Relative Entropy and Relative Entropy of Entanglement for Infinite-Dimensional Systems

In this paper, firstly, we derive some inequalities about the relative entropy for infinite-dimensional quantum systems. Secondly, we propose a new measurement based on the relative entropy of entanglement for infinite-dimensional systems with bounded mean energy, and give a lower bound on this entanglement measure. Lastly, we generalize this measure to multi-partite quantum systems.     

Entanglement Entropy Scaling Law in the Ground State of Supersymmetric Fermion Lattice Model

Journal of Experimental and Theoretical Physics - We investigate the entanglement entropy in the ground state of the supersymmetric fermion lattice model. The Hamiltonian of this model is block...     

Relative Entropy and Squashed Entanglement

We are interested in the properties and relations of entanglement measures. Especially, we focus on the squashed entanglement and relative entropy of entanglement, as well as their analogues and variants. Our first result is a monogamy-like inequality involving the relative entropy of entanglement and its one-way LOCC variant. The proof is accomplished by exploring the properties of relative entropy in the context of hypothesis testing via one-way LOCC operations, and by making use of an argument resembling that by Piani on the faithfulness of regularized relative entropy of entanglement. Following this, we obtain a commensurate and faithful lower bound for squashed entanglement, in the form of one-way LOCC relative entropy of entanglement. This gives a strengthening to the strong subadditivity of von Neumann entropy. Our result improves the trace-distance-type bound derived in Brandão et al. (Commun Math Phys, 306:805–830, 2011), where faithfulness of squashed entanglement was first proved. Applying Pinsker’s inequality, we are able to recover the trace-distance-type bound, even with slightly better constant factor. However, the main improvement is that our new lower bound can be much larger than the old one and it is almost a genuine entanglement measure. We evaluate exactly the relative entropy of entanglement under various restricted measurement classes, for maximally entangled states. Then, by proving asymptotic continuity, we extend the exact evaluation to their regularized versions for all pure states. Finally, we consider comparisons and separations between some important entanglement measures and obtain several new results on these, too.     

The Fisher-Hartwig Formula and Entanglement Entropy

Toeplitz matrices have applications to different problems of statistical mechanics. Recently it was used for calculation of entanglement entropy in spin chains. In the paper we review these recent developments. We use the Fisher-Hartwig formula, as well as the recent results concerning the asymptotics of the block Toeplitz determinants, to calculate entanglement entropy of large block of spins in the ground state of XY spin chain.     

Holographic Entanglement Entropy of the BTZ Black Hole

We investigate quantum entanglement of gravitational configurations in 3D AdS gravity using the AdS/CFT correspondence. We derive explicit formulas for the holographic entanglement entropy (EE) of the BTZ black hole, conical singularities and regularized AdS₃. The leading term in the large temperature expansion of the holographic EE of the BTZ black hole reproduces exactly its Bekenstein-Hawking entropy S _BH , whereas the subleading term behaves as ln S _BH . We also show that the leading term of the holographic EE for the BTZ black hole can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus. This result indicates that black hole EE is not a fundamental feature of the underlying theory of quantum gravity but emerges when the semiclassical notion of spacetime geometry is used to describe the black hole.     

Entropy and Entanglement Bounds for Reduced Density Matrices of Fermionic States

Unlike bosons, fermions always have a non-trivial entanglement. Intuitively, Slater determinantal states should be the least entangled states. To make this intuition precise we investigate entropy and entanglement of fermionic states and prove some extremal and near extremal properties of reduced density matrices of Slater determinantal states.     

Topological Entanglement Entropy from the Holographic Partition Function

We study the entropy of chiral 2+01-dimensional topological phases, where there are both gapped bulk excitations and gapless edge modes. We show how the entanglement entropy of both types of excitations can be encoded in a single partition function. This partition function is holographic because it can be expressed entirely in terms of the conformal field theory describing the edge modes. We give a general expression for the holographic partition function, and discuss several examples in depth, including abelian and non-abelian fractional quantum Hall states, and $p+ip$ superconductors. We extend these results to include a point contact allowing tunneling between two points on the edge, which causes thermodynamic entropy associated with the point contact to be lost with decreasing temperature. Such a perturbation effectively breaks the system in two, and we can identify the thermodynamic entropy loss with the loss of the edge entanglement entropy. From these results, we obtain a simple interpretation of the non-integer ‘ground state degeneracy’ which is obtained in 1+1-dimensional quantum impurity problems: its logarithm is a 2+1-dimensional topological entanglement entropy.     

Bounded Entanglement Entropy in the Quantum Ising Model

Journal of Statistical Physics - A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model...     

Fluctuations of Entanglement Entropy in Noninertial Frames

The fluctuation and relative fluctuation of entanglement entropy of a bipartite system for Dirac fields in noninertial frames are investigated. It is shown that the fluctuation and relative fluctuation of entanglement entropy are observer-dependent, which depend on their observed frames. It is found that both the fluctuation and relative fluctuation of entanglement entropy become more noticeable with the increasing of the subsystem's acceleration. We also find that the entanglement entropy always has fluctuation regardless of the initial state parameter and for any magnitude of the acceleration. We argue that the statistical mean for the measurement of entanglement entropy relates to the accelerated motion of the observer.     

Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy

Let ?? ₁₂ be a bipartite density matrix. We prove lower bounds for the entanglement of formation E_f(?? ₁₂) and the squashed entanglement E_sq(?? ₁₂) in terms of the conditional entropy S ₁₂ ? S ₁ and prove that these bounds are sharp by constructing a new class of states whose entanglements can be computed, and for which the bounds are saturated.     

Properties of Entropy and Entanglement of Two-Mode Nonlinear Coherent States

In this communication, two-mode nonlinear coherent states are reviewed and special cases are given.Starting from a three-level atom coupled to two modes of radiation with any form of nonlinearities of the two-modefields, we derive a Raman-coupled effective Hamiltonian by a unitary transformation, evaluated perturbatively in couplingconstants. We use the quantum entropy to measure the degree of entanglement in the time development of an effectivetwo-level atom interacting with two-mode nonlinear-coherent states. The results show that the nonlinearity effect yieldsthe superstructure of atomic Rabi oscillations and the effect of the Stark shift changes the quasiperiod of the field entropyevolution and entanglement between the particle and the field. A possible experimental test of a new effect is proposed.     

Unitarity as Preservation of Entropy and Entanglement in Quantum Systems

The logical structure of Quantum Mechanics (QM) and its relation to other fundamental principles of Nature has been for decades a subject of intensive research. In particular, the question whether the dynamical axiom of QM can be derived from other principles has been often considered. In this contribution, we show that unitary evolutions arise as a consequences of demanding preservation of entropy in the evolution of a single pure quantum system, and preservation of entanglement in the evolution of composite quantum systems. ⁶ We would also like to dedicate this work to the memory of Asher Peres, whose contributions and sharp comments guided the first steps of the present article.     

Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

Journal of Statistical Physics - We consider a macroscopic disordered system of free d-dimensional lattice fermions whose one-body Hamiltonian is a Schrödinger operator H with ergodic...     

Stability of Pairwise Entanglement in Decoherence Environment

Consider the dynamics of a bipartite entangled system in the decoherence environment, we investigate the stability of pairwise entanglement under decoherence.We find that with the same initial entanglement, the lifetime of entanglement in pure states and some mixed states is the longest.We call these special entangled states as Decoherence Path States (DPS).Besides, we present simple analytic evolution equations of the entanglement in these states.The lifetimes can also be obtained easily.Furthermore, we also study the stability of the nearest neighbor entanglement in the ground state of an antiferromagnetic spin-1/2 ring.Coincidentally, the conclusion is that it is as stable as Decoherence Path States.Thus the nearest neighbor entanglement in the ground state is not maximized but it is the most stable.This interesting result links the energy and entanglement in a spin system from a new point of view.     

Expanding the Area of Gravitational Entropy

I describe how gravitational entropy is intimately connected with the concept of gravitational heat, expressed as the difference between the total and free energies of a given gravitational system. From this perspective one can compute these thermodyanmic quantities in settings that go considerably beyond Bekenstein's original insight that the area of a black hole event horizon can be identified with thermodynamic entropy. The settings include the outsides of cosmological horizons and spacetimes with NUT charge. However the interpretation of gravitational entropy in these broader contexts remains to be understood.     

Corrected Entropy Law for Charged and Rotating Black Strings

The primary objective in this work is to study the corrected entropy law for charged and rotating black strings in asymptotically anti-de Sitter spacetime. By employing, the Hamilton-Jacobi approach, fermions tunneling beyond semiclassical approximation is investigated. The correction has been done by taking the proportionality parameters of quantum correction of action I _i to the semiclassical action I ₀ as 2π times the inverse of the black string horizon area. Moreover, with the aid of corrected Hawking temperature we finally compute the corrected area law, which includes the logarithmic term and inverse area terms.     

Relative Entropy of Entanglement of a Kind of Two-Qubit Entangled States

We strictly prove that some block diagonalizable two-qubit entangled state with six non-zero elements reaches its quantum relative entropy entanglement by a separable state having the same matrix structure. The entangled state comprises local filtering result state as a special case.     

Form Factors of Branch-Point Twist Fields in Quantum Integrable Models and Entanglement Entropy

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.