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Journal of Physics B Atomic Molecular and Optical Physics

Weighted graph states and applications to spin chains, lattices and gases

Ludwig Hartmann, John Calsamiglia, Wolfgang Dür, Hans J. Briegel

Published Apr 18, 2007• 38 citations
Publisher

Abstract

Weighted graph states naturally arise when spin systems interact via an Ising-type interaction. First, we abstractly define the class of weighted graph states and demonstrate its computational accessibility. We show how reduced density matrices of a small number of spins (≈10) can be computed from arbitrarily large systems using weighted graph techniques and projected entangled pair techniques, and we discuss various entanglement measures accessible from these reduced density matrices. Second, we apply these findings to spin chains and lattices with long-range interactions and analytically derive area laws for the scaling of block-wise entanglement. Then, we turn to disordered spin systems, spin gases, which are connected to random weighted graph states and which share their entanglement properties. Finally, we use a spin gas as a bath that introduces decoherence in single as well as multipartite spin systems. The microscopic, exact decoherence model we obtain can operate in different regimes and exhibit non-Markovian features as well as spatially correlated noise effects.

DOI
10.1088/0953-4075/40/9/s01

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Weighted graph states naturally arise when spin systems interact via an Ising-type interaction. First, we abstractly define the class of weighted graph states and demonstrate its computational accessibility. We show how reduced density matrices of a small number of spins (≈10) can be computed from arbitrarily large systems using weighted graph techniques and projected entangled pair techniques, and we discuss various entanglement measures accessible from these reduced density matrices. Second, we apply these findings to spin chains and lattices with long-range interactions and analytically derive area laws for the scaling of block-wise entanglement. Then, we turn to disordered spin systems, spin gases, which are connected to random weighted graph states and which share their entanglement properties. Finally, we use a spin gas as a bath that introduces decoherence in single as well as multipartite spin systems. The microscopic, exact decoherence model we obtain can operate in different regimes and exhibit non-Markovian features as well as spatially correlated noise effects.
The cluster state acquired by evolving the nearest-neighbor (NN) Ising model from a completely separable state is the resource for measurement-based quantum computation. Instead of an NN system, a variable-range power-law interacting Ising model can generate a weighted graph state (WGS) whose genuine multipartite entanglement may reveal intrinsic characteristics of the evolving Hamiltonian. We establish that the generalized geometric measure (GGM) in the time evolved state with an arbitrary number of qubits is sensitive to falloff rates and the range of interactions of the evolving Hamiltonian. We report that the time-derivative and time-averaged GGM at a particular time can detect the transition points present in the falloff rates of the interaction strength, separating different regions, namely, long-range, quasilocal, and local ones in one- and two-dimensional lattices with deformation. Moreover, we illustrate that in the quasilocal and local regimes there exists a minimum coordination number in the evolving Ising model for a fixed total number of qubits which can mimic the GGM of the long-range model. In order to achieve a finite-size subsystem from the entire system, we design a local measurement strategy that allows a WGS of an arbitrary number of qubits to be reduced to a local unitarily equivalent WGS having fewer qubits with modified weights.
We investigate entanglement properties of N initially disentangled spins, embedded in a ring or d-dimensional lattice. The spins interact via some long-range Ising-type interaction. A description in terms of generalized valence bond solids allows us, for arbitrarily large N, to efficiently calculate reduced density operators of up to ten particles. From these density operators we can compute many quantities, for example the bipartite entanglement S_L between a block of L neighboring spins and the remaining system, lower and upper bounds on the localizable entanglement, and higher order correlation functions. We vary the distance dependence of the interaction and investigate how this variation affects the scaling of the entropy S_L as L grows. We provide a sufficient condition when S_L saturates. In other cases, we find that the entanglement length diverges. For special configurations, we determine S_L analytically for all L in the limit N to infinity.
We study the dynamics of entanglement in spin gases. A spin gas consists of a (large) number of interacting particles whose random motion is described classically while their internal degrees of freedom are described quantum-mechanically. We determine the entanglement that occurs naturally in such systems for specific types of quantum interactions. At the same time, these systems provide microscopic models for non–Markovian decoherence: the interaction of a group of particles with other particles belonging to a background gas are treated exactly, and differences between collective and non–collective decoherence processes are studied. We give quantitative results for the Boltzmann gas and also for a lattice gas, which could be realized by neutral atoms hopping in an optical lattice. These models can be simulated efficiently for systems of mesoscopic sizes (N ~ 105).
We investigate the entanglement dynamics of quantum states defined on graphs with non-local Ising interactions governed by the Katz kernel of the underlying network. The interaction pattern is physically motivated by a gapped fermionic mediator propagating on the same graph, whose perturbative elimination yields an effective Katz-weighted Ising Hamiltonian. Using the Entanglement Distance, we derive an exact analytical expression for the entanglement generated from an initially separable state and apply it to representative deterministic graph families. We then characterize the dynamics in different propagation regimes. In the weak-Katz regime, the dynamics admits a systematic motif expansion with triangles entering at first order order and four-cycles, local degree structure, and overlappin triangles appearing at second order. In the strong-propagation regime, the interaction is instead dominated by the principal adjacency mode and by the localization properties of its eigenvector. For Erdős--Rényi graphs, the weak-propagation expansion can be averaged analytically, revealing a locally tree-like contribution in the sparse regime and saturation of the Entanglement Distance density in the dense regime. Our results connect entanglement dynamics with both the walk-based and spectral structure of complex networks.
We analyze a microscopic decoherence model in which the total system is described as a spin gas. A spin gas consists of NNN classically moving particles with additional, interacting quantum degrees of freedom (e.g., spins). For various multipartite entangled probe states, we analyze the decoherence induced by interactions between the probe and environmental spins in such spin gases. We can treat mesoscopic environments ($\ensuremath{\approx}{10}^{5}$ particles). We present results for a lattice gas, which could be realized by neutral atoms hopping in an optical lattice, and show the effects of non-Markovian and correlated noise, as well as finite-size effects.
We consider N initially disentangled spins, embedded in a ring or d-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit N-->infinity. We provide a sufficient condition when bipartite entanglement between blocks of L neighboring spins and the remaining system saturates and determine S(L) analytically for special configurations. We find an unbounded increase of S(L) as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large N, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.