We consider systems of interacting spins and study the entanglement that can be localized, on average, between two separated spins by performing local measurements on the remaining spins. This concept of localizable entanglement (LE) leads naturally to notions like entanglement length and entanglement fluctuations. For both spin-
1∕2 and spin-1 systems, we prove that the LE of a pure quantum state can be lower bounded by connected correlation functions. We further propose a scheme, based on matrix-product states and the Monte Carlo method, to efficiently calculate the LE for quantum states of a large number of spins. The virtues of LE are illustrated for various spin models. In particular, characteristic features of a quantum phase transition such as a diverging entanglement length can be observed. We also give examples for pure quantum states exhibiting a diverging entanglement length but finite correlation length. We have numerical evidence that the ground state of the antiferromagnetic spin-1 Heisenberg chain can serve as a perfect quantum channel. Furthermore, we apply the numerical method to mixed states and study the entanglement as a function of temperature.
We begin by deriving bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure Finitely Correlated States (FCS). The bounds we derive become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our result permits a more efficient calculation, numerically and in some cases even analytically, of the entanglement in arbitrary finitely correlated quantum spin chains. We continue the study of entanglement in the setting of ground states of Hamiltonians with a spectral gap. In particular, for
V a finite subset of
Zd, we let
HV denote a Hamiltonian on
V with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap. For a density matrix
ρA that describes the finite-volume restriction to a region
A of the unique ground state, we provide a detailed version of a proof by M. Hastings, that the entropy of
ρA is bounded by a uniform constant
C. Moreover, we provide a detailed generalization of the 1-dimensional construction of Hastings' approximation to the ground state in dimensions 2 and higher. Finally, we turn our attention to the study of a conjecture central to Quantum Information Theory, the multiplicativity of the maximal output Schatten
p-norm of quantum channels, for
p>1. In particular, we study the output 2-norm for a special class of quantum channels, the depolarized Werner-Holevo channels, and show that multiplicativity holds for a product of two identical channels in this class.
We study the relation between the recently defined localizable entanglement and generalized correlations in quantum spin systems. Differently from the current belief, the localizable entanglement is always given by the average of a generalized string. Using symmetry arguments we show that in most spin-1/2 and spin-1 systems the localizable entanglement reduces to the spin-spin or string correlations, respectively. We prove that a general class of spin-1 systems, which includes the Heisenberg model, can be used as a perfect quantum channel. These conclusions are obtained in analytic form and confirm some results found previously on numerical grounds.
Antiferromagnetic spin chains play an important role in condensed matter and statistical mechanics. Recently
XXX spin chain was discussed in relation to information theory. Here we consider localizable entanglement. It is how much entanglement can be localized on two spins by performing local measurements on other individual spins (in a system of many interacting spins). We consider the ground state of antiferromagnetic spin chain. We study localizable entanglement [represented by concurrence] between two spins. It is a function of the distance. We start with isotropic spin chain. Then we study effects of anisotropy and magnetic field. We conclude that anisotropy increases the localizable entanglement. We discovered high sensitivity to a magnetic field in cases of high symmetry. We also evaluated concurrence of these two spins before the measurement to illustrate that the measurement raises the concurrence.
We consider pure quantum states of N>>1 spins or qubits and study the average entanglement that can be localized between two separated spins by performing local measurements on the other individual spins. We show that all classical correlation functions provide lower bounds to this localizable entanglement, which follows from the observation that classical correlations can always be increased by doing appropriate local measurements on the other qubits. We analyze the localizable entanglement in familiar spin systems and illustrate the results on the hand of the Ising spin model, in which we observe characteristic features for a quantum phase transition such as a diverging entanglement length.
The authors show that every translation invariant valence bond state on a one-dimensional quantum spin chain arises as the unique ground state of a certain family of finite-range interactions. For each interaction in this family they show the existence of a nonzero spectral gap above the ground state energy. A special example of this structure is a state recently studied by Affleck et al. (1987). For the Hamiltonian studied by these authors they estimate the gap, and prove that it lies between 1/3 and 10/27.