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arXiv:quant-ph/0512214 · Physical Review A

Infinite qubit rings with maximal nearest-neighbor entanglement: The Bethe ansatz solution

Uffe V. Poulsen, Tim Meyer, Dagmar Bruß, Maciej Lewenstein

Published May 31, 2006• 6 citations
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Abstract

We search for translationally invariant states of qubits on a ring that maximize the nearest-neighbor entanglement. This problem was initially studied by O'Connor and Wootters [Phys. Rev. A 63, 052302 (2001)]. We first map the problem to the search for the ground state of a spin-1/2 Heisenberg $XXZ$ model. Using the exact Bethe ansatz solution in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, we prove the correctness of the assumption of O'Connor and Wootters that the state of maximal entanglement does not have any pair of neighboring spins ``down'' (or, alternatively, spins ``up''). For sufficiently small fixed magnetization, however, the assumption does not hold: we identify the region of magnetizations for which the states that maximize the nearest-neighbor entanglement necessarily contain pairs of neighboring spins ``down.''

DOI
10.1103/physreva.73.052326

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We search for translationally invariant states of qubits on a ring that maximize the nearest-neighbor entanglement. This problem was initially studied by O'Connor and Wootters [Phys. Rev. A 63, 052302 (2001)]. We first map the problem to the search for the ground state of a spin-1/2 Heisenberg XXZXXZXXZ model. Using the exact Bethe ansatz solution in the limit N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, we prove the correctness of the assumption of O'Connor and Wootters that the state of maximal entanglement does not have any pair of neighboring spins down'' (or, alternatively, spins up''). For sufficiently small fixed magnetization, however, the assumption does not hold: we identify the region of magnetizations for which the states that maximize the nearest-neighbor entanglement necessarily contain pairs of neighboring spins ``down.''
Consider a ring of N qubits in a translationally invariant quantum state. We ask to what extent each pair of nearest neighbors can be entangled. Under certain assumptions about the form of the state, we find a formula for the maximum possible nearest-neighbor entanglement. We then compare this maximum with the entanglement achieved by the ground state of an antiferromagnetic ring consisting of an even number of spin-1/2 particles. We find that, though the antiferromagnetic ground state typically does not maximize the nearest-neighbor entanglement relative to all other states, it does so relative to other states having zero z component of spin.
We study translationally invariant rings of qubits with a finite number of sites N, and determine the maximal nearest-neighbor entanglement for a fixed z-component of the total spin. For small numbers of sites we present analytical results. We establish a relation between the maximal nearest-neighbor concurrence and the ground state energy of an XXZ spin model. This connection allows us to calculate the concurrence numerically for N≤24. We point out some interesting finite-size effects. Finally, we generalize our results beyond nearest neighbors.
We consider translationally invariant states of an infinite one-dimensional chain of qubits or spin-12\frac{1}{2}21​ particles. We maximize the entanglement shared by nearest neighbors via a variational approach based on finitely correlated states. We find an upper bound of nearest-neighbor concurrence equal to C=0.434095\mathcal{C}=0.434\phantom{\rule{0.2em}{0ex}}095C=0.434095 which is 0.090.09%0.09 away from the bound CW=0.434467{\mathcal{C}}_{\mathrm{W}}=0.434\phantom{\rule{0.2em}{0ex}}467CW​=0.434467 obtained by a completely different procedure. The obtained state-maximizing nearest-neighbor entanglement seems to approximate the maximally entangled mixed states. Further we investigate in detail several other properties of the so-obtained optimal state.
We study the entanglement of thermal and ground states in the Heisenberg XX\mathrm{XX}XX qubit rings with a magnetic field. A general result is found that for even-number rings, pairwise entanglement between nearest-neighbor qubits is independent of both the sign of exchange interaction constants and the sign of magnetic fields. As an example we study the entanglement in the four-qubit model and find that the ground state of this model without magnetic fields is shown to be a four-body maximally entangled state measured by the square N-bit concurrence.
The entanglement-sharing properties of an infinite spin-chain are studied when the state of the chain is a pure, translation-invariant state with a matrix-product structure (Klümper A., Schadschneider A. and Zittartz J., J. Phys. A , 24 (1991) L955; Z. Phys. B , 87 (1992) 281; Europhys. Lett. , 24 (1993) 293). We study the entanglement properties of such states by means of their finitely correlated structure (Fannes M., Nachtergaele B. and Werner R. F., Comm. Math. Phys. , 144 (1992) 443; Europhys. Lett. , 10 (1989) 633; J. Phys. A , 24 (1991) L185). These states are recursively constructed by means of an auxiliary density matrix ρ on a matrix algebra ℬ and a completely positive map : ⊗ ℬ → ℬ, where is the spin 2 × 2 matrix algebra. General structural results for the infinite chain are therefore obtained by explicit calculations in (finite) matrix algebras. In particular, we study not only the entanglement shared by nearest-neighbours, but also, differently from previous works (Wootters W. K., Contemp. Math. , 305 (2002) 299) the entanglement shared between connected regions of the spin-chain. This range of possible applications is illustrated and the maximal concurrence = 1/√2 (Coffman V., Kundu J. and Wootters W. K., Phys. Rev. A , 61 (2000) 052306) for the entanglement of connected regions can actually be reached.
Consider an infinite collection of qubits arranged in a line, such that every pair of nearest neighbors is entangled: an "entangled chain." In this paper we consider entangled chains with translational invariance and ask how large one can make the nearest neighbor entanglement. We find that it is possible to achieve an entanglement of formation equal to 0.285 ebits between each pair of nearest neighbors, and that this is the best one can do under certain assumptions about the state of the chain.