As two of the most important entanglement measures--the entanglement of formation and the entanglement of distillation--have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures--the relative entropy of entanglement and the geometric measure of entanglement--are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
Quantum entropy is an important measure for describing the uncertainty of a quantum state, more uncertainty in subsystems implies stronger quantum entanglement between subsystems. Our goal in this work is to quantify bipartite entanglement using both von Neumann entropy and its complementary dual. We first propose a type of dual entropy from Shannon entropy. We define $S^{t}$-entropy entanglement based on von Neumann entropy and its complementary dual. This implies an analytic formula for two-qubit systems. We show that the monogamy properties of the $S^{t}$-entropy entanglement and the entang…
Entanglement Rényi-$α$ entropy is an entanglement measure. It generalizes the entanglement of formation, and they coincide when $α$ tends to 1. We derive analytical lower and upper bounds for the entanglement Rényi-$α$ entropy of arbitrary dimensional bipartite quantum systems. We also demonstrate the application our bound for some concrete examples. Moreover, we establish the relation between entanglement Rényi-$α$ entropy and some other entanglement measures.
Because of the difficulty in getting the analytic formula of relative entropy of entanglement, it becomes troublesome to study the monogamy relations of relative entropy of entanglement for three-qubit pure states. However, we find that all generalized W states have the monogamous property for relative entropy of entanglement by calculating the relative entropy of entanglement for the reduced states of the generalized W states in three-qubit systems.
In the previous paper \cite{FYK}, we mainly studied the mathematical properties of Tsallis relative entropy with respect to the density operators. As an application of it, we adopt a parametrically extended entanglement-measure due to Tsallis relative entropy in order to measure the degree of entanglement. Then we study its properies with respect to the parameter $q$ appearing in Tsallis entropies. In addition, the relation between it and the relative entropy of entanglement is studied.
We propose a simple approach to the calculation of the entanglement entropy of a spherically symmetric quantum system composed of two separate regions. We consider bound states of the system described by a wave function that is scale invariant and vanishes exponentially at infinity. Our result is in accordance with the holographic bound on entropy and shows that entanglement entropy scales with the area of the boundary.
Previously proposed measures of entanglement, such as entanglement of formation and assistance, are shown to be special cases of the relative entropy of entanglement. The difference between these measures for an ensemble of mixed states is shown to depend on the availability of classical information about particular members of the ensemble. Based on this, relations between relative entropy of entanglement and mutual information are derived.
We show that restricted shareability of multi-qubit entanglement can be fully characterized by unified-$(q,s)$ entropy. We provide a two-parameter class of bipartite entanglement measures, namely unified-$(q,s)$ entanglement with its analytic formula in two-qubit systems for $q\geq 1$, $0\leq s \leq1$ and $qs\leq3$. Using unified-$(q,s)$ entanglement, we establish a broad class of the monogamy inequalities of multi-qubit entanglement for $q\geq2$, $0\leq s \leq1$ and $qs\leq3$.
We show that an entanglement measure called relative entropy of entanglement satisfies a strong continuity condition. If two states are close to each other then so are their entanglements per particle pair in this measure. It follows in particular, that the measure is appropriate for the description of entanglement manipulations in the limit of an infinite number of pairs of particles.
We calculate the relative entropy of entanglement for rotationally invariant states of spin-1/2 and arbitrary spin-$j$ particles or of spin-1 particle and spin-$j$ particle with integer $j$. A lower bound of relative entropy of entanglement and an upper bound of distillable entanglement are presented for rotationally invariant states of spin-1 particle and spin-$j$ particle with half-integer $j$.
The effects of boundary impurities on the entanglement entropy in an antiferromagnetic Heisenberg opened spin-$1/2$ chain are investigated. The method of density-matrix renormalization-group is used to obtain the bipartite entanglement. The entropy increases when the length of the subsystem increases. It will approach to a constant when system length is very large. With the same impurity interaction, qutrit impurities of spin-$1$ can increase the entanglement entropy.
We present the modified relative entropy of entanglement (MRE) in order to both improve the computability for the relative entropy of entanglement and avoid the problem that the entanglement of formation seems to be greater than entanglement of distillation. For two qubit system we derive out an explicit and "weak" closed expression of MRE that depends on the pure state decompositions in the case of mixed states. For more qubit system, we obtain an algorithm to calculate MRE in principle. MRE significantly improves the computability of relative entropy of entanglement and decreases the depende…
We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal dimensions. Expanding the von Neumann entropy around its maximum we show that the first order only depends on the participation ratio, while higher orders involve other multifractal exponents. These results can be applied to entanglement behavior near the Anderson transition.
We show that the restricted sharability and distribution of multi-qubit entanglement can be characterized by Tsallis-$q$ entropy. We first provide a class of bipartite entanglement measures named Tsallis-$q$ entanglement, and provide its analytic formula in two-qubit systems for $1 \leq q \leq 4$. For $2 \leq q \leq 3$, we show a monogamy inequality of multi-qubit entanglement in terms of Tsallis-$q$ entanglement, and we also provide a polygamy inequality using Tsallis-$q$ entropy for $1 \leq q \leq 2$ and $3 \leq q \leq 4$.
In this paper, we study the monogamy inequality of Tsallis-q entropy entanglement. We first provide an analytic formula of Tsallis-q entropy entanglement in two-qubit systems for $\frac{5-\sqrt{13}}{2}\leq q\leq\frac{5+\sqrt{13}}{2}.$ The analytic formula of Tsallis-q entropy entanglement in $2\otimes d$ system is also obtained and we show that Tsallis-q entropy entanglement satisfies a set of hierarchical monogamy equalities. Furthermore, we prove the squared Tsallis-q entropy entanglement follows a general inequality in the qubit systems. Based on the monogamy relations, a set of multipartit…
In this four-part prospectus, we first give a brief introduction to the motivation for studying entanglement entropy and some recent development. Then follows a summary of our recent work about entanglement entropy in states with traditional long-range order. After that we demonstrate calculation of entanglement entropy in both one-dimensional spin-less fermionic systems as well as bosonic systems via different approaches, and connect them using one-dimensional bosonization. In the last part, we briefly sketch the idea of bosonization in high-dimensions, and discuss the possibility and advanta…
We generalize the symmetric multi-qubit states to their q-analogs, whose basis vectors are identified with the q-Dicke states. We study the entanglement entropy in these states and find that entanglement is extruded towards certain regions of the system due to the inhomogeneity aroused by q-deformation. We also calculate entanglement entropy in ground states of a related q-deformed Lipkin-Meshkov-Glick model and show that the singularities of entanglement can correctly signify the quantum phase transition points for different strengths of q-deformation.
We derive a family of entanglement criteria for continuous variable systems based on the Rényi entropy of complementary distributions. We show that these entanglement witnesses can be more sensitive than those based on second-order moments, as well as previous tests involving the Shannon entropy [Phys. Rev. Lett. \textbf{103}, 160505 (2009)]. We extend our results to include the case of discrete sampling, and develop another set of entanglement tests using the discrete Tsallis entropy. We provide several numerical results which show that our criteria can be used to identify entanglement in a n…
Using Rényi-$α$ entropy to quantify bipartite entanglement, we prove monogamy of entanglement in multi-qubit systems for $α\geq 2$. We also conjecture a polygamy inequality of multi-qubit entanglement with strong numerical evidence for $0.83-ε\leq α\leq 1.43+ε$ with $0<ε<0.01$.
The Wehrl entropy is an entropy associated to the Husimi quasi-probability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Białynicki-Birula and Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how a Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.
We present the modified relative entropy of entanglement for multi-party systems by a given relative density matrix which is spanned by a linear combination of the direct products of so-called basis of relative density matrices and reduced density matrices for every party. For three qubit system in a pure state we derive out the explicit and closed expression of our relative density matrix which is a function of the components of the state vector of three qubits. Through computing the modified relative entropy of entanglement of some important and interesting examples, we display the main beha…
We provide generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy. Using quantum Tsallis entropy of order $q$, we first provide a generalized monogamy inequality of multi-qubit entanglement for $q=2$ or $3$. This generalization encapsulates multi-qubit CKW-type inequality as a special case. We further provide a generalized polygamy inequality of multi-qubit entanglement in terms of Tsallis-$q$ entropy for $1 \leq q \leq2$ or $3 \leq q \leq 4$, which also contains the multi-qubit polygamy inequality as a special case.
Properties of the max- relative entropy of entanglement are investigated, and its significance as an upper bound to the one shot rate for perfect entanglement dilution, under a particular class of quantum operations, is discussed. It is shown that it is in fact equal to another known entanglement monotone, namely the log robustness. It is known that the latter is not asymptotically continuous and it is not known whether it is weakly additive. However, by suitably modifying the max- relative entropy of entanglement we obtain a quantity which is seen to satisfy both these properties. In fact, th…
We extend Vedral and Plenio's theorem (theorem 3 in Phys. Rev. A 57, 1619) to a more general case, and obtain the relative entropy of entanglement for a class of mixed states, this result can also follow from Rains' theorem 9 in Phys. Rev. A 60, 179.
We establish relations between tripartite pure state entanglement and additivity properties of the bipartite relative entropy of entanglement. Our results pertain to the asymptotic limit of local manipulations on a large number of copies of the state. We show that additivity of the relative entropy would imply that there are at least two inequivalent types of asymptotic tripartite entanglement. The methods used include the application of some useful lemmas that enable us to analytically calculate the relative entropy for some classes of bipartite states.
In this work, a canonical method to compute entanglement entropy is proposed. We show that for two-dimensional conformal theories defined in a torus, a choice of moduli space allows the typical entropy operator of the TFD to provide the entanglement entropy of the degrees of freedom defined in a segment and their complement. In this procedure, it is not necessary to make an analytic continuation from the Rényi entropy and the von Neumann entanglement entropy is calculated directly from the expected value of an entanglement entropy operator. We also propose a model for the evolution of the enta…
The ground state entanglement entropy is studied in a many-body bipartite quantum system with either a single or multiple conserved quantities. It is shown that the entanglement entropy exhibits a universal power-law behaviour at large $R$ -- the occupancy ratio between the two subsystems. Single and multiple conserved quantities lead to different power-law exponents, suggesting the entanglement entropy can serve to detect hidden conserved quantities. Moreover, occupancy measurements allow to infer the bipartite entanglement entropy. All the above results are generalized for the Rényi entropy.
Motivated by recent discussions of entanglement in the context of high energy scattering, we consider the relation between the entanglement entropy of a highly excited state of a quantum system and the classical entanglement entropy of the corresponding classical system. We show on the example of two weakly coupled harmonic oscillators, that the two entropies are equal. Quantum mechanically, the reduced density matrix which yields this entropy is close to the maximally entangled state. We thus observe that the nature of entanglement in this type of state is purely classical.
We study the properties of entanglement in two-dimensional topologically ordered phases of matter. Such phases support anyons, quasiparticles with exotic exchange statistics. The emergent nonlocal state spaces of anyonic systems admit a particular form of entanglement that does not exist in conventional quantum mechanical systems. We study this entanglement by adapting standard notions of entropy to anyonic systems. We use the algebraic theory of anyon models (modular tensor categories) to illustrate the nonlocal entanglement structure of anyonic systems. Using this formalism, we present a gen…
We compute a variational approximation to the entanglement entropy for states of the Gross-Neveu model. Further, we examine the functional dependence of the entanglement entropy on the coupling and number of colors in the theory. Our results display non-peturbative behavior. It is shown that the entanglement entropy is monotonically decreasing and convex with respect to the coupling. We also show how the behavior of the entanglement entropy under renormalization group transformations is related to the beta function of the Gross-Neveu model, and compare these renormalization group results to ou…
In this note a new method for computing the entanglement entropy of a CFT holographically is explored. It consists of finding a bulk background with a boundary metric that has the conical singularities needed to compute the entanglement entropy in the usual QFT definition. An explicit calculation is presented for d=2.
Two significant consequences of quantum fluctuations are entanglement and criticality. Entangled states may not be critical but a critical state shows signatures of universality in entanglement. A surprising result found here is that the entanglement entropy may become arbitrarily large and negative near the dissociation of a bound pair of quantum particles. Although apparently counter-intuitive, it is shown to be consistent and essential for the phase transition, by mapping to a classical problem of DNA melting. We associate the entanglement entropy to a subextensive part of the entropy of DN…
We study the Von Neumann and Rényi entanglement entropy of long-range harmonic oscillators (LRHO) by both theoretical and numerical means. We show that the entanglement entropy in massless harmonic oscillators increases logarithmically with the sub-system size as $S=\frac{c_{eff}}{3}\log l$. Although the entanglement entropy of LRHO's shares some similarities with the entanglement entropy at conformal critical points we show that the Rényi entanglement entropy presents some deviations from the expected conformal behavior. In the massive case we demonstrate that the behavior of the entanglement…
We prove a conjecture by Bravyi on an upper bound on entanglement rates of local Hamiltonians. We then use this bound to prove the stability of the area law for the entanglement entropy of quantum spin systems under adiabatic and quasi-adiabatic evolutions.
We study the change in topological entanglement entropy that occurs when a two-dimensional system in a topologically ordered phase undergoes a transition to another such phase due to the formation of a Bose condensate. We also consider the topological entanglement entropy of systems with domains in different topological phases, and of phase boundaries between these domains. We calculate the topological entropy of these interfaces and derive two fundamental relations between the interface topological entropy and the bulk topological entropies on both sides of the interface.
We study holographic entanglement entropy in dS/CFT and introduce time-like entanglement entropy in CFTs. Both of them take complex values in general and are related with each other via an analytical continuation. We argue that they are correctly understood as pseudo entropy. We find that the imaginary part of pseudo entropy implies an emergence of time in dS/CFT.
In this paper we discuss behaviors of entanglement entropy between two interacting CFTs and its holographic interpretation using the AdS/CFT correspondence. We explicitly perform analytical calculations of entanglement entropy between two free scalar field theories which are interacting with each other in both static and time-dependent ways. We also conjecture a holographic calculation of entanglement entropy between two interacting $\mathcal{N}=4$ super Yang-Mills theories by introducing a minimal surface in the S$^5$ direction, instead of the AdS$_5$ direction. This offers a possible general…
The entanglement entropy in Kondo impurity systems is studied analytically using conformal field theory. From the impurity contribution to the scaling corrections of the entanglement entropy we extract information about the screening cloud profile for general non-Fermi-liquid fixed points. By also considering the finite-temperature corrections to scaling of the von Neumann entropy we point out a direct connection between the large-distance screening cloud profile and thermodynamic observables such as the specific heat.
We investigate the von Neumann entropy of a block of subsystem for the valence-bond solid (VBS) state with general open boundary conditions. We show that the effect of the boundary on the von Neumann entropy decays exponentially fast in the distance between the subsystem considered and the boundary sites. Further, we show that as the size of the subsystem increases, its von Neumann entropy exponentially approaches the summation of the von Neumann entropies of the two ends, the exponent being related to the size. In contrast to critical systems, where boundary effects to the von Neumann entropy…
We show that the Jaynes principle is indeed a proper inference scheme when applied to compound systems and will correctly produce the entangled maximum entropy states compatible with appropriate data. This is accomplished by including the dispersion of the entanglement along with its mean value as constraints and an application of the uncertainty principle. We also construct a "thermodynamic-like" description of the entanglement arising out of the maximum entropy principle.
The entanglement entropy of a noninteracting fermionic system confined to a two-dimensional honeycomb lattice on a torus is calculated. We find that the entanglement entropy can characterize Lifshitz phase transitions without a local order parameter. In the noncritical phase and critical phase with a nodal Fermi surface, the entanglement entropy satisfies an area law. The leading subarea term is a constant in the gapped phase rather than a logarithmic additive term in the gapless regime. The tuning of chemical potential allows for a nonzero Fermi surface, whose variation along a particular dir…
In this paper, we review how to obtain the central charge of a critical entanglement Hamiltonian through the nested entanglement entropy which was first introduced in [J. Lou et al. PRB 84, 245128 (2011)]. The critical phenomena of the entanglement Hamiltonian can be identified by the central charge obtained by the nested entanglement entropy. We review our previous studies[J. Lou et al. PRB 84, 245128 (2011), S. Tanaka et al. PRA 87, 214401 (2013)] in which we investigated certain entanglement nature of two-dimensional valence-bond-solid (VBS) state and quantum hard-square models on square an…
A charged entanglement entropy is a new measure which probes quantum entanglement between different charge sectors. We study symmetry protected topological (SPT) phases in 2+1 dimensional space-time by using this charged entanglement entropy. SPT phases are short range entangled states without topological order and hence cannot be detected by the topological entanglement entropy. We demonstrate that the universal part of the charged entanglement entropy is non-zero for non-trivial SPT phases and therefore it is a useful measure to detect short range entangled topological phases. We also discus…
We calculate the entanglement entropy of strongly correlated low-dimensional fermions in metallic, superfluid and antiferromagnetic insulating phases. The entanglement entropy reflects the degrees of freedom available in each phase for storing and processing information, but is found not to be a state function in the thermodynamic sense. The role of critical points, smooth crossovers and Hilbert space restrictions in shaping the dependence of the entanglement entropy on the system parameters is illustrated for metallic, insulating and superfluid systems. The dependence of the spin susceptibili…
The entanglement entropy associated with a spatial boundary in quantum field theory is UV divergent, with the leading term proportional to the area of the boundary. For a class of quantum states defined by a path integral, the Callan-Wilczek formula gives a geometrical definition of the entanglement entropy. We show that, for this class of quantum states, the entanglement entropy is rendered UV-finite by precisely the counterterms required to cancel the UV divergences in the gravitational effective action. In particular, the leading contribution to the entanglement entropy is given by the reno…
Entanglement entropy in topologically ordered matter phases has been computed extensively using various methods. In this paper, we study the entanglement entropy of topological phases in two-spaces from a new perspective---the perspective of quasiparticle fluctuations. In this picture, the entanglement spectrum of a topologically ordered system is identified with the spectrum of quasiparticle fluctuations of the system, and the entanglement entropy measures the maximal quasiparticle fluctuations on the EB. As a consequence, entanglement entropy corresponds to the thermal entropy of the quasipa…
We continue the study of entanglement entropy for a QFT through a perturbative expansion of the path integral definition of the reduced density matrix. The universal entanglement entropy for a CFT perturbed by a relevant operator is calculated to second order in the coupling. We also explore the geometric dependence of entanglement entropy for a deformed planar entangling surface, finding surprises at second order.
Over the last three decades entanglement entropy has been obtained for quantum fields propagating in genus zero topologies (Spheres). For scalar fields propagating in these topologies, it has been shown that the entanglement entropy scales as area. In the last few years non-trivial topologies are increasingly relevant for different areas. For instance, in describing quantum phases, it has been realized that long-range entangled states are described by topological order. If quantum entanglement can plausibly provide explanation for these, it then imperative to obtain entanglement entropy in the…
We analyse the entanglement entropy properties of a two-dimensional p-wave superconductor with Rashba spin-orbit coupling, which displays a rich phase-space that supports non-trivial topological phases, as the chemical potential and the Zeeman term are varied. We show that the entanglement entropy and its derivatives clearly signal the topological transitions and provides a sensible signature of each topological phase. We separately analyse the contributions to the entanglement entropy that are proportional to or independent of the perimeter of the system, as a function of the Hamiltonian coup…
The time evolution of entanglement tracks how information propagates in interacting quantum systems. We study entanglement entropy in CFT$_2$ with a time-dependent Hamiltonian. We perturb by operators with time-dependent source functions and use the replica trick to calculate higher order corrections to entanglement entropy. At first order, we compute the correction due to a metric perturbation in AdS$_3$/CFT$_2$ and find agreement on both sides of the duality. Past first order, we find evidence of a universal structure of entanglement propagation to all orders. The central feature is that int…
We study entanglement entropy of excited states in two dimensional conformal field theories (CFTs). Especially we consider excited states obtained by acting primary operators on a vacuum. We show that under its time evolution, entanglement entropy increases by a finite constant when the causality condition is satisfied. Moreover, in rational CFTs, we prove that this increased amount of (both Renyi and von-Neumann) entanglement entropy always coincides with the log of quantum dimension of the primary operator.
We study bipartite post measurement entanglement entropy after selective measurements in quantum chains. We first study the quantity for the critical systems that can be described by conformal field theories. We find a connection between post measurement entanglement entropy and the Casimir energy of floating objects. Then we provide formulas for the post measurement entanglement entropy for open and finite temperature systems. We also comment on the Affleck-Ludwig boundary entropy in the context of the post measurement entanglement entropy. Finally, we also provide some formulas regarding mod…
The Quantum Null Energy Condition (QNEC) relates energy to the second variation of entropy in relativistic quantum field theory. We use the QNEC inequality to bound entanglement entropy in quenches. At early times the entanglement entropy grows quadratically in time, and the QNEC provides an upper bound on the prefactor. We demonstrate that the bound is tight, by showing that it is saturated in certain quench protocols: boundary state quenches in conformal field theories in any dimensions. In higher than two dimensions we compute entanglement entropy using AdS/CFT. Our results are the first pu…
