Entanglement and Quantum Phase Transition Revisited

We show that, for an exactly solvable quantum spin model, a discontinuity in the first derivative of the ground state concurrence appears in the absence of quantum phase transition. It is opposed to the popular belief that the non-analyticity property of entanglement (ground state concurrence) can be used to determine quantum phase transitions. We further point out that the analyticity property of the ground state concurrence in general can be more intricate than that of the ground state energy. Thus there is no one-to-one correspondence between quantum phase transitions and the non-analyticity property of the concurrence. Moreover, we show that the von Neumann entropy, as another measure of entanglement, can not reveal quantum phase transition in the present model. Therefore, in order to link with quantum phase transitions, some other measures of entanglement are needed.Comment: RevTeX 4, 4 pages, 1 EPS figures. some modifications in the text. Submitted to Phys. Rev.

Universality in the entanglement structure of ferromagnets

Systems of exchange-coupled spins are commonly used to model ferromagnets. The quantum correlations in such magnets are studied using tools from quantum information theory. Isotropic ferromagnets are shown to possess a universal low-temperature density matrix which precludes entanglement between spins, and the mechanism of entanglement cancellation is investigated, revealing a core of states resistant to pairwise entanglement cancellation. Numerical studies of one-, two-, and three-dimensional lattices as well as irregular geometries showed no entanglement in ferromagnets at any temperature or magnetic field strength.Comment: 4 pages, 2 figure

Scaling of entanglement between separated blocks in spin chains at criticality

We compute the entanglement between separated blocks in certain spin models showing that at criticality this entanglement is a function of the ratio of the separation to the length of the blocks and can be written as a product of a power law and an exponential decay. It thereby interpolates between the entanglement of individual spins and blocks of spins. It captures features of correlation functions at criticality as well as the monogamous nature of entanglement. We exemplify invariant features of this entanglement to microscopic changes within the same universality class. We find this entanglement to be invariant with respect to simultaneous scale transformations of the separation and the length of the blocks. As a corollary, this study estimates the entanglement between separated regions of those quantum fields to which the considered spin models map at criticality.Comment: 4 pages, 3 figures; comments welcom

Violation of the entropic area law for Fermions

We investigate the scaling of the entanglement entropy in an infinite translational invariant Fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. Thus, in contrast to analogous Bosonic systems, the entropic area law is violated for Fermions. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven, that the presented scaling law holds whenever the Fermi surface is finite. This is in particular true for all ground states of Hamiltonians with finite range interactions.Comment: 5 pages, 1 figur

Area law and vacuum reordering in harmonic networks

We review a number of ideas related to area law scaling of the geometric entropy from the point of view of condensed matter, quantum field theory and quantum information. An explicit computation in arbitrary dimensions of the geometric entropy of the ground state of a discretized scalar free field theory shows the expected area law result. In this case, area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations. Furthermore, the explicit control on all the eigenvalues of the reduced density matrix allows for a verification of entropy loss along the renormalization group trajectory driven by the mass term. A further result of our computation shows that single-copy entanglement also obeys area law scaling, majorization relations and decreases along renormalization group flows.Comment: 15 pages, 6 figures; typos correcte

Configuration-Space Location of the Entanglement between Two Subsystems

In this paper we address the question: where in configuration space is the entanglement between two particles located? We present a thought-experiment, equally applicable to discrete or continuous-variable systems, in which one or both parties makes a preliminary measurement of the state with only enough resolution to determine whether or not the particle resides in a chosen region, before attempting to make use of the entanglement. We argue that this provides an operational answer to the question of how much entanglement was originally located within the chosen region. We illustrate the approach in a spin system, and also in a pair of coupled harmonic oscillators. Our approach is particularly simple to implement for pure states, since in this case the sub-ensemble in which the system is definitely located in the restricted region after the measurement is also pure, and hence its entanglement can be simply characterised by the entropy of the reduced density operators. For our spin example we present results showing how the entanglement varies as a function of the parameters of the initial state; for the continuous case, we find also how it depends on the location and size of the chosen regions. Hence we show that the distribution of entanglement is very different from the distribution of the classical correlations.Comment: RevTex, 12 pages, 9 figures (28 files). Modifications in response to journal referee

GHZ extraction yield for multipartite stabilizer states

Let $|\Psi>$ be an arbitrary stabilizer state distributed between three remote parties, such that each party holds several qubits. Let $S$ be a stabilizer group of $|\Psi>$. We show that $|\Psi>$ can be converted by local unitaries into a collection of singlets, GHZ states, and local one-qubit states. The numbers of singlets and GHZs are determined by dimensions of certain subgroups of $S$. For an arbitrary number of parties $m$ we find a formula for the maximal number of $m$-partite GHZ states that can be extracted from $|\Psi>$ by local unitaries. A connection with earlier introduced measures of multipartite correlations is made. An example of an undecomposable four-party stabilizer state with more than one qubit per party is given. These results are derived from a general theoretical framework that allows one to study interconversion of multipartite stabilizer states by local Clifford group operators. As a simple application, we study three-party entanglement in two-dimensional lattice models that can be exactly solved by the stabilizer formalism.Comment: 12 pages, 1 figur

Mixed-state fidelity and quantum criticality at finite temperature

We extend to finite temperature the fidelity approach to quantum phase transitions (QPTs). This is done by resorting to the notion of mixed-state fidelity that allows one to compare two density matrices corresponding to two different thermal states. By exploiting the same concept we also propose a finite-temperature generalization of the Loschmidt echo. Explicit analytical expressions of these quantities are given for a class of quasi-free fermionic Hamiltonians. A numerical analysis is performed as well showing that the associated QPTs show their signatures in a finite range of temperatures.Comment: 7 pages, 4 figure

Optimal minimal measurements of mixed states

The optimal and minimal measuring strategy is obtained for a two-state system prepared in a mixed state with a probability given by any isotropic a priori distribution. We explicitly construct the specific optimal and minimal generalized measurements, which turn out to be independent of the a priori probability distribution, obtaining the best guesses for the unknown state as well as a closed expression for the maximal mean averaged fidelity. We do this for up to three copies of the unknown state in a way which leads to the generalization to any number of copies, which we then present and prove.Comment: 20 pages, no figure

Entanglement and Quantum Phases in the Anisotropic Ferromagnetic Heisenberg Chain in the Presence of Domain Walls

We discuss entanglement in the spin-1/2 anisotropic ferromagnetic Heisenberg chain in the presence of a boundary magnetic field generating domain walls. By increasing the magnetic field, the model undergoes a first-order quantum phase transition from a ferromagnetic to a kink-type phase, which is associated to a jump in the content of entanglement available in the system. Above the critical point, pairwise entanglement is shown to be non-vanishing and independent of the boundary magnetic field for large chains. Based on this result, we provide an analytical expression for the entanglement between arbitrary spins. Moreover the effects of the quantum domains on the gapless region and for antiferromagnetic anisotropy are numerically analysed. Finally multiparticle entanglement properties are considered, from which we establish a characterization of the critical anisotropy separating the gapless regime from the kink-type phase.Comment: v3: 7 pages, including 4 figures and 1 table. Published version. v2: One section (V) added and references update