applied to reconstruct asymptotically AdS geometries dual to arbitrary general CFT states provided the modular Hamiltonian is available. We discuss several obvious extensions to the case of higher spacetime dimensions as well as some future applications, in particular, for constructing metric beyond the causal wedge of a boundary region. In the process, we also extend the construction of \cite{Kabat:2017mun} to incorporate the first order perturbative locality for AdS scalars.

Decay of a thermofield-double state in chaotic quantum systems

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite $N$ for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.

Microcanonical path integrals and the holography of small black hole interiors

We use a microcanonical path integral closely related to that introduced by Brown and York in 1992 to add new entries to the AdS/CFT dictionary concerning the interiors of small black holes. Stationary points of such path integrals are also stationary points of more standard canonical-type path integrals with fixed boundary metric, but the condition for dominance is now maximizing Hubeny-Rangamani-Takayanagi entropy at fixed energy. As a result, such path integrals can bring to the fore saddles that fail to dominate in more familiar contexts. We use this feature to argue that the standard Kruskal-like two-sided extension of small AdS black holes with energy E0 is dual to a microcanonical version of the thermofield double state for AdS black holes that maximize the microcanonical bulk entropy at this energy. We also comment on entanglement in such states and on quantum effects that become large when the energy-width of the microcanonical ensemble is sufficiently small.

Chaos and relative entropy

One characteristic feature of a chaotic system is the quick delocalization of quantum information (fast scrambling). One therefore expects that in such a system a state quickly becomes locally indistinguishable from its perturbations. In this paper we study the time dependence of the relative entropy between the reduced density matrices of the thermofield double state and its perturbations in two dimensional conformal field theories. We show that in a CFT with a gravity dual, this relative entropy exponentially decays until the scrambling time. This decay is not uniform. We argue that the early time exponent is universal while the late time exponent is sensitive to the butterfly effect. This large c answer breaks down at the scrambling time, therefore we also study the relative entropy in a class of spin chain models numerically. We find a similar universal exponential decay at early times, while at later times we observe that the relative entropy has large revivals in integrable models, whereas there are no revivals in non-integrable models.