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arXiv:quant-ph/0009041 · Physical Review A

Entangled rings

Kevin M O'Connor, William K. Wootters

Published Apr 13, 2001• 366 citations
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Abstract

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.

DOI
10.1103/physreva.63.052302

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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 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 $N\ensuremath{\rightarrow}\ensuremath{\infty}$, 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.''
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.
We investigate the entanglement properties of finite spin rings, with noncollinear Ising interaction between nearest neighbors. The orientations of the Ising axes are determined either by the spin position within the ring (model AAA) or by the direction of the bond (model BBB). In both cases, the considered spin Hamiltonians have a point-group symmetry instead of the translation invariance that characterizes spin rings with collinear Ising interaction. The ground state of these models exhibit remarkable entanglement properties, resembling Greenberger-Horne-Zeilinger (GHZ)-like states in the absence of an applied magnetic field (model BBB). Besides, the application of a homogeneous magnetic field allows to modify qualitatively the character of the ground-state entanglement, switching from multipartite to pairwise quantum correlations (both models AAA and BBB).
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 consider the two-spin-subsystem entanglement for eigenstates of the Hamiltonian $H={\ensuremath{\sum}}_{1\ensuremath{\le}j<k\ensuremath{\le}N}{(\frac{1}{{r}_{j,k}})}^{\ensuremath{\alpha}}{\ensuremath{\sigma}}_{j}\ensuremath{\cdot}{\ensuremath{\sigma}}_{k}$ for a ring of NNN spin-1/2 particles with associated spin vector operator $(\ensuremath{\hbar}/2){\ensuremath{\sigma}}_{j}$ for the jjjth spin. Here rj,k{r}_{j,k}rj,k​ is the chord distance between sites jjj and kkk. The case $\ensuremath{\alpha}=2$ corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for $\ensuremath{\alpha}\ensuremath{\ne}2$. Two-spin-subsystem entanglement shows high sensitivity and distinguishes $\ensuremath{\alpha}=2$ from $\ensuremath{\alpha}\ensuremath{\ne}2$. There is no entanglement beyond nearest neighbors for all eigenstates when $\ensuremath{\alpha}=2$. Whereas for $\ensuremath{\alpha}\ensuremath{\ne}2$ one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of $\ensuremath{\alpha}$ which depends on the energy. The ground state (which is a singlet only for even NNN) does not have entanglement beyond nearest neighbors, and the nearest-neighbor entanglement is virtually independent of the range of the interaction controlled by $\ensuremath{\alpha}$.