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arXiv:quant-ph/0001114

Entangled Chains

William K. Wootters

Published Jan 31, 2000• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0001114

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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 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.
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.''
Investigating translationally invariant qudit spin chains with a low local dimension, we ask what is the best possible tradeoff between the scaling of the entanglement entropy of a large block and the inverse-polynomial scaling of the spectral gap. Restricting ourselves to Hamiltonians with a "rewriting" interaction, we find the pair-flip model, a family of spin chains with nearest neighbor, translationally invariant, frustration-free interactions, with a very entangled ground state and an inverse-polynomial spectral gap. For a ground state in a particular invariant subspace, the entanglement entropy across a middle cut scales as log⁡n\log nlogn for qubits (it is equivalent to the XXX model), while for qutrits and higher, it scales as n\sqrt{n}n​. Moreover, we conjecture that this particular ground state can be made unique by adding a small translationally-invariant perturbation that favors neighboring letter pairs, adding a small amount of frustration, while retaining the entropy scaling.
We consider the problem of detecting entanglement and nonlocality in one-dimensional (1D) infinite, translation-invariant (TI) systems when just near-neighbor information is available. This issue is deeper than one might think a priori, since, as we show, there exist instances of local separable states (classical boxes) which admit only entangled (nonclassical) TI extensions. We provide a simple characterization of the set of local states of multiseparable TI spin chains and construct a family of linear witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Similarly, we prove that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Using an algorithm based on matrix product states, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state. All our results can be easily adapted to detect entanglement and nonlocality in large (finite, not TI) 1D condensed matter systems.