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arXiv:quant-ph/0311058 · Europhysics Letters (EPL)

Ground-state entanglement in interacting bosonic graphs

P Giorda, Paolo Zanardi A

Published Oct 1, 2004• 13 citations
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Abstract

We consider a collection of bosonic modes corresponding to the vertices of a graph Γ. Quantum tunneling can occur only along the edges of Γ and a local self-interaction term is present. Quantum entanglement of one vertex with respect to the rest of the graph (mode entanglement) is analyzed in the ground state of the system as a function of the tunneling amplitude τ. The topology of Γ plays a major role in determining the tunneling amplitude τ max that leads to the maximum value of the mode entanglement. Whereas in most of the cases one finds the intuitively expected result τ max = ∞ , we show that there exists a family of graphs for which the optimal value of τ is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the crossover between insulating and superfluid ground states.

DOI
10.1209/epl/i2004-10129-2

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We consider a collection of bosonic modes corresponding to the vertices of a graph Γ. Quantum tunneling can occur only along the edges of Γ and a local self-interaction term is present. Quantum entanglement of one vertex with respect to the rest of the graph (mode entanglement) is analyzed in the ground state of the system as a function of the tunneling amplitude τ. The topology of Γ plays a major role in determining the tunneling amplitude τ max that leads to the maximum value of the mode entanglement. Whereas in most of the cases one finds the intuitively expected result τ max = ∞ , we show that there exists a family of graphs for which the optimal value of τ is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the crossover between insulating and superfluid ground states.
We consider a collection of bosonic modes corresponding to the vertices of a graph Gamma.\\Gamma.Gamma. Quantum tunneling can occur only along the edges of Gamma\\GammaGamma and a local self-interaction term is present. Quantum entanglement of one vertex with respect the rest of the graph is analyzed in the ground-state of the system as a function of the tunneling amplitude tau.\\tau.tau. The topology of Gamma\\GammaGamma plays a major role in determining the tunneling amplitude tau∗\\tau^*tau∗ which leads to the maximum ground-state entanglement. Whereas in most of the cases one finds the intuitively expected result tau∗=infty\\tau^*=\\inftytau∗=infty we show that it there exists a family of graphs for which the optimal value oftau\\tautau is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the cross-over between insulating and superfluid ground states
We analyze the quantum entanglement properties of bosonic particles hopping over graph structures. Mode entanglement of a graph vertex with respect to the rest of the graph is generated, starting from a product state, by turning on for a finite time a tunneling along the graph edges. The maximum achieved during the dynamical evolution by this bipartite entanglement characterizes the entangling power of a given hopping Hamiltonian. We studied this entangling power as a function of the self-interaction parameters, i.e., nonlinearities, for all the graphs up to four vertices and for two different natural choices of the initial state. The role of graph topology and self-interaction strengths in optimizing entanglement generation is extensively studied by means of exact numerical simulations and by perturbative calculations
We study a set of L spatial bosonic modes localized on a graph $\ensuremath{\Gamma}.$ The particles are allowed to tunnel from vertex to vertex by hopping along the edges of $\ensuremath{\Gamma}.$ We analyze how, in the exact many-body eigenstates of the system, i.e., Bose-Einstein condensates over single-particle eigenfunctions, the bipartite quantum entanglement of a graph vertex with respect to the rest of the graph depends on the topology of $\ensuremath{\Gamma}.$
The ground and thermal states of a quadratic Hamiltonian representing the interaction of bosonic modes or particles are always Gaussian states. We investigate the entanglement properties of these states for the case where the interactions are represented by harmonic forces acting along the edges of symmetric graphs---i.e., one-, two-, and three-dimensional rectangular lattices, mean-field clusters, and platonic solids. We determine the entanglement of formation (EOF) as a function of the interaction strength, calculate the maximum EOF in each case, and compare these values with the bounds found previously for quadratic Hamiltonians.
We study how bi-partite quantum entanglement (measured in terms of a concurrence) can be shared in multi-qubit systems. We introduce a concept of the entangled graph such that each qubit of a multi-partite system is associated with a vertex while a bi-partite entanglement between two specific qubits is represented by an edge. We prove that any entangled graph can be associated with a pure state of a multi-qubit system. We also derive bounds on the concurrence for some weighted entangled graphs (the weight corresponds to the value of concurrence associated with the given edge).
Bi-partite entanglement in multi-qubit systems cannot be shared freely. The rules of quantum mechanics impose bounds on how multi-qubit systems can be correlated. In this paper we utilize a concept of entangled graphs with weighted edges in order to analyze pure quantum states of multi-qubit systems. Here qubits are represented by vertexes of the graph while the presence of bi-partite entanglement is represented by an edge between corresponding vertexes. The weight of each edge is defined to be the entanglement between the two qubits connected by the edge, as measured by the concurrence. We prove that each entangled graph with entanglement bounded by a specific value of the concurrence can be represented by a pure multi-qubit state. In addition we present a logic network with O(N2) elementary gates that can be used for preparation of the weighted entangled graphs of N qubits.