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arXiv:1211.3877 · Physical Review Letters

Characterizing Genuine Multisite Entanglement in Isotropic Spin Lattices

Himadri Shekhar Dhar, Aditi Sen, Ujjwal Sen

Published Aug 14, 2013• 21 citations
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Abstract

We consider a class of large superposed states, obtained from dimer coverings on spin-1/2 isotropic lattices, whose potential usefulness ranges from organic molecules to quantum computation. We show that they are genuinely multiparty entangled, irrespective of the geometry and dimension of the isotropic lattice. We then present an efficient method to characterize the genuine multisite entanglement in the case of isotropic square spin-1/2 lattices, with short-range dimer coverings. We use this iterative analytical method to calculate the multisite entanglement of finite-sized lattices, which through finite-size scaling, enables us to obtain the estimate of the multisite entanglement of the infinite square lattice. The method can be a useful tool to investigate other single- and multisite properties of such states.

DOI
10.1103/physrevlett.111.070501

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We consider a class of large superposed states, obtained from dimer coverings on spin-1/2 isotropic lattices, whose potential usefulness ranges from organic molecules to quantum computation. We show that they are genuinely multiparty entangled, irrespective of the geometry and dimension of the isotropic lattice. We then present an efficient method to characterize the genuine multisite entanglement in the case of isotropic square spin-1/2 lattices, with short-range dimer coverings. We use this iterative analytical method to calculate the multisite entanglement of finite-sized lattices, which through finite-size scaling, enables us to obtain the estimate of the multisite entanglement of the infinite square lattice. The method can be a useful tool to investigate other single- and multisite properties of such states.
We introduce an analytical iterative method, the density matrix recursion method, to generate arbitrary reduced density matrices of superpositions of short-range dimer coverings on periodic or non-periodic quantum spin-1/2 ladder lattices, with an arbitrary number of legs. The method can be used to calculate bipartite as well as multipartite physical properties, including bipartite and multi-partite entanglement. We apply this technique to distinguish between even- and odd-legged ladders. Specifically, we show that while genuine multi-partite entanglement decreases with increasing system size for the even-legged ladder states, it does the opposite for odd-legged ones.
We devise a method based on the tensor-network formalism to calculate genuine multisite entanglement in ground states of infinite spin chains containing spin-1/2 or spin-1 quantum particles. The ground state is obtained by employing an infinite time-evolving block decimation method acting upon an initial matrix product state for the infinite spin system. We explicitly show how such infinite matrix product states with translational invariance provide a natural framework to derive the generalized geometric measure, a computable measure of genuine multisite entanglement, in the thermodynamic limit of quantum many-body systems with both spin-1/2 and higher-spin particles.
We employ a genuine multipartite entanglement measure, the generalized geometric measure, for investigating the quantum phase transition in an infinite quantum spin-1/2 chain with two-spin as well as three-spin interactions. We show that in contrast to bipartite entanglement of the ground state, the genuine multiparty one unerringly indicates the quantum phase transition in the system. The system lends itself to a complementary behavior between bipartite and genuine multipartite entanglements. We also investigate the trend of the multipartite entanglement in finite chains, in which we perform a finite-size scaling to confirm the signature of the quantum phase transition already in finite systems. The finite-size investigation renders the phenomenon to be observable in real systems.
We formulate an analytical recursive method to generate the wave function of doped short-range resonating valence bond (RVB) states as a tool to efficiently estimate multisite entanglement as well as other physical quantities in doped quantum spin ladders. We prove that doped RVB ladder states are always genuine multipartite entangled. Importantly, our results show that within specific doping concentration and model parameter regimes, the doped RVB state essentially characterizes the trends of genuine multiparty entanglement in the exact ground states of the Hubbard model with large on-site interactions, in the limit that yields the $t\text{\ensuremath{-}}J$ Hamiltonian.
Quantum networks are an integral component in performing efficient computation and communication tasks that are not accessible using classical systems. A key aspect in designing an effective and scalable quantum network is generating entanglement between its nodes, which is robust against defects in the network. We consider an isotropic quantum network of spin-1/2 particles with a finite fraction of defects, where the corresponding wave function of the network is rotationally invariant under the action of local unitaries. By using quantum information-theoretic concepts like strong subadditivity of von Neumann entropy and approximate quantum telecloning, we prove analytically that in the presence of defects, caused by loss of a finite fraction of spins, the network, composed of a fixed numbers of lattice sites, sustains genuine multisite entanglement and at the same time may exhibit finite moderate-range bipartite entanglement, in contrast to the network with no defects.
We define a multiparty entanglement measure, called generalized geometric measure, that can detect and quantify genuine multiparty entanglement for any number of parties. The quantum phase transitions in exactly solvable models like the anisotropic XY model can be detected by this measure. We find that the multisite measure can be a useful tool to detect quantum phenomena in more complex systems like quasi 2D and 2D frustrated Heisenberg antiferromagnets. We propose an order parameter, called bound generalized geometric measure, in the spirit of bound quantum states, that can recognize the gapless and gapped phases of the frustrated models by its sign. The vanishing of the order parameter therefore signals the transition between such phases.