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arXiv:1007.4136 · Physical Review B

Even-odd effects of Heisenberg chains on long-range interaction and entanglement

Sangchul Oh, Mark G. Friesen, Xuedong Hu

Published Oct 8, 2010• 28 citations
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Abstract

A strongly coupled Heisenberg chain provides an important channel for quantum communication through its many-body ground state. Yet, the nature of the effective interactions and the ability to mediate long-range entanglement differs significantly for chains of opposite parity. Here, we contrast the characters of even- and odd-size chains when they are coupled to external qubits. Additional parity effects emerge in both cases, depending on the positions of the attached qubits. Some striking results include (i) the emergence of maximal entanglement and (ii) spin-chain Ruderman-Kittel-Kasuya-Yosida interactions for qubits attached to an even chain, and (iii) the ability of chains of either parity to mediate qubit entanglement that is undiminished by distance.

DOI
10.1103/physrevb.82.140403

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A strongly coupled Heisenberg chain provides an important channel for quantum communication through its many-body ground state. Yet, the nature of the effective interactions and the ability to mediate long-range entanglement differs significantly for chains of opposite parity. Here, we contrast the characters of even- and odd-size chains when they are coupled to external qubits. Additional parity effects emerge in both cases, depending on the positions of the attached qubits. Some striking results include (i) the emergence of maximal entanglement and (ii) spin-chain Ruderman-Kittel-Kasuya-Yosida interactions for qubits attached to an even chain, and (iii) the ability of chains of either parity to mediate qubit entanglement that is undiminished by distance.
A strongly coupled spin chain can mediate long-distance effective couplings or entanglement between remote qubits and can be used as a quantum data bus. We study how the fidelity of a spin-1/2 Heisenberg chain as a spin bus is affected by static random exchange couplings and magnetic fields. We find that, while nonuniform exchange couplings preserve the isotropy of the qubit effective couplings, they cause the energy levels, the eigenstates, and the magnitude of the couplings to vary locally. On the other hand, random local magnetic fields lead to an avoided level crossing for the bus ground-state manifold and cause the effective qubit couplings to be anisotropic. Interestingly, the total magnetic moment of the ground state of an odd-size bus may not be parallel to the average magnetic field. Its alignment depends on both the direction of the average field and the field distribution, in contrast with the ground state of a single spin which always aligns with the applied magnetic field to minimize the Zeeman energy. Lastly, we calculate sensitivities of the spin bus to such local variations, which are potentially useful for evaluating decoherence when dynamical fluctuations in the exchange coupling or magnetic field are considered.
Recent numerical results showed that spin chains are able to produce long-distance entanglement (LDE). We develop a formalism that allows the computation of LDE for weakly interacting probes with gapped many-body systems. At zero temperature, a dc response function determines the ability of the physical system to generate genuine quantum correlations between the probes. We show that the biquadratic Heisenberg spin-1 chain is able to produce LDE in the thermodynamic limit and that the finite antiferromagnetic Heisenberg chain maximally entangles two spin-1/21/21/2 probes very far apart. These results support the current perspective of using quantum spin chains as entanglers or quantum channels in quantum information devices.
We examine the entanglement entropy of the even half of a translationally invariant finite chain or lattice in its ground state. This entropy measures the entanglement between the even and odd halves (each forming a ``comb'' of n/2n/2n/2 sites) and can be expected to be extensive for short-range couplings away from criticality. We first consider bosonic systems with quadratic couplings, where analytic expressions for arbitrary dimensions can be provided. The bosonic treatment is then applied to finite spin chains and arrays by means of the random-phase approximation. Results for first-neighbor anisotropic XY\mathit{XY}XY couplings indicate that, while at strong magnetic fields this entropy is strictly extensive, at weak fields important deviations arise, stemming from parity-breaking effects and the presence of a factorizing field (in the vicinity of which it becomes size-independent and identical to the entropy of a contiguous half). Exact numerical results for small spin sss chains are shown to be in agreement with the bosonic random-phase approximation prediction.
The effective Heisenberg interaction is generated by weakly interacting qubits with two strongly coupled chains. The system of two coupled chains can be regarded as a quantum bus. Universal quantum computation is enabled by restricting the bus to its ground state. The effective interaction is primarily determined by the energy spectrum of the bus. The interactions alternate between antiferromagnetic and ferromagnetic ones. The long-range interaction can also be attained in coupled chains with infinite size. The quantum gate operations with high precision can be realized when the quantum fluctuations are taken into account.
We conjecture that nonequilibrium boundary conditions generically trigger long-range order in nonequilibrium steady states of locally interacting quantum chains. Our result is based on large scale density matrix renormalization group simulations of several models of quantum spin-1/2 chains which are driven far from equilibrium by coupling to a pair of unequal Lindblad reservoirs attached locally to the ends of the chain. In particular, we find a phase transition from exponentially decaying to long-range spin-spin correlations in an integrable Heisenberg XXZ chain by changing the anisotropy parameter. Long-range order also typically emerges after breaking the integrability of the model.