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We develop a mathematical formalism that allows to study decoherence with a great level generality, so as to make it appear as a geometrical phenomenon between reservoirs of dimensions. It enables us to give quantitative estimates of the level of decoherence induced by a purely random environment on a system according to their respectives sizes, and to exhibit some links with entanglement entropy.

By entangling soft massless particles one can create an arbitrarily large amount of entanglement entropy that carries an arbitrarily small amount of energy. Dropping this entropy into the black hole (b.h.) one can increase the b.h. entropy by an amount that violates Bekenstein bound or any other reasonable bound, leading to a version of b.h. information paradox that does not involve Hawking radiation. Among many proposed solutions of the standard b.h. information paradox with Hawking radiation, only a few can also resolve this version without the Hawking radiation. The assumption that both versions should be resolved in the same way significantly helps to reduce the space of possible resolutions., Comment: 11 pages, revised and improved, accepted for publication in Universe

We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between $\mathsf{QMA}$ and $\mathsf{QMA(2)}$, a long standing question in quantum complexity theory., 58 pages, v2: typos corrected, Section 6.1-6.3 revised, added some new results

In this article, we investigate various physical implications of quantum circuit complexity using squeezed state formalism of Primordial Gravitational Waves (PGW). Recently quantum information theoretic concepts, such as entanglement entropy, and complexity are playing a pivotal role to understand the dynamics of quantum system even in the diverse fields such as, high energy physics and cosmology. This paper is devoted in studying quantum circuit complexity of PGW for various cosmological models, such as de Sitter, inflation, radiation, reheating, matter, bouncing, cyclic and black hole gas model etc. We compute complexity measure using both Covariance and Nielsen's wave function method for three different choices of quantum initial vacua: Motta-Allen, $\alpha$ and Bunch-Davies. Besides computing circuit complexity, we have also computed Von-Neumann entanglement entropy. By making the comparison of complexity with entanglement entropy, we are able to probe various features regarding the dynamics of evolution for different cosmological models. Because entanglement entropy is independent of the squeezing angle, we are able to understand more details of the system using Nielsen's measure of complexity which is dependent on both squeezing parameter and angle. This implies that quantum complexity could indeed be a useful probe to study quantum features in cosmological scale. Quantum complexity is also becoming a powerful technique to understand the chaotic behaviour and random fluctuations of quantum fields. Using the growth of complexity, we are able to compute quantum Lyapunov exponent for various cosmological models and comment on it's chaotic nature., Comment: 113 pages, 18 figures, 1 appendix, Revised version, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: "Role of Black Holes in Testing Modified Theories of Gravity")

We study the quantum information spreading in one-dimensional free-fermion systems in the presence of localized thermal baths. We employ a nonlocal Lindblad master equation to describe the system-bath interaction, in the sense that the Lindblad operators are written in terms of the Bogoliubov operators of the closed system, and hence are nonlocal in space. The statistical ensemble describing the steady state is written in terms of a convex combination of the Fermi-Dirac distributions of the baths. Due to the singularity of the free-fermion dispersion, the steady-state mutual information exhibits singularities as a function of the system parameters. While the mutual information generically satisfies an area law, at the singular points it exhibits logarithmic scaling as a function of subsystem size. By employing the Fisher-Hartwig theorem, we derive the prefactor of the logarithmic scaling, which depends on the parameters of the baths and plays the role of an effective "central charge". This is upper bounded by the central charge governing ground-state entanglement scaling. We provide numerical checks of our results in the paradigmatic tight-binding chain and the Kitaev chain., Comment: 17 pages, 11 figures. Added two figures. Close to published version

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator $\rho_S$. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of $\rho_S$. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from $\rho_S$. Using examples where the SOI are generated using representations of $SU(2)$, $SO(3)$, and $SO(N)$, we show two features of the CG. (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grainied space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to maximum entanglement between system and environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function w.r.t the purity of $\rho_S$. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG., Comment: 12 pages, 10 figures. arXiv admin note: text overlap with arXiv:2111.07836

Recent advancements in our understanding of ordered magnets call for a quantification of their entanglement content on an equal footing with classical thermodynamic quantities, such as the total magnetic moment. We evaluate the entanglement entropy (EE) between the two sublattices of a bipartite ordered antiferromagnet finding it to scale with volume. Thus, the EE density becomes an intensive property and is evaluated to be a universal dimensionality-dependent constant when exchange is the dominant interaction. Our analytic results are validated against the DMRG-based analysis of a one-dimensional (1D) system, finding good agreement. Further, our evaluated EE per bond provides a useful shortcut towards obtaining the central-cut EE in 1D, and the area law in higher-dimensional magnets., Comment: 6 pages, 5 figures

We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement entropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models., 31 pages, 4 figures, 1 table

We numerically demonstrate that, although it is critical, the two-box symmetric $\mathrm{SU}(3)$ chain possesses edge states in the adjoint representation whose excitation energy scales with the number of sites $N_s$ as $1/(N_s \log N_s)$, in close analogy to those found in half-integer $\mathrm{SU}(2)$ chains with spin $S\ge 3/2$. We further show that these edge states dominate the entanglement entropy of finite chains, explaining why it has been impossible so far to verify with DMRG simulations the field theory prediction that this model is in the $\mathrm{SU}(3)_1$ universality class. Finally, we show that these edge states are very efficiently screened by attaching adjoint representations at the ends of the chain, leading to an estimate of the central charge consistent within 1\% with the prediction $c=2$ for $\mathrm{SU}(3)_1$., 9 pages including SM

Notions of circuit complexity and cost play a key role in quantum computing and simulation where they capture the (weighted) minimal number of gates that is required to implement a unitary. Similar notions also become increasingly prominent in high energy physics in the study of holography. While notions of entanglement have in general little implications for the quantum circuit complexity and the cost of a unitary, in this note, we discuss a simple such relationship when both the entanglement of a state and the cost of a unitary take small values, building on ideas on how values of entangling power of quantum gates add up. This bound implies that if entanglement entropies grow linearly in time, so does the cost. The implications are two-fold: It provides insights into complexity growth for short times. In the context of quantum simulation, it allows to compare digital and analog quantum simulators. The main technical contribution is a continuous-variable small incremental entangling bound., 7 pages, 1 figure, typos corrected

This is a continuation of our previous works on entanglement entropy (EE) in interacting field theories. In arXiv:2103.05303, we have proposed the notion of $\mathbb{Z}_M$ gauge theory on Feynman diagrams to calculate EE in quantum field theories and shown that EE consists of two particular contributions from propagators and vertices. As shown in the next paper arXiv:2105.02598, the purely non-Gaussian contributions from interaction vertices can be interpreted as renormalized correlation functions of composite operators. In this paper, we will first provide a unified matrix form of EE containing both contributions from propagators and (classical) vertices, and then extract further non-Gaussian contributions based on the framework of the Wilsonian renormalization group. It is conjectured that the EE in the infrared is given by a sum of all the vertex contributions in the Wilsonian effective action., Comment: 29 pages, 10 figures; typos corrected, published version in Symmetry (v2)

We study in detail various information theoretic quantities with the intent of distinguishing between different charged sectors in fractionalized states of large-$N$ gauge theories. For concreteness, we focus on a simple holographic $(2+1)$-dimensional strongly coupled electron fluid whose charged states organize themselves into fractionalized and coherent patterns at sufficiently low temperatures. However, we expect that our results are quite generic and applicable to a wide range of systems, including non-holographic. The probes we consider include the entanglement entropy, mutual information, entanglement of purification and the butterfly velocity. The latter turns out to be particularly useful, given the universal connection between momentum and charge diffusion in the vicinity of a black hole horizon. The RT surfaces used to compute the above quantities, though, are largely insensitive to the electric flux in the bulk. To address this deficiency, we propose a generalized entanglement functional that is motivated through the Iyer-Wald formalism, applied to a gravity theory coupled to a $U(1)$ gauge field. We argue that this functional gives rise to a coarse grained measure of entanglement in the boundary theory which is obtained by tracing over (part) of the fractionalized and cohesive charge degrees of freedom. Based on the above, we construct a candidate for an entropic $c$-function that accounts for the existence of bulk charges. We explore some of its general properties and their significance, and discuss how it can be used to efficiently account for charged degrees of freedom across different energy scales., 30 pages (+4 appendices), 9 figures. v2: typos corrected, references added. v3: General form of the generalized functional highlighted in the main text. Matches published version

It was shown recently, building on work of Alexakis, Balehowksy, and Nachman that the geometry of (some portion of) a manifold with boundary is uniquely fixed by the areas of a foliation of two-dimensional disk-shaped surfaces anchored to the boundary. In the context of AdS/CFT, this implies that (a portion of) a four-dimensional bulk geometry can be fixed uniquely from the entanglement entropies of disk-shaped boundary regions, subject to several constraints. In this Note, we loosen some of these constraints, in particular allowing for the bulk foliation of extremal surfaces to be local and removing the constraint of disk topology; these generalizations ensure uniqueness of more of the deep bulk geometry by allowing for e.g. surfaces anchored on disconnected asymptotic boundaries, or HRT surfaces past a phase transition. We also explore in more depth the generality of the local foliation requirement, showing that even in a highly dynamical geometry like AdS-Vaidya it is satisfied., Comment: 22 pages, 6 figures

Random quantum states and operations are of fundamental and practical interests. In this work, we investigate the entanglement properties of random hypergraph states, which generalize the notion of graph states by applying generalized controlled-phase gates on an initial reference product state. In particular, we study the two ensembles generated by random Controlled-Z(CZ) and Controlled-Controlled-Z(CCZ) gates, respectively. By applying tensor network representation and combinational counting, we analytically show that the average subsystem purity and entanglement entropy for the two ensembles feature the same volume law, but greatly differ in typicality, namely the purity fluctuation is small and universal for the CCZ ensemble while it is large for the CZ ensemble. We discuss the implications of these results for the onset of entanglement complexity and quantum chaos., Comment: 14+8 pages,9 figures,published version

Using the fact that we only observe those modes which exit the Hubble horizon during inflation, one can calculate the entanglement entropy of such long-wavelength perturbations by tracing out unobservable sub-Hubble fluctuations they are coupled to. On requiring that this perturbative entanglement entropy, which increases with time, obey the covariant entropy bound for an accelerating background, we find an upper bound on the duration of inflation. This presents a new perspective on the (meta-)stability of de Sitter spacetime and an associated lifetime for it., Comment: 6 pages, Comments welcome; v2: slightly expanded, title changed

We show that quantum entanglement between causally separated regions of a nucleon in antineutrino-nucleon scattering manifests itself as a thermal component in the resulting pion momentum distribution. For antineutrino scattering coherently from the (carbon) nucleus as a whole, this thermal component is absent, as expected by our quantum entanglement thermalization proposition. These phenomena, which have been observed in proton-proton collisions at the Large Hadron Collider, and in electromagnetic deep inelastic scattering, are now for the first time shown to exist in electroweak interactions as well., Comment: 8 pages, 3 figures

We show that one-body entanglement, which is a measure of the deviation of a pure fermionic state from a Slater determinant (SD) and is determined by the mixedness of the single-particle density matrix (SPDM), can be considered as a quantum resource. The associated theory has SDs and their convex hull as free states, and number conserving fermion linear optics operations (FLO), which include one-body unitary transformations and measurements of the occupancy of single-particle modes, as the basic free operations. We first provide a bipartitelike formulation of one-body entanglement, based on a Schmidt-like decomposition of a pure N-fermion state, from which the SPDM [together with the (N-1)-body density matrix] can be derived. It is then proved that under FLO operations the initial and postmeasurement SPDMs always satisfy a majorization relation, which ensures that these operations cannot increase, on average, the one-body entanglement. It is finally shown that this resource is consistent with a model of fermionic quantum computation which requires correlations beyond antisymmetrization. More general free measurements and the relation with mode entanglement are also discussed., Facultad de Ciencias Exactas, Instituto de Física La Plata

This work considers entropy generation and relaxation in quantum quenches in the Ising and $3$-state Potts spin chains. In the absence of explicit symmetry breaking we find universal ratios involving R\'enyi entropy growth rates and magnetisation relaxation for small quenches. We also demonstrate that the magnetisation relaxation rate provides an observable signature for the "dynamical Gibbs effect" which is a recently discovered characteristic non-monotonous behaviour of entropy growth linked to changes in the quasi-particle spectrum., Comment: Version 2: minor changes, abstract updated, `Submission to SciPost'; 28 pages, pdflatex, pdf figures. Version 3: clarifications and references added, typos corrected

We calculate the expectation value of the Wilson line in the pure AdS$_3$ Einstein gravity theory and also the entanglement entropy in the boundary theory. Our one-loop calculation of entanglement entropy shows a shift of the central charge 26. Finally, we show that the Wilson line provides the equivalent description to the boundary entanglement entropy. This equivalence leads to a concrete example of the building of "minimum surface=entanglement entropy"., 17 pages, minor changes

When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry., Comment: ver 1: 8 pages, 2 figures, ver 2: substantially updated, 19 pages, and 2 figures

In this work we consider two complex scalar fields distinguished by their masses coupled to constant background electric and magnetic fields in the $(3+1)$-dimensional Minkowski spacetime and subsequently investigate a few measures quantifying the quantum correlations between the created particle-antiparticle Schwinger pairs. Since the background magnetic field itself cannot cause the decay of the Minkowski vacuum, our chief motivation here is to investigate the interplay between the effects due to the electric and magnetic fields. We start by computing the entanglement entropy for the vacuum state of a single scalar field. Second, we consider some maximally entangled states for the two-scalar field system and compute the logarithmic negativity and the mutual information. Qualitative differences of these results pertaining to the charge content of the states are emphasised. Based upon these results, we suggest some possible effects of a background magnetic field on the degradation of entanglement between states in an accelerated frame, for charged quantum fields., v2; 17 pages, 8 figures

For a class of $D=5$ holographic models we construct boomerang RG flow solutions that start in the UV at an $AdS_5$ vacuum and end up at the same vacuum in the IR. The RG flows are driven by deformations by relevant operators that explicitly break translation invariance. For specific models, such that they admit another $AdS_5$ solution, $AdS_5^c$, we show that for large enough deformations the RG flows approach an intermediate scaling regime with approximate conformal invariance governed by $AdS^c_5$. For these flows we calculate the holographic entanglement entropy and the entropic $c$-function for the RG flows. The latter is not monotonic, but it does encapsulate the degrees of freedom in each scaling region. For a different set of models, we find boomerang RG flows with intermediate scaling governed by an $AdS_2\times\mathbb{R}^3$ solution which breaks translation invariance. Furthermore, for large enough deformations these models have interesting and novel thermal insulating ground states for which the entropy vanishes as the temperature goes to zero, but not as a power-law. Remarkably, the thermal diffusivity and the butterfly velocity for these new insulating ground states are related via $D=Ev^2_B/(2\pi T)$, with $E(T)\to 0.5$ as $T\to 0$., Comment: 30 pages, 19 figures

The continuous Multi Scale Entanglement Renormalization Anstaz (cMERA) consists of a variational method which carries out a real space renormalization scheme on the wavefunctionals of quantum field theories. In this work we calculate the entanglement entropy of the half space for a free scalar theory through a Gaussian cMERA circuit. We obtain the correct entropy written in terms of the optimized cMERA variational parameter, the local density of disentanglers. Accordingly, using the entanglement entropy production per unit scale, we study local areas in the bulk of the tensor network in terms of the differential entanglement generated along the cMERA flow. This result spurs us to establish an explicit relation between the cMERA variational parameter and the radial component of a dual AdS geometry through the Ryu-Takayanagi formula. Finally, based on recent formulations of non-Gaussian cMERA circuits, we argue that the entanglement entropy for the half space can be written as an integral along the renormalization scale whose measure is given by the Fisher information metric of the cMERA circuit. Consequently, a straightforward relation between AdS geometry and the Fisher information metric is also established., Comment: 18 pages, 1 figure. Some references added in which v2 is the final version published in Fortschritte der Physik. Progress of Physics

In conformal field theories (CFTs) of dimension $d>3$, two-dimensional (2d) conformal defects are characterised in part by central charges defined via the defect's contribution to the trace anomaly. However, in general for interacting CFTs these central charges are difficult to calculate. For superconformal 2d defects in supersymmetric (SUSY) CFTs (SCFTs), we show how to compute these defect central charges from the SUSY partition function either on $S^d$ with defect along $S^2$, or on $S^1 \times S^{d-1}$ with defect along $S^1 \times S^1$. In the latter case we propose that defect central charges appear in an overall normalisation factor, as part of the SUSY Casimir energy. For 2d half-BPS defects in 4d ${\mathcal N}=2$ SCFTs and in the 6d ${\mathcal N}=(2,0)$ SCFT we obtain novel, exact results for defect central charges using existing results for partition functions computed using SUSY localisation, SUSY indices, and correspondences to 2d Liouville, Toda, and $q$-deformed Yang-Mills theories. Some of our results for defect central charges agree with those obtained previously via holography, showing that the latter are not just large-$N$ and/or strong-coupling limits, but are exact. Our methods can be straightforwardly extended to other superconformal defects, of various codimension, as we demonstrate for a 4d defect in the 6d ${\mathcal N}=(2,0)$ SCFT., 40 pages + 2 appendices, 2 PDF figures; v2 minor clarifications, added references, version published in JHEP

We explore the connection between the area law for entanglement and geometry by representing the entanglement entropies corresponding to all $2^N$ bipartitions of an $N$-party pure quantum system by means of a (generalized) adjacency matrix. In the cases where the representation is exact, the elements of that matrix coincide with the mutual information between pairs of sites. In others, it provides a very good approximation, and in all the cases it yields a natural {\em entanglement contour} which is similar to previous proposals.Moreover, for one-dimensional conformal invariant systems, the generalized adjacency matrix is given by the two-point correlator of an {\em entanglement current} operator. We conjecture how this entanglement current may give rise to a metric entirely built from entanglement., Comment: 13 pages, 11 figures, close to the published version

The quantum state exchange is a quantum communication task in which two users exchange their respective quantum information in the asymptotic setting. In this work, we consider a one-shot version of the quantum state exchange task, in which the users hold a single copy of the initial state, and they exchange their parts of the initial state by means of entanglement-assisted local operations and classical communication. We first derive lower bounds on the least amount of entanglement required for carrying out this task, and provide conditions on the initial state such that the protocol succeeds with zero entanglement cost. Based on these results, we reveal two counter-intuitive phenomena in this task, which make it different from a conventional SWAP operation. One tells how the users deal with their symmetric information in order to reduce the entanglement cost. The other shows that it is possible for the users to gain extra shared entanglement after this task., 7 pages, 3 figures

The von Neumann entropy is a key quantity in quantum information theory and, roughly speaking, quantifies the amount of quantum information contained in a state when many identical and independent i.i.d. copies of the state are available, in a regime that is often referred to as being asymptotic. In this work, we provide a new operational characterization of the von Neumann entropy which neither requires an i.i.d. limit nor any explicit randomness. We do so by showing that the von Neumann entropy fully characterizes single-shot state transitions in unitary quantum mechanics, as long as one has access to a catalyst - an ancillary system that can be re-used after the transition - and an environment which has the effect of dephasing in a preferred basis. Building upon these insights, we formulate and provide evidence for the catalytic entropy conjecture, which states that the above result holds true even in the absence of decoherence. If true, this would prove an intimate connection between single-shot state transitions in unitary quantum mechanics and the von Neumann entropy. Our results add significant support to recent insights that, contrary to common wisdom, the standard von Neumann entropy also characterizes single-shot situations and opens up the possibility for operational single-shot interpretations of other standard entropic quantities. We discuss implications of these insights to readings of the third law of quantum thermodynamics and hint at potentially profound implications to holography., Comment: 4+5 pages, 1 figure

Several realistic spin-orbital models for transition metal oxides go beyond the classical expectations and could be understood only by employing the quantum entanglement. Experiments on these materials confirm that spin-orbital entanglement has measurable consequences. Here, we capture the essential features of spin-orbital entanglement in complex quantum matter utilizing 1D spin-orbital model which accommodates SU(2)xSU(2) symmetric Kugel-Khomskii superexchange as well as the Ising on-site spin-orbit coupling. Building on the results obtained for full and effective models in the regime of strong spin-orbit coupling, we address the question whether the entanglement found on superexchange bonds always increases when the Ising spin-orbit coupling is added. We show that (i) quantum entanglement is amplified by strong spin-orbit coupling and, surprisingly, (ii) almost classical disentangled states are possible. We complete the latter case by analyzing how the entanglement existing for intermediate values of spin-orbit coupling can disappear for higher values of this coupling., Proceedings paper [Quantum Complex Matter Conference, June 2020, Frascati] reporting on the research first presented in Physical Review Research 2, 013353 (2020) [arXiv:1911.12180]

We explore the holographic properties of non-perturbative vacuum decay in Anti-de Sitter ($\mathrm{AdS}$) geometries. To this end, we consider a gravitational theory in a metastable $\mathrm{AdS}_3$ state, which decays into an $\mathrm{AdS}_3$ of lower vacuum energy via bubble nucleation, and we employ the Ryu-Takayanagi conjecture to compute the entanglement entropy $S_\text{ent}$ in its alleged holographic dual. Our analysis leads us to infer that the nucleation and growth of a vacuum bubble correspond, in the boundary theory, to the introduction of a relevant deformation and a subsequent Renormalization Group (RG) flow, where $S_\text{ent}$ provides a $c$-function. We provide further evidence for the claim and comment on the holographic interpretation of off-centred or multiple bubbles. We also frame the issue in the formalism of Holographic Integral Geometry, discussing the consequences on the structure of the holographic RG flow and recovering the standard holographic RG as a limiting case., Comment: 26 pages, 10 figures; references and comments added

The area law conjecture states that the entanglement entropy of a region of space in the ground state of a gapped, local Hamiltonian only grows like the surface area of the region. We show that, for any quantum state that fulfills an area law, the reduced quantum state of a region of space can be unitarily compressed into a thickened surface of the region. If the interior of the region is lost after this compression, the full quantum state can be recovered to high precision by a quantum channel only acting on the thickened surface. The thickness of the boundary scales inversely proportional to the error for arbitrary spin systems and logarithmically with the error for quasi-free bosonic systems. Our results can be interpreted as a single-shot operational interpretation of the area law. The result for spin systems follows from a simple inequality showing that probability distributions with low entropy can be approximated by distributions with small support, which we believe to be of independent interest. We also discuss an emergent approximate correspondence between bulk and boundary operators and the relation of our results to tensor network states., 4+5 pages, 1 figure; v2: improved presentation, removed erroneous lemma from appendix (results unchanged), close to published version

In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Renyi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient $\sigma$ for the EE and $\sigma_n$ for the Renyi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for $\sigma$ and $\sigma_n$ for all $n=2,3,\dots$. The results for $\sigma$ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that $\sigma/C_T = \pi^2/24$ in all 3d CFTs, where $C_T$ is the central charge. For the Renyi entropies, the ratios $\sigma_n/C_T$ do not indicate similar universality. However, in the limit $n \to \infty$, the asymptotic values satisfy a simple relationship and equal $1/(4\pi^2)$ times the asymptotic values of the free energy of free scalars/fermions on the $n$-covered 3-sphere., Comment: 10 pages, 2 figures. v2: typos corrected, references added, asymptotics updated

We study the Zeeman effect on entanglement of non-equilibrium finite-spin systems with external fields using a method based on thermofield dynamics (TFD). For this purpose, the extended density matrices and extended entanglement entropies of two systems with either non-competing or competing external fields are calculated according to the dissipative von Neumann equation, and the numerical results are compared. Consequently, through the "twin-peaks" oscillations of the quantum entanglement, we have illustrated the Zeeman effect on the entanglement of non-equilibrium finite-spin systems with competing external fields in the TFD algorithm., Comment: Title and abstract were corrected

We compute the modular Hamiltonians of regions having the future horizon lying on a null plane. For a CFT this is equivalent to regions with boundary of arbitrary shape lying on the null cone. These Hamiltonians have a local expression on the horizon formed by integrals of the stress tensor. We prove this result in two different ways, and show that the modular Hamiltonians of these regions form an infinite dimensional Lie algebra. The corresponding group of unitary transformations moves the fields on the null surface locally along the null generators with arbitrary null line dependent velocities, but act non locally outside the null plane. We regain this result in greater generality using more abstract tools on algebraic quantum field theory. Finally, we show that modular Hamiltonians on the null surface satisfy a Markov property that leads to the saturation of the strong sub-additive inequality for the entropies and to the strong super-additivity of the relative entropy., 35 pages, 4 figures. v3: typos corrected, comments and references added

We use an estimator of quantum criticality based on the entanglement entropy to discuss the ground state properties of the 1D anisotropic Kondo necklace model. We found that the T=0 phase diagram of the model is described by a critical line separating an antiferromagnetic phase from a Kondo singlet state. Moreover we calculate the conformal anomaly on the critical line and obtained that c tends to 0.5 as the thermodynamic limit is reached. We conclude that these transitions belong to Ising universality class being, therefore, second order transitions instead of infinite order as claimed before., Comment: 11 pages, 5 figures

We consider the entanglement entropy for a spacetime region and its spacelike complement in the framework of algebraic quantum field theory. For a M\"obius covariant local net satisfying a certain nuclearity property, we consider the von Neumann entropy for type I factors between local algebras and introduce an entropic quantity. Then we implement a cutoff on this quantity with respect to the conformal Hamiltonian and show that it remains finite as the distance of two intervals tends to zero. We compare our definition to others in the literature., Comment: 23 pages

The renormalization of entanglement entropy of quantum field theories is investigated in the simplest setting with a $\lambda \phi^4$ scalar field theory. The 3+1 dimensional spacetime is separated into two regions by an infinitely flat 2-dimensional interface. The entanglement entropy of the system across the interface has an elegant geometrical interpretation using the replica trick, which requires putting the field theory on a curved spacetime background. We demonstrate that the theory, and hence the entanglement entropy, is renormalizable at order $\lambda$ once all the relevant operators up to dimension-4 are included in the action. This exercise has a one-to-one correspondence to entanglement entropy interpretation of the black hole entropy which suggests that our treatment is sensible. Our study suggests that entanglement entropy is renormalizable and is a physical quantity., Comment: 7 pages,3 figures

6 pags., fig. -- PACS Number(s): 04.60.−m, 11.10.−z, 04.60.Bc, We argue that the requirement of a finite entanglement entropy of quantum degrees of freedom across a boundary surface is closely related to the phenomenon of running spectral dimension, universal in approaches to quantum gravity. If quantum geometry hinders diffusion, for instance, when its structure at some given scale is discrete or too rough, then the spectral dimension of spacetime vanishes at that scale and the entropy density blows up. A finite entanglement entropy is a key ingredient in deriving Einstein gravity in a semi-classical regime of a quantum-gravitational theory and, thus, our arguments strengthen the role of running dimensionality as an imprint of quantum geometry with potentially observable consequences., The work of G. C. is under a Ramón y Cajal contract and is supported by the I+D Grant FIS2014-54800-C2-2-P.

Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to non-unitary quantum mechanics, which has seen growing interest from areas as diverse as open quantum systems, non-interacting electronic disordered systems, or non-unitary conformal field theory (CFT). We propose and investigate such an extension here, by focussing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of R\'enyi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well understood cases. The example of the $sl(2|1)$ alternating spin chain for percolation is discussed in detail., Comment: 11 pages, 3 figures

Since euclidean global AdS_2 space represented as a strip has two boundaries, the state / operator correspondence in the dual CFT_1 reduces to the standard map from the operators acting on a single copy of the Hilbert space to states in the tensor product of two copies of the Hilbert space. Using this picture we argue that the corresponding states in the dual string theory living on AdS_2 x K are described by twisted version of the Hartle-Hawking states, the twists being generated by a large unitary group of symmetries that this string theory must possess. This formalism makes natural the dual interpretation of the black hole entropy, -- as the logarithm of the degeneracy of ground states of the quantum mechanics describing the low energy dynamics of the black hole, and also as an entanglement entropy between the two copies of the same quantum theory living on the two boundaries of global AdS_2 separated by the event horizon., Comment: LaTeX file, 23 pages, 3 figures

We study a process of equilibration of holographic dark energy (HDE) with the cosmic horizon around the dark-energy dominated epoch. This process is characterized by a huge amount of information conveyed across the horizon, filling thereby a large gap in entropy between the system on the brink of experiencing a sudden collapse to a black hole and the black hole itself. At the same time, even in the absence of interaction between dark matter and dark energy, such a process marks a strong jump in the entanglement entropy, measuring the quantum-mechanical correlations between the horizon and its interior. Although the effective quantum field theory (QFT) with a peculiar relationship between the UV and IR cutoffs, a framework underlying all HDE models, may formally account for such a huge shift in the number of distinct quantum states, we show that the scope of such a framework becomes tremendously restricted, devoiding it virtually any application in other cosmological epochs or particle-physics phenomena. The problem of negative entropies for the non-phantom stuff is also discussed., 10 pages, version to appear in PLB

In order to understand the detailed mechanism by which a fundamental discreteness can provide a finite entanglement entropy, we consider the entanglement entropy of two classes of free massless scalar fields on causal sets that are well approximated by causal diamonds in Minkowski spacetime of dimensions 2,3 and 4. The first class is defined from discretised versions of the continuum retarded Green functions, while the second uses the causal set's retarded nonlocal d'Alembertians parametrised by a length scale $l_k$. In both cases we provide numerical evidence that the area law is recovered when the double-cutoff prescription proposed in arXiv:hep-th/1611.10281 is imposed. We discuss in detail the need for this double cutoff by studying the effect of two cutoffs on the quantum field and, in particular, on the entanglement entropy, in isolation. In so doing, we get a novel interpretation for why these two cutoff are necessary, and the different roles they play in making the entanglement entropy on causal sets finite., Comment: 36 pages, 14 figures

We investigate the entanglement entropy in gravity duals of confining large $N_c$ gauge theories using the proposal of arXiv:hep-th/0603001, arXiv:hep-th/0605073. Dividing one of the directions of space into a line segment of length $l$ and its complement, the entanglement entropy between the two subspaces is given by the classical action of the minimal bulk hypersurface which approaches the endpoints of the line segment at the boundary. We find that in confining backgrounds there are generally two such surfaces. One consists of two disconnected components localized at the endpoints of the line segment. The other contains a tube connecting the two components. The disconnected surface dominates the entropy for $l$ above a certain critical value $l_{\rm crit}$ while the connected one dominates below that value. The change of behavior at $l=l_{\rm crit}$ is reminiscent of the finite temperature deconfinement transition: for $l < l_{\rm crit}$ the entropy scales as $N_c^2$, while for $l > l_{\rm crit}$ as $N_c^0$. We argue that a similar transition should occur in any field theory with a Hagedorn spectrum of non-interacting bound states. The requirement that the entanglement entropy has a phase transition may be useful in constraining gravity duals of confining theories., 22 pages, 6 figures, v1. references added

We construct condensate states encoding the continuum spherically symmetric quantum geometry of an horizon in full quantum gravity, i.e. without any classical symmetry reduction, in the group field theory formalism. Tracing over the bulk degrees of freedom, we show how the resulting reduced density matrix manifestly exhibits an holographic behavior. We derive a complete orthonormal basis of eigenstates for the reduced density matrix of the horizon and use it to compute the horizon entanglement entropy. By imposing consistency with the horizon boundary conditions and semiclassical thermodynamical properties, we recover the Bekenstein--Hawking entropy formula for any value of the Immirzi parameter. Our analysis supports the equivalence between the von Neumann (entanglement) entropy interpretation and the Boltzmann (statistical) one., Comment: 6 pages; published version

The first law for the holographic entanglement entropy of spheres in a boundary CFT with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a family of boundary CFTs. The new contribution to the first law is found to be the product of the variation $\delta a$ of the A-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension $\delta a^*$ to include odd dimensional boundaries, times the ratio $S/a^*$. Since $a^*$ is a measure of the number of degrees of freedom $N$ per unit volume of the boundary CFT, this new term has the form $\mu\delta N$, where the chemical potential $\mu$ is given by the entanglement entropy per degree of freedom., Comment: 18 pages

We analyze the 1+1 CFT states dual to hot (time-symmetric) 2+1 multiboundary AdS wormholes. These are black hole geometries with high local temperature, $n \ge 1$ asymptotically-AdS$_3$ regions, and arbitrary internal topology. The dual state at $t=0$ is defined on $n$ circles. We show these to be well-described by sewing together tensor networks corresponding to thermofield double states. As a result, the entanglement is spatially localized and bipartite: away from particular boundary points ("vertices") any small connected region $A$ of the boundary CFT is entangled only with another small connected region $B$, where $B$ may lie on a different circle or may be a different part of the same circle. We focus on the pair-of-pants case, from which more general cases may be constructed. We also discuss finite-temperature corrections, where we note that the states involve a code subspace in each circle., Comment: 32 pages, 17 figures

We study and construct spacetimes, dubbed planar AdS-dS-wormholes, satisfying the null energy condition and having two asymptotically AdS boundaries connected through a (non-traversable) inflating wormhole. As for other wormholes, it is natural to expect dual descriptions in terms of two disconnected CFTs in appropriate entangled states. But for our cases certain expected bulk entangling surfaces used by the Hubeny-Rangamani-Takayanagi (HRT) prescription to compute CFT entropy do not exist. In particular, no real codimension-2 extremal surface can run from one end of the wormhole to the other. According to HRT, the mutual information between any two finite-sized subregions (one in each CFT) must then vanish at leading order in large $N$ -- though the leading-order mutual information per unit area between the two CFTs taken as wholes may be nonzero. Some planar AdS-dS-wormholes also fail to have plane-symmetric surfaces that would compute the total entropy of either CFT. We suggest this to remain true of less-symmetric surfaces so that the HRT entropy is ill-defined and some modified prescription is required. It may be possible to simply extend HRT or the closely-related maximin construction by a limiting procedure, though complex extremal surfaces could also play an important role., Comment: 27+10 pages. v2: minor modifications to address referee comments. v3: fixed typos

We discuss the possible relevance of complex codimension-two extremal surfaces to the the Ryu-Takayanagi holographic entanglement proposal and its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization. Such surfaces live in a complexified bulk spacetime defined by analytic continuation. We identify surfaces of this type for BTZ, Schwarzschild-AdS, and Schwarzschild-Lifshitz planar black holes. Since the dual CFT interpretation for the imaginary part of their areas is unclear, we focus on a straw man proposal relating CFT entropy to the real part of the area alone. For Schwarzschild-AdS and Schwarzschild-Lifshitz, we identify families where the real part of the area agrees with qualitative physical expectations for the appropriate CFT entropy and, in addition, where it is smaller than the area of corresponding real extremal surfaces. It is thus plausible that the CFT entropy is controlled by these complex extremal surfaces., Comment: 28+5 pages. v2: Addressed referee comments

We prove the generalized Covariant Entropy Bound, $\Delta S\leq (A-A')/4G\hbar$, for light-sheets with initial area $A$ and final area $A'$. The entropy $\Delta S$ is defined as a difference of von Neumann entropies of an arbitrary state and the vacuum, with both states restricted to the light-sheet under consideration. The proof applies to free fields, in the limit where gravitational backreaction is small. We do not assume the null energy condition. In regions where it is violated, we find that the bound is protected by the defining property of light-sheets: that their null generators are nowhere expanding., Comment: 19 pages, 3 figures; v2: references added

We study the vacuum-subtracted von Neumann entropy of a segment on a null plane. We argue that for interacting quantum field theories in more than two dimensions, this entropy has a simple expression in terms of the expectation value of the null components of the stress tensor on the null interval. More explicitly $\Delta S = 2\pi \int d^{d-2}y \int_0^1 dx^+\, g(x^+)\, \langle T_{++}\rangle$, where $g(x^+)$ is a theory-dependent function. This function is constrained by general properties of quantum relative entropy. These constraints are enough to extend our recent free field proof of the quantum Bousso bound to the interacting case. This unusual expression for the entropy as the expectation value of an operator implies that the entropy is equal to the modular Hamiltonian, $\Delta S = \langle \Delta K \rangle $, where $K$ is the operator in the right hand side. We explain how this equality is compatible with a non-zero value for $\Delta S$. Finally, we also compute explicitly the function $g(x^+)$ for theories that have a gravity dual., Comment: 35 pages, 6 figures

We identify conditions for the entanglement entropy as a function of spatial region to be compatible with causality in an arbitrary relativistic quantum field theory. We then prove that the covariant holographic entanglement entropy prescription (which relates entanglement entropy of a given spatial region on the boundary to the area of a certain extremal surface in the bulk) obeys these conditions, as long as the bulk obeys the null energy condition. While necessary for the validity of the prescription, this consistency requirement is quite nontrivial from the bulk standpoint, and therefore provides important additional evidence for the prescription. In the process, we introduce a codimension-zero bulk region, named the entanglement wedge, naturally associated with the given boundary spatial region. We propose that the entanglement wedge is the most natural bulk region corresponding to the boundary reduced density matrix., Comment: 42 pages, 11 figures. v2: minor clarifications + refs added