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ℏ → 0 limit of the entanglement entropy
Abstract
Entangled quantum states share properties that do not have classical analogs; in particular, they show correlations that can violate Bell inequalities. It is, therefore, an interesting question to see what happens to entanglement measures—such as the entanglement entropy for a pure state—taking the semiclassical limit, where the naive expectation is that they may become singular or zero. This conclusion is, however, incorrect. In this paper, we determine the ℏ→0 limit of the bipartite entanglement entropy for a one-dimensional system of N quantum particles in an external potential and we explicitly show that this limit is finite. Moreover, if the particles are fermionic, we show that the ℏ→0 limit of the bipartite entanglement entropy coincides with the Shannon entropy of N bits.
... We point out that the only difference with respect to (50) is the presence of an additional flux (−1) n−1 between the n-th and the first replica, a factor that was already introduced in [41] and employed for instance in [9] in the calculation of the VEV of the Ising twist field. For fermionic theories we need to define another twist operator which implements explicitly the fermionic partial transposition, and from now on we only consider n even, denoted also by n e . ...
... These papers in turn extend work on entanglement measures for zero-density excited states carried out in [26,27,28,29]. Other important contributions to this study are [44,45,46,47,48,49,50] In line with the results of [30,31] for the Rényi entropies, also here we expected and indeed found that the contribution of a finite number of excitations to the symmetry resolved (logarithmic) negativity is given by a simple formula, a polynomial on the variables r A , r B and r = 1 − r A − r B , which represent the relative sizes of two subsystems A and B and their complement, respectively. For the symmetry resolved moments of the negativity, this polynomial will also depend on a parameter related to the internal symmetry of the theory, which in this paper we take to be U (1). ...
... Here we find that, first, this redefinition is easy to implement in the context of twist operators, and, second, that once implemented it leads to a result which is the same as for bosons (even if the intermediate steps and starting point of the computation are different). This ties in well with the idea that the universal part of the entanglement associated with these types of excitation has a semiclassical interpretation (as recently explored in [50]), thus the statistics of excitations plays no role, even if it does play an important role for nonuniversal contributions, which are non-trivial when we consider excited states of QFT. ...
In two recent works, we studied the symmetry resolved R\'enyi entropies of quasi-particle excited states in quantum field theory. We found that the entropies display many model-independent features which we discussed and analytically characterised. In this paper we extend this line of investigation by providing analytical and numerical evidence that a similar universal behavior arises for the symmetry resolved negativity. In particular, we compute the ratio of charged moments of the partially transposed reduced density matrix as an expectation value of twist operators. These are ``fused" versions of the more traditionally used branch point twist fields and were introduced in a previous work. The use of twist operators allows us to perform the computation in an arbitrary number of spacial dimensions. We show that, in the large-volume limit, only the commutation relations between the twist operators and local fields matter, and computations reduce to a purely combinatorial problem. We address some specific issues regarding fermionic excitations, whose treatment requires the notion of partial time-reversal transformation, and we discuss the differences and analogies with their bosonic counterpart. We find that although the operation of partial transposition requires a redefinition for fermionic theories, the ratio of the negativity moments between an excited state and the ground state is universal and identical for fermions and bosons as well as for a large variety of very different states, ranging from simple qubit states to the excited states of free quantum field theories. Our predictions are tested numerically on a 1D Fermi chain.
... These papers in turn extend work on entanglement measures for zero-density excited states carried out in [29][30][31][32]. Other important contributions to this study are [48][49][50][51][52][53][54] In line with the results of [33,34] for the Rényi entropies, also here we expected and indeed found that the contribution of a finite number of excitations to the symmetry JHEP06(2023)074 The left/right panels show the real/imaginary part of E n (α) − E n,0 (α). The size of the chain is L = 400, and we plot the results as functions of r A ∈ (0, 1/2). ...
... Here we find that, first, this redefinition is easy to implement in the context of twist operators, and, second, that once implemented it leads to a result which is the same as for bosons (even if the intermediate steps and starting point of the computation are different). This ties in well with the idea that the universal part of the entanglement associated with these types of excitation has a semiclassical interpretation (as recently explored in [54]), thus the statistics of excitations plays no role, even if it does play an important role for non-universal contributions, which are non-trivial when we consider excited states of QFT. ...
A bstract
In two recent works, we studied the symmetry resolved Rényi entropies of quasi-particle excited states in quantum field theory. We found that the entropies display many model-independent features which we discussed and analytically characterised. In this paper we extend this line of investigation by providing analytical and numerical evidence that a similar universal behavior arises for the symmetry resolved negativity. In particular, we compute the ratio of charged moments of the partially transposed reduced density matrix as an expectation value of twist operators. These are “fused” versions of the more traditionally used branch point twist fields and were introduced in a previous work. The use of twist operators allows us to perform the computation in an arbitrary number of spacial dimensions. We show that, in the large-volume limit, only the commutation relations between the twist operators and local fields matter, and computations reduce to a purely combinatorial problem. We address some specific issues regarding fermionic excitations, whose treatment requires the notion of partial time-reversal transformation, and we discuss the differences and analogies with their bosonic counterpart. We find that although the operation of partial transposition requires a redefinition for fermionic theories, the ratio of the negativity moments between an excited state and the ground state is universal and identical for fermions and bosons as well as for a large variety of very different states, ranging from simple qubit states to the excited states of free quantum field theories. Our predictions are tested numerically on a 1D Fermi chain.
... However, it was shown in [26] for several examples, that the formulae should hold for interacting and even higher-dimensional theories, as long as excitations are localised. These results have been confirmed and then extended in many ways by later works, [31][32][33][34][35][36][37] and also interpreted within a semiclassical picture [38,39]. ...
In this work, we revisit a problem we addressed in previous publications with various collaborators, that is, the computation of the symmetry resolved entanglement entropies of zero-density excited states in infinite volume. The universal nature of the charged moments of these states has already been noted previously. Here, we investigate this problem further, by writing general formulae for the entropies of excited states consisting of an arbitrary number of subsets of identical excitations. When the initial state is written in terms of qubits with appropriate probabilistic coefficients, we find the final formulae to be of a combinatorial nature too. We analyse some of their features numerically and analytically and find that for qubit states consisting of particles of the same charge, the symmetry resolved entropies are independent of region size relative to system size, even if the number and configuration entropies are not.
... Quasiparticles are interesting collective excitations in integrable many-body systems that often provide simple intuitive explanations for complex phenomena [1]. One such example is the entanglement entropy in quasiparticle states of integrable quantum spin chains, which displays intriguing universal features in the scaling limit [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. It was found that under certain limits, the entanglement entropy in quasiparticle states displays universal behaviors that can be explained by a semiclassical quasiparticle picture [5,6]. ...
We investigate the Shannon entropy of the total system and its subsystems, as well as the subsystem Shannon mutual information, in quasiparticle excited states of free bosonic and fermionic chains and the ferromagnetic phase of the spin-1/2 XXX chain. For single-particle and double-particle states, we derive various analytical formulas for free bosonic and fermionic chains in the scaling limit. These formulas are also applicable to certain magnon excited states in the XXX chain in the scaling limit. We also calculate numerically the Shannon entropy and mutual information for triple-particle and quadruple-particle states in bosonic, fermionic, and XXX chains. We discover that Shannon entropy, unlike entanglement entropy, typically does not separate for quasiparticles with large momentum differences. Moreover, in the limit of large momentum difference, we obtain universal quantum bosonic and fermionic results that are generally distinct and cannot be explained by a semiclassical picture.
... However, this naive inference is not correct. The semiclassical limit of entanglement needs to be taken carefully [66][67][68], where one first obtains an SK path integral for entanglement entropy, e.g., second Rényi entropy [69], in the quantum model and then takes the semiclassical limit. This results in effective dynamical equations for entanglement [70] different from Eq. (3). ...
We discuss the dynamics of integrable and nonintegrable chains of coupled oscillators under continuous weak position measurements in the semiclassical limit. We show that, in this limit, the dynamics is described by a standard stochastic Langevin equation, and a measurement-induced transition appears as a noise- and dissipation-induced chaotic-to-nonchaotic transition akin to stochastic synchronization. In the nonintegrable chain of anharmonically coupled oscillators, we show that the temporal growth and the ballistic light-cone spread of a classical out-of-time correlator characterized by the Lyapunov exponent and the butterfly velocity are halted above a noise or below an interaction strength. The Lyapunov exponent and the butterfly velocity both act like order parameter, vanishing in the nonchaotic phase. In addition, the butterfly velocity exhibits a critical finite-size scaling. For the integrable model, we consider the classical Toda chain and show that the Lyapunov exponent changes nonmonotonically with the noise strength, vanishing at the zero noise limit and above a critical noise, with a maximum at an intermediate noise strength. The butterfly velocity in the Toda chain shows a singular behavior approaching the integrable limit of zero noise strength.
... Depending on the particular quantum state of a given many-body system, the entanglement entropy exhibits various behaviors in the scaling limit [33][34][35][36][37]. In specific scenarios, the entanglement entropy in quasiparticle states of integrable quantum spin chains displays intriguing universal features in the scaling limit [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. A quasiparticle state could be represented by the momenta K of the excited quasiparticles, denoted as |K . ...
In this paper, we investigate the Shannon entropy of the total system and its subsystems, as well as the subsystem Shannon mutual information, in quasiparticle excited states of free bosonic and fermionic chains and the ferromagnetic phase of the spin-1/2 XXX chain. Our focus is on single-particle and double-particle states, and we derive various analytical formulas for free bosonic and fermionic chains in the scaling limit. These formulas are also applicable to magnon excited states in the XXX chain under certain conditions. We discover that, unlike entanglement entropy, Shannon entropy does not separate when two quasiparticles have a large momentum difference. Moreover, in the large momentum difference limit, we obtain universal results for quantum spin chains that cannot be explained by a semiclassical picture of quasiparticles.
... In particular, a series of works by Rajabpour and collaborators [59][60][61][62][63][64] has expanded previous work in various directions: by obtaining finite volume corrections, new formulae for systems where quasiparticles are not localised, and finally by establishing that the formulae indeed hold for generic magnon states, thus also in interacting theories, in [64]. Similar formulae have also been found for interacting higher-dimensional theories in [65] and even in the presence of an external potential, arising from a semiclassical limit [66]. Indeed, the formulae found in [55] were not entirely unexpected as they can be derived for semiclassical systems [67], however their wide range of applicability, well beyond the semiclassical regime, as well as their derivation in the context of QFT were new. ...
A bstract
The excess entanglement resulting from exciting a finite number of quasiparticles above the ground state of a free integrable quantum field theory has been investigated quite extensively in the literature. It has been found that it takes a very simple form, depending only on the number of excitations and their statistics. There is now mounting evidence that such formulae also apply to interacting and even higher-dimensional quantum theories. In this paper we study the entanglement content of such zero-density excited states focusing on the symmetry resolved entanglement, that is on 1+1D quantum field theories that possess an internal symmetry. The ratio of charged moments between the excited and grounds states, from which the symmetry resolved entanglement entropy can be obtained, takes a very simple and universal form, which in addition to the number and statistics of the excitations, now depends also on the symmetry charge. Using form factor techniques, we obtain both the ratio of moments and the symmetry resolved entanglement entropies in complex free theories which possess U(1) symmetry. The same formulae are found for simple qubit states.
... These results in turn have been extended in a series of works [66][67][68][69][70][71] to deal with finite volume corrections and non-localised excitations. More recently, some of the α = 0 results were recovered as a semiclassical limit in the presence of an interaction potential [72]. This semiclassical picture had already been invoked much earlier, see for ...
A bstract
In a recent paper we studied the entanglement content of zero-density excited states in complex free quantum field theories, focusing on the symmetry resolved entanglement entropy (SREE). By zero-density states we mean states consisting of a fixed, finite number of excitations above the ground state in an infinite-volume system. The SREE is defined for theories that possess an internal symmetry and provides a measure of the contribution to the total entanglement of each symmetry sector. In our work, we showed that the ratio of Fourier-transforms of the SREEs (i.e. the ratio of charged moments) takes a very simple and universal form for these states, which depends only on the number, statistics and symmetry charge of the excitations as well as the relative size of the entanglement region with respect to the whole system’s size. In this paper we provide numerical evidence for our formulae by computing functions of the charged moments in two free lattice theories: a 1D Fermi gas and a complex harmonic chain. We also extend our results in two directions: by showing that they apply also to excited states of interacting theories (i.e. magnon states) and by developing a higher dimensional generalisation of the branch point twist field picture, leading to results in (interacting) higher-dimensional models.
... This tells us that entanglement entropies in (4.1) and (4.3) are not uniformly convergent. It is worth considering the limit → 0 of our results (see also the recent analysis in [127]), which is determined by the behaviour of the dimensionless parameter η in this limit. From (2.38) and (2.32), one realises that different results can be obtained for entanglement entropies when → 0, depending on the quantities that are kept constant in this limit. ...
A bstract
We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent z = 2. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic C function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent z , we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.
... The universal excess Rényi and entanglement entropies could be written out by a simple semiclassical quasiparticle picture with the quantum effects of distinguishability and indistinguishability of the excited quasiparticles. The same universal formulas could be obtained in the classical limit of a one-dimensional quantum gas in presence of an external potential [62]. By relaxing the limit that quasiparticle momentum differences are large, we have obtained additional contributions to the Rényi and entanglement entropies in [56,58]. ...
A bstract
We investigate the subsystem Schatten distance, trace distance and fidelity between the quasiparticle excited states of the free and the nearest-neighbor coupled fermionic and bosonic chains and the ferromagnetic phase of the spin-1/2 XXX chain. The results support the scenario that in the scaling limit when one excited quasiparticle has a large energy it decouples from the ground state and when two excited quasiparticles have a large momentum difference they decouple from each other. From the quasiparticle picture, we get the universal subsystem distances that are valid when both the large energy condition and the large momentum difference condition are satisfied, by which we mean each of the excited quasiparticles has a large energy and the momentum difference of each pair of the excited quasiparticles is large. In the free fermionic and bosonic chains, we use the subsystem mode method and get efficiently the subsystem distances, which are also valid in the coupled fermionic and bosonic chains if the large energy condition is satisfied. Moreover, under certain limit the subsystem distances from the subsystem mode method are even valid in the XXX chain. We expect that the results can be also generalized for other integrable models.
... These results in turn have been extended in a series of works [62][63][64][65][66][67] to deal with finite volume corrections and non-localised excitations. More recently, some of the α = 0 results were recovered as a semiclassical limit in the presence of an interaction potential [68]. This semiclassical picture had already been invoked much earlier, see for instance [69]. ...
In a recent paper we studied the entanglement content of zero-density excited states in complex free quantum field theories, focusing on the symmetry resolved entanglement entropy (SREE). The SREE is defined for theories that possess an internal symmetry and provides a measure of the contribution to the total entanglement of each symmetry sector. In our work, we showed that the ratio of Fourier-transforms of the SREEs (i.e. the ratio of charged moments) takes a very simple and universal form for these states, which depends only on the number, statistics and symmetry charge of the excitations as well as the relative size of the entanglement region with respect to the whole system's size. In this paper we provide numerical evidence for our formulae by computing functions of the charged moments in two free lattice theories: a 1D Fermi gas and a complex harmonic chain. We also extend our results in two directions: by showing that they apply also to excited states of interacting theories (i.e. magnon states) and by developing a higher dimensional generalisation of the branch point twist field picture, leading to results in (interacting) higher-dimensional models.
... In particular, a series of works by Rajabpour and collaborators [54][55][56][57][58][59] has expanded previous work in various directions: by obtaining finite volume corrections, new formulae for systems where quasiparticles are not localised, and finally by establishing that the formulae indeed hold for generic magnonic states, thus also in interacting theories, in [59]. Similar formulae have also been found for interacting higher dimensional theories in [60] and even in the presence of an external potential, arising from a semiclassical limit [61]. Indeed, the formulae found in [50] were not entirely unexpected as they can be derived for semiclassical systems [62], however their wide range of applicability, well beyond the semiclassical regime, as well as their derivation in the context of QFT were new. ...
The excess entanglement resulting from exciting a finite number of quasiparticles above the ground state of a free integrable quantum field theory has been investigated quite extensively in the literature. It has been found that it takes a very simple form, depending only on the number of excitations and their statistics. There is now mounting evidence that such formulae also apply to interacting and even higher-dimensional theories. In this paper we study the entanglement content of such zero-density excited states focusing on the symmetry resolved entanglement, that is on quantum field theories that possess an internal symmetry. The ratio of charged moments between the excited and grounds states, from which the symmetry resolved entanglement entropy can be obtained, takes a very simple and universal form, which in addition to the number and statistics of the excitations, now depends also on the symmetry charge. Using form factor techniques, we obtain both the ratio of moments and the symmetry resolved entanglement entropies in complex free theories which possess U(1) symmetry. The same formulae are found for simple qubit states.
... In this same scenario, additional novel possibilities emerge, such as the consideration of nonseparable-in-time interactions, the emergence of quantum time operators and energytime uncertainty relations, and the rigorous definition of entanglement in time: in the same way as standard second quantization is required for the notion of a reduced density matrix of a space interval, and hence for entanglement in space [67], the present second quantized spacetime states formalism is a natural scenario to accommodate the notion of entanglement in time. At the same time, the conventional "sum over histories", previously only accessible through classical computations, now admits the application of quantum protocols for trace evaluation (2.4). ...
We present a spacetime Hilbert space formulation of Feynman path integrals (PIs). It relies on a tensor product structure in time which provides extended representations of dynamical quantum observables through a spacetime quantum action operator. As a consequence, the ``sum over paths'' of the different PI formulations naturally arise within the same Hilbert space, with each one associated with a different quantum trajectory basis. New insights on PI-based results naturally follow, including exact discretizations and a non-trivial approach to the continuum limit.
In this short review, we present the key definitions, ideas and techniques involved in the study of symmetry resolved entanglement measures, with a focus on the symmetry resolved entanglement entropy. In order to be able to define such entanglement measures, it is essential that the theory under study possess an internal symmetry. Then, symmetry-resolved entanglement measures quantify the contribution to a particular entanglement measure that can be associated to a chosen symmetry sector. Our review focuses on conformal (gapless/massless/critical) and integrable (gapped/massive) quantum field theories, where the leading computational technique employs symmetry fields known as (composite) branch point twist fields.
A bstract
In this paper we study the increment of the entanglement entropy and of the (replica) logarithmic negativity in a zero-density excited state of a free massive bosonic theory, compared to the ground state. This extends the work of two previous publications by the same authors. We consider the case of two disconnected regions and find that the change in the entanglement entropy depends only on the combined size of the regions and is independent of their connectivity. We subsequently generalize this result to any number of disconnected regions. For the replica negativity we find that its increment is a polynomial with integer coefficients depending only on the sizes of the two regions. The logarithmic negativity turns out to have a more complicated functional structure than its replica version, typically involving roots of polynomials on the sizes of the regions. We obtain our results by two methods already employed in previous work: from a qubit picture and by computing four-point functions of branch point twist fields in finite volume. We test our results against numerical simulations on a harmonic chain and find excellent agreement.
We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions N ≫ 1, the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the 'Airy kernel') and hard edge (described by the 'Bessel kernel') universality classes. We further discuss extensions of these results to higher dimensions d ≥ 2 and to finite temperature. Finally, we discuss correlations in the phase space, i.e. in the space of positions and momenta, characterized by the so called Wigner function.
A bstract
We evaluate the entanglement entropy of a single connected region in excited states of one-dimensional massive free theories with finite numbers of particles, in the limit of large volume and region length. For this purpose, we use finite-volume form factor expansions of branch-point twist field two-point functions. We find that the additive contribution to the entanglement due to the presence of particles has a simple “qubit” interpretation, and is largely independent of momenta: it only depends on the numbers of groups of particles with equal momenta. We conjecture that at large momenta, the same result holds for any volume and region lengths, including at small scales. We provide accurate numerical verifications.
We study quantum information aspects of the fermionic entanglement negativity recently introduced in [Phys. Rev. B 95, 165101 (2017)] based on the fermionic partial transpose. In particular, we show that it is an entanglement monotone under the action of local quantum operations and classical communications (LOCC) which preserve the local fermion-number parity and satisfies other common properties expected for an entanglement measure of mixed states. We present fermionic analogs of tripartite entangled states such as W and GHZ states and compare the results of bosonic and fermionic partial transpose in various fermionic states, where we explain why the bosonic partial transpose fails in distinguishing separable states of fermions. Finally, we explore a set of entanglement quantities which distinguish different classes of entangled states of a system with two and three fermionic modes. In doing so, we prove that vanishing entanglement negativity is a necessary and sufficient condition for separability of fermionic modes with respect to the bipartition into one mode and the rest. We further conjecture that the entanglement negativity of inseparable states which mix local fermion-number parity is always non-vanishing.
The one-particle density matrix of the one-dimensional Tonks-Girardeau gas
with inhomogeneous density profile is calculated, thanks to a recent
observation that relates this system to a two-dimensional conformal field
theory in curved space. The result is asymptotically exact in the limit of
large particle density and small density variation, and holds for arbitrary
trapping potentials. In the particular case of a harmonic trap, we recover a
formula obtained by Forrester et al. [Phys. Rev. A 67, 043607 (2003)] from a
different method.
We propose a definition of partial transpose for a fermionic density matrix based on time-reversal transformation in the Grassmannian coherent state representation of fermions. Using this definition, we present a path-integral picture and develop methods to compute the entanglement negativity associated with the partial time-reversal. For noninteracting fermions, we derive an exact formula for the entanglement negativity which can be straightforwardly computed for any lattice models. We benchmark our method against two examples: Majorana chain and Su-Schrieffer-Heeger model, where we also provide analytical expressions for the negativity in the fixed-point limits of topological phases as well as at the criticality. The results at the criticality match with previous conformal field theory calculations. Our definition of partial transpose is intrinsically fermionic and goes beyond previous definitions by capturing not only the singlet bonds but also the Majorana bonds between two subsystems. Finally, we uncover the first concrete microscopic calculation of the entanglement negativity in the random singlet phase of disordered spin chains.
This book provides a comprehensive overview of developments in the field of holographic entanglement entropy. Within the context of the AdS/CFT correspondence, it is shown how quantum entanglement is computed by the area of certain extremal surfaces. The general lessons one can learn from this connection are drawn out for quantum field theories, many-body physics, and quantum gravity. An overview of the necessary background material is provided together with a flavor of the exciting open questions that are currently being discussed.
The book is divided into four main parts. In the first part, the concept of entanglement, and methods for computing it, in quantum field theories is reviewed. In the second part, an overview of the AdS/CFT correspondence is given and the holographic entanglement entropy prescription is explained. In the third part, the time-dependence of entanglement entropy in out-of-equilibrium systems, and applications to many body physics are explored using holographic methods. The last part focuses on the connection between entanglement and geometry. Known constraints on the holographic map, as well as, elaboration of entanglement being a fundamental building block of geometry are explained.
The book is a useful resource for researchers and graduate students interested in string theory and holography, condensed matter and quantum information, as it tries to connect these different subjects linked by the common theme of quantum entanglement.
Conformal field theory (CFT) has been extremely successful in describing
large-scale universal effects in one-dimensional (1D) systems at quantum
critical points. Unfortunately, its applicability in condensed matter physics
has been limited to situations in which the bulk is uniform because CFT
describes low-energy excitations around some energy scale, taken to be constant
throughout the system. However, in many experimental contexts, such as quantum
gases in trapping potentials and in several out-of-equilibrium situations,
systems are strongly inhomogeneous. We show here that the powerful CFT methods
can be extended to deal with such 1D situations, providing a few concrete
examples for non-interacting Fermi gases. The system's inhomogeneity enters the
field theory action through parameters that vary with position; in particular,
the metric itself varies, resulting in a CFT in curved space. This approach
allows us to derive exact formulas for entanglement entropies which were not
known by other means.
The Eigenstate Thermalization Hypothesis (ETH) posits that the reduced
density matrix for a subsystem corresponding to an excited eigenstate is
"thermal". Here we expound on this hypothesis by asking: for which class of
operators, local or non-local, is ETH satisfied? We show that this question is
directly related to a seemingly unrelated question: is the Hamiltonian of a
system encoded within a single eigenstate? We formulate a strong form of ETH
where in the thermodynamic limit, the reduced density matrix of a subsystem
corresponding to a pure, finite energy density eigenstate asymptotically
becomes equal to the thermal reduced density matrix, as long as certain
conditions on the ratio of the subsystem size to the total system size are
satisfied. This allows one to access the properties of the underlying
Hamiltonian at arbitrary energy densities/temperatures using just a single
eigenstate. We provide support for our conjecture by performing an exact
diagonalization study of a non-integrable 1D lattice quantum model with only
energy conservation. For this particular model, we provide evidence that the
aforementioned strong form of ETH holds up to finite size corrections, which
enables us to extract the free energy density and correlators of operators at
various temperatures using a single eigenstate. We also study a particle number
conserving model at infinite temperature which also substantiates our
conjecture.
We study bipartite entanglement entropies in the ground and excited states of free-fermion models, where a staggered potential, μs, induces a gap in the spectrum. Ground-state entanglement entropies satisfy the "area law", and the "area-law" coefficient is found to diverge as a logarithm of the staggered potential, when the system has an extended Fermi surface at μs=0. On the square lattice, we show that the coefficient of the logarithmic divergence depends on the Fermi surface geometry and its orientation with respect to the real-space interface between subsystems and is related to the Widom conjecture as enunciated by Gioev and Klich [ Phys. Rev. Lett. 96 100503 (2006)]. For point Fermi surfaces in two-dimension, the "area-law" coefficient stays finite as μs→0. The von Neumann entanglement entropy associated with the excited states follows a "volume law" and allows us to calculate an entropy density function sV(e), which is substantially different from the thermodynamic entropy density function sT(e), when the lattice is bipartitioned into two equal subsystems but approaches the thermodynamic entropy density as the fraction of sites in the larger subsystem, that is integrated out, approaches unity.
We study the Renyi entanglement entropy of an interval in a periodic spin
chain, for a general eigenstate of a free, translational invariant Hamiltonian.
In order to compute analytically the entropy we use two technical tools. The
first one is used to reduce logarithmically the complexity of the problem and
the second one to compute the R\'enyi entropy of the chosen subsystem. We
introduce new strategies to perform the computations and derive new expressions
for the entropy of these general states. Finally we show the perfect agreement
of the analytical computations and the numerical results.
I give a simple argument that demonstrates that the state
∣0⟩∣1⟩+∣1⟩∣0⟩ , with ∣0⟩
denoting a state with 0 particles (or photons) and ∣1⟩ a
one-particle state, is entangled in spite of recent claims to the
contrary. I also discuss viewpoints on the old controversy about whether
the above state can be said to display single-particle or single-photon
nonlocality.
Rényi and von Neumann entropies quantifying the amount of entanglement in ground states
of critical spin chains are known to satisfy a universal law which is given by the conformal field
theory (CFT) describing their scaling regime. This law can be generalized to excitations
described by primary fields in CFT, as was done by Alcaraz et al in 2011 (see reference [1], of which this work is a completion). An alternative derivation is presented,
together with numerical verifications of our results in different models belonging to the
c = 1, 1/2
universality classes. Oscillations of the Rényi entropy in excited states are also
discussed.
In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable
quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle
spectrum of the theory and not on the details of the scattering matrix. We employ the “replica trick” whereby the entropy
is obtained as the derivative with respect to n of the trace of the nth power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are
naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the nth power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function
in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation.
Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations
satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide
solutions, which we specialize both to the Ising and sinh-Gordon models.
We numerically analyze the dynamical generation of quantum entanglement in a
system of 2 interacting particles, started in a coherent separable state, for
decreasing values of . As the entanglement entropy,
computed at any finite time, converges to a finite nonzero value. The limit law
that rules the time dependence of entropy is well reproduced by purely
classical computations. Its general features may then be explained by simple
classical arguments, which expose the different ways entanglement is generated
in systems which are classically chaotic or regular.
We continue the study of the entanglement entropy of two disjoint intervals
in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We
compute Tr\rho_A^n for any integer n for the Ising universality class and the
final result is expressed as a sum of Riemann-Siegel theta functions. These
predictions are checked against existing numerical data. We provide a
systematic method that gives the full asymptotic expansion of the scaling
function for small four-point ratio (i.e. short intervals). These formulas are
compared with the direct expansion of the full results for free compactified
boson and Ising model. We finally provide the analytic continuation of the
first term in this expansion in a completely analytic form.
We study the entanglement entropy of a block of contiguous spins in excited
states of spin chains. We consider the XY model in a transverse field and the
XXZ Heisenberg spin-chain. For the latter, we developed a numerical application
of algebraic Bethe Ansatz. We find two main classes of states with logarithmic
and extensive behavior in the dimension of the block, characterized by the
properties of excitations of the state. This behavior can be related to the
locality properties of the Hamiltonian having a given state as ground state. We
also provide several details of the finite size scaling.
In this review we first introduce the general methods to calculate the entanglement entropy for free fields, within the Euclidean and the real time formalisms. Then we describe the particular examples which have been worked out explicitly in two, three and more dimensions. Comment: 41 pages, 15 figures, references added, minor changes
We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr\rho_A^n for any integer n is calculated as the four-point function of a particular type of twist fields and the final result is expressed in a compact form in terms of the Riemann-Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data. Comment: 34 pages, 7 figures. V2: Results for small x behavior added, typos corrected and refs added
Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations. We investigate, through a microscopic calculation, the scaling properties of entanglement in spin chain systems, both near and at a quantum critical point. Our results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories. We explore some of the implications of this connection.
Physical interactions in quantum many-body systems are typically local:
Individual constituents interact mainly with their few nearest neighbors. This
locality of interactions is inherited by a decay of correlation functions, but
also reflected by scaling laws of a quite profound quantity: The entanglement
entropy of ground states. This entropy of the reduced state of a subregion
often merely grows like the boundary area of the subregion, and not like its
volume, in sharp contrast with an expected extensive behavior. Such "area laws"
for the entanglement entropy and related quantities have received considerable
attention in recent years. They emerge in several seemingly unrelated fields,
in the context of black hole physics, quantum information science, and quantum
many-body physics where they have important implications on the numerical
simulation of lattice models. In this Colloquium we review the current status
of area laws in these fields. Center stage is taken by rigorous results on
lattice models in one and higher spatial dimensions. The differences and
similarities between bosonic and fermionic models are stressed, area laws are
related to the velocity of information propagation, and disordered systems,
non-equilibrium situations, classical correlation concepts, and topological
entanglement entropies are discussed. A significant proportion of the article
is devoted to the quantitative connection between the entanglement content of
states and the possibility of their efficient numerical simulation. We discuss
matrix-product states, higher-dimensional analogues, and states from
entanglement renormalization and conclude by highlighting the implications of
area laws on quantifying the effective degrees of freedom that need to be
considered in simulations.
If two separated observers are supplied with entanglement, in the form of n pairs of particles in identical partly-entangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximally-entangled pairs of particles, for example Einstein-Podolsky-Rosen singlets, similarly shared between the two observers. The concentration process asymptotically conserves {\em entropy of entanglement}---the von Neumann entropy of the partial density matrix seen by either observer---with the yield of singlets approaching, for large n, the base-2 entropy of entanglement of the initial partly-entangled pure state. Conversely, any pure or mixed entangled state of two systems can be produced by two classically-communicating separated observers, drawing on a supply of singlets as their sole source of entanglement.
The recent interest in aspects common to quantum information and condensed matter has prompted a prosperous activity at the border of these disciplines that were far distant until few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermionic and bosonic model systems. Both bipartite and multipartite entanglement will be considered. At equilibrium we emphasize on how entanglement is connected to the phase diagram of the underlying model. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.
The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.
It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models via the process of decoherence. In this paper, we propose a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information, and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum, as suggested by previous conjectures in the literature. General analytical predictions are confirmed by detailed numerical calculations for two paradigmatic models, relevant in atomic and optical experiments, which exhibit a regular-to-chaotic transition: the quantum kicked top and the Dicke model.
This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case.
We investigate the quantum entanglement content of quasiparticle excitations in extended many-body systems. We show that such excitations give an additive contribution to the bipartite von Neumann and Rényi entanglement entropies that takes a simple, universal form. It is largely independent of the momenta and masses of the excitations and of the geometry, dimension, and connectedness of the entanglement region. The result has a natural quantum information theoretic interpretation as the entanglement of a state where each quasiparticle is associated with two qubits representing their presence within and without the entanglement region, taking into account quantum (in)distinguishability. This applies to any excited state composed of finite numbers of quasiparticles with finite de Broglie wavelengths or finite intrinsic correlation length. This includes particle excitations in massive quantum field theory and gapped lattice systems, and certain highly excited states in conformal field theory and gapless models. We derive this result analytically in one-dimensional massive bosonic and fermionic free field theories and for simple setups in higher dimensions. We provide numerical evidence for the harmonic chain and the two-dimensional harmonic lattice in all regimes where the conditions above apply. Finally, we provide supporting calculations for integrable spin chain models and other interacting cases without particle production. Our results point to new possibilities for creating entangled states using many-body quantum systems.
Entanglement entropy plays a variety of roles in quantum field theory, including the connections between quantum states and gravitation through the holographic principle. This article provides a review of entanglement entropy from a mixed viewpoint of field theory and holography. A set of basic methods for the computation is developed and illustrated with simple examples such as free theories and conformal field theories. The structures of the ultraviolet divergences and the universal parts are determined and compared with the holographic descriptions of entanglement entropy. The utility of quantum inequalities of entanglement are discussed and shown to derive the C-theorem that constrains renormalization group flows of quantum field theories in diverse dimensions.
In quantum statistical mechanics, it is of fundamental interest to understand how close the entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.
Long-distance entanglement distribution is essential for both foundational tests of quantum physics and scalable quantum networks. Owing to channel loss, however, the previously achieved distance was limited to ∼100 kilometers. Here we demonstrate satellite-based distribution of entangled photon pairs to two locations separated by 1203 kilometers on Earth, through two satellite-to-ground downlinks with a summed length varying from 1600 to 2400 kilometers. We observed a survival of two-photon entanglement and a violation of Bell inequality by 2.37 ± 0.09 under strict Einstein locality conditions. The obtained effective link efficiency is orders of magnitude higher than that of the direct bidirectional transmission of the two photons through telecommunication fibers.
A central problem in quantum mechanics is the calculation of the overlap, that is, the scalar product between two quantum states. In the semiclassical limit (Bohr's correspondence principle) we visualize this quantity as the area of overlap between two bands in phase space. In the case of more than one overlap the contributing amplitudes have to be combined with a phase difference again determined by an area in phase space. In this sense the familiar double‐slit interference experiment is generalized to an interference in phase space. We derive this concept by the WKB approximation, illustrate it by the example of Franck‐Condon transitions in diatomic molecules, and compare it with and contrast it to Wigner's concept of pseudo‐probabilities in phase space.
Entanglement is one of the most intriguing features of quantum mechanics. It describes non-local correlations between quantum objects, and is at the heart of quantum information sciences. Entanglement is now being studied in diverse fields ranging from condensed matter to quantum gravity. However, measuring entanglement remains a challenge. This is especially so in systems of interacting delocalized particles, for which a direct experimental measurement of spatial entanglement has been elusive. Here, we measure entanglement in such a system of itinerant particles using quantum interference of many-body twins. Making use of our single-site-resolved control of ultracold bosonic atoms in optical lattices, we prepare two identical copies of a many-body state and interfere them. This enables us to directly measure quantum purity, Rényi entanglement entropy, and mutual information. These experiments pave the way for using entanglement to characterize quantum phases and dynamics of strongly correlated many-body systems.
We consider oscillators evolving subject to a periodic driving force that
dynamically entangles them, and argue that this gives the linearized evolution
around periodic orbits in a general chaotic Hamiltonian dynamical system. We
show that the entanglement entropy, after tracing over half of the oscillators,
generically asymptotes to linear growth at a rate given by the sum of the
positive Lyapunov exponents of the system. These exponents give a classical
entropy growth rate, in the sense of Kolmogorov, Sinai and Pesin. We also
calculate the dependence of this entropy on linear mixtures of the oscillator
Hilbert space factors, to investigate the dependence of the entanglement
entropy on the choice of coarse-graining. We find that for almost all choices
the asymptotic growth rate is the same.
List of papers on quantum philosophy by J. S. Bell Preface Acknowledgements Introduction by Alain Aspect 1. On the problem of hidden variables in quantum mechanics 2. On the Einstein-Rosen-Podolsky paradox 3. The moral aspects of quantum mechanics 4. Introduction to the hidden-variable question 5. Subject and object 6. On wave packet reduction in the Coleman-Hepp model 7. The theory of local beables 8. Locality in quantum mechanics: reply to critics 9. How to teach special relativity 10. Einstein-Podolsky-Rosen experiments 11. The measurement theory of Everett and de Broglie's pilot wave 12. Free variables and local causality 13. Atomic-cascade photons and quantum-mechanical nonlocality 14. de Broglie-Bohm delayed choice double-slit experiments and density matrix 15. Quantum mechanics for cosmologists 16. Bertlmann's socks and the nature of reality 17. On the impossible pilot wave 18. Speakable and unspeakable in quantum mechanics 19. Beables for quantum field theory 20. Six possible worlds of quantum mechanics 21. EPR correlations and EPR distributions 22. Are there quantum jumps? 23. Against 'measurement' 24. La Nouvelle cuisine.
One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th anniversary edition includes an introduction from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction. Quantum mechanics and computer science are introduced before moving on to describe what a quantum computer is, how it can be used to solve problems faster than 'classical' computers and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering.
We investigate entanglement properties of the excited states of the spin-1/2
Heisenberg (XXX) chain with isotropic antiferromagnetic interactions, by
exploiting the Bethe ansatz solution of the model. We consider eigenstates
obtained from both real and complex solutions ("strings") of the Bethe
equations. Physically, the former are states of interacting magnons, whereas
the latter contain bound states of groups of particles. We first focus on the
low-density regime, i.e., with few particles in the chain. Using exact results
and semiclassical arguments, we derive an upper bound S_MAX for the
entanglement entropy. This exhibits an intermediate behavior between
logarithmic and extensive, and it is saturated for highly-entangled states. As
a function of the eigenstate energy, the entanglement entropy is organized in
bands. Their number depends on the number of blocks of contiguous
Bethe-Takahashi quantum numbers. In presence of bound states a significant
reduction in the entanglement entropy occurs, reflecting that a group of bound
particles behaves effectively as a single particle. Interestingly, the
associated entanglement spectrum shows edge-related levels. Upon increasing the
particle density, the semiclassical bound S_MAX becomes inaccurate. For
highly-entangled states S_A\propto L_c, with L_c the chord length, signaling
the crossover to extensive entanglement. Finally, we consider eigenstates
containing a single pair of bound particles. No significant entanglement
reduction occurs, in contrast with the low-density regime.
It is shown that for solvable fermionic and bosonic lattice systems, the reduced density matrices can be determined from the properties of the correlation functions. This provides the simplest way to these quantities which are used in the density-matrix renormalization group method.
Bell's 1964 theorem, which states that the predictions of quantum theory
cannot be accounted for by any local theory, represents one of the most
profound developments in the foundations of physics. In the last two decades,
Bell's theorem has been a central theme of research from a variety of
perspectives, mainly motivated by quantum information science, where the
nonlocality of quantum theory underpins many of the advantages afforded by a
quantum processing of information. The focus of this review is to a large
extent oriented by these later developments. We review the main concepts and
tools which have been developed to describe and study the nonlocality of
quantum theory, and which have raised this topic to the status of a full
sub-field of quantum information science.
The properties of the entanglement entropy (EE) of two particle excited
states in a one-dimensional ring are studied. For a clean system we show
analytically that as long as the momenta of the two particles are not close,
the EE is twice the value of the EE of any single particle state. For almost
identical momenta the EE is lower than this value. The introduction of disorder
is numerically shown to lead to a decrease in the median EE of a two particle
excited state, while interactions (which have no effect for the clean case)
mitigate the decrease. For a ring which is of the same size as the localization
length, interaction increase the EE of a typical two particle excited state
above the clean system EE value.
Bell System Technical Journal, also pp. 623-656 (October)
In statistical physics, useful notions of entropy are defined with respect to some coarse-graining procedure over a microscopic model. Here we consider some special problems that arise when the microscopic model is taken to be relativistic quantum field theory. These problems are associated with the existence of an infinite number of degrees of freedom per unit volume. Because of these the microscopic entropy can, and typically does, diverge for sharply localized states. However, the difference in the entropy between two such states is better behaved, and for most purposes it is the useful quantity to consider. In particular, a renormalized entropy can be defined as the entropy relative to the ground state. We make these remarks quantitative and precise in a simple model situation: the states of a conformal quantum field theory excited by a moving mirror. From this work, we attempt to draw some lessons concerning the “information problem” in black hole physics.
We investigate entanglement for a composite closed system endowed with a
scaling property allowing to keep the dynamics invariant while the effective
Planck constant hbar_eff of the system is varied. Entanglement increases as
hbar_eff goes to 0. Moreover for sufficiently low hbar_eff the evolution of the
quantum correlations, encapsulated for example in the quantum discord, can be
obtained from the mutual information of the corresponding \emph{classical}
system. We show this behavior is due to the local suppression of path
interferences in the interaction that generates the entanglement. This behavior
should be generic for quantum systems in the classical limit.
We present a measure of entanglement that can be computed effectively for any mixed state of an arbitrary bipartite system. We show that it does not increase under local manipulations of the system, and use it to obtain a bound on the teleportation capacity and on the distillable entanglement of mixed states.
If a physical system contains a single particle, and if two distant detectors test the presence of linear superpositions of one-particle and vacuum states, a violation of classical locality can occur. It is due to the creation of a two-particle component by the detecting process itself. Comment: final version in PRL 74 (1995) 4571; 76 (1996) 2205 (erratum)
A quantum system consisting of two subsystems is separable if its density matrix can be written as rho=Sigma(A)W(A) rho(A)' x rho(A)' and rho(A) '' are density matrices for the two subsystems, and the positive weights w(A) satisfy Sigma w(A)=1. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of rho, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.
We compute the entropy of entanglement between the first N spins and the rest of the system in the ground states of a general class of quantum spin chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like kappalog(2N+kappa as N-->infinity, where kappa and kappa are determined explicitly. In an important class of systems, kappa is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for kappa therefore provides an explicit formula for the central charge.
We carry out a systematic study of entanglement entropy in relativistic
quantum field theory. This is defined as the von Neumann entropy S_A=-Tr rho_A
log rho_A corresponding to the reduced density matrix rho_A of a subsystem A.
For the case of a 1+1-dimensional critical system, whose continuum limit is a
conformal field theory with central charge c, we re-derive the result
S_A\sim(c/3) log(l) of Holzhey et al. when A is a finite interval of length l
in an infinite system, and extend it to many other cases: finite systems,finite
temperatures, and when A consists of an arbitrary number of disjoint intervals.
For such a system away from its critical point, when the correlation length \xi
is large but finite, we show that S_A\sim{\cal A}(c/6)\log\xi, where \cal A is
the number of boundary points of A. These results are verified for a free
massive field theory, which is also used to confirm a scaling ansatz for the
case of finite-size off-critical systems, and for integrable lattice models,
such as the Ising and XXZ models, which are solvable by corner transfer matrix
methods. Finally the free-field results are extended to higher dimensions, and
used to motivate a scaling form for the singular part of the entanglement
entropy near a quantum phase transition.