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arXiv:quant-ph/0411123 · Physical Review A

Localizable entanglement

M. Popp, Frank Verstraete, Miguel Angel Martin-Delgado, J. I. Cirac

Published Apr 4, 2005• 244 citations
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Abstract

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.

DOI
10.1103/physreva.71.042306

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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∕21∕21∕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 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.
We prove the existence of gapped quantum Hamiltonians whose ground states exhibit an infinite entanglement length, as opposed to their finite correlation length. Using the concept of entanglement swapping, the localizable entanglement is calculated exactly for valence bond and finitely correlated states, and the existence of the so-called string-order parameter is discussed. We also report on evidence that the ground state of an antiferromagnetic chain can be used as a perfect quantum channel if local measurements on the individual spins can be implemented.
This thesis is devoted to the theoretical characterization and generation of strongly correlated many-body states from the perspective of quantum information theory.We first study the entanglement properties of general many-body states and define a new measure for multipartite entanglement, the "Localizable Entanglement (LE)".It quantifies the maximal entanglement that can be localized, on average, between two parties by performing local measurements on the other parties.This measure provides knowledge of the "entanglement length", which is of particular importance in the context of quantum repeaters for long-distance quantum communication.We prove upper and lower bounds on LE in terms of two-body correlation functions and develop a method for the numerical computation of LE in one-dimensional spin systems.We analyze the localizable entanglement in various spin systems and observe characteristic features for a quantum phase transition such as a diverging entanglement length.In the second part of the thesis we focus on a specific many-body system: ultra-cold atoms in an optical lattice.This system is a promising candidate for the first physical realization of a "quantum simulator", but the temperature in current experiments is still too high.We propose, analyze and compare several schemes to cool the atoms close to the ground state of the strong interaction regime.In particular, we devise an algorithmic cooling protocol that combines occupation number filtering with spin-dependent lattice shifts.In addition, we propose two different physical realizations of filtering and also design protocols that generate an ensemble of quantum registers for quantum computation.In the third part we propose how to create entanglement in small atomic clouds by deforming and rotating the harmonic trapping potential.The resulting states are entangled in their motional degrees of freedom and can be identified with strongly correlated fractional quantum Hall states.For the case of two, three and four atoms we show how to adiabatically transform the unentangled ground state into the desired entangled state.We further discuss characteristic features of these states and propose how to create and detect them experimentally using an optical lattice setup. A Proof of the lower bound on LEA.1 Theorem on connected correlation functions and local measurements . . . . . . . . .
Antiferromagnetic spin chains play an important role in condensed matter and statistical mechanics. Recently XXX\mathit{XXX}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.
Antiferromagnetic spin chains play an important role in condensed matter physics and statistical mechanics. Recently XXX spin chain was discussed in relation to the information theory. We consider here localizable entanglement, introduced recently by F.Verstraete, M.Popp and J.I.Cirac. That is how much entanglement can be localized on two spins on average by performing local measurements on the other individual spins in a system of many interacting spins. We consider the ground state in antiferromagnetic spin chains and study localizable entanglement between two spins as a function of the distance. We start with isotropic spin chain. Then we study effect of anisotropy and magnetic field. We conclude that anisotropy increases localizable entanglement. We found an explicit dependence of critical exponents in XXZ spin chain on magnetic field. We discovered that the cases of high symmetry corresponds to high sensitivity of magnetic field. We also calculated the concurrence before the measurement to illustrate that the measurment raises the concurrence.
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.