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arXiv:quant-ph/0503133 · Physical Review A

Direct versus measurement-assisted bipartite entanglement in multiqubit systems and their dynamical generation in spin systems

Mátyás Koniorczyk, Peter Rapčan, Vladimír Bužek

Published Aug 17, 2005• 15 citations
arXivHTMLPDF

Abstract

We consider multiqubit systems and relate quantitatively the problems of generating cluster states with high values of concurrence of assistance, and of generating states with maximal bipartite entanglement. We prove an upper bound for the concurrence of assistance. We consider dynamics of spin-$1∕2$ systems that model qubits, with different couplings and possible presence of magnetic field, to investigate the appearance of the discussed entanglement properties. We find that states with maximal bipartite entanglement can be generated by an isotropic $XY$ Hamiltonian, and their generation can be controlled by the initial state of one of the spins. The same Hamiltonian is capable of creating states with high concurrence of assistance from a suitably chosen initial state. We show that the production of graph states using the Ising Hamiltonian is controllable via a single-qubit rotation of one spin-$1∕2$ subsystem in the initial multiqubit state. We show that the property of Ising dynamics to convert a product-state basis into a special maximally entangled basis is temporally enhanced by the application of a suitable magnetic field. Similar basis transformations are found to be feasible in the case of isotropic $XY$ couplings with a magnetic field.

DOI
10.1103/physreva.72.022321

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We consider multiqubit systems and relate quantitatively the problems of generating cluster states with high values of concurrence of assistance, and of generating states with maximal bipartite entanglement. We prove an upper bound for the concurrence of assistance. We consider dynamics of spin-1∕21∕21∕2 systems that model qubits, with different couplings and possible presence of magnetic field, to investigate the appearance of the discussed entanglement properties. We find that states with maximal bipartite entanglement can be generated by an isotropic XYXYXY Hamiltonian, and their generation can be controlled by the initial state of one of the spins. The same Hamiltonian is capable of creating states with high concurrence of assistance from a suitably chosen initial state. We show that the production of graph states using the Ising Hamiltonian is controllable via a single-qubit rotation of one spin-1∕21∕21∕2 subsystem in the initial multiqubit state. We show that the property of Ising dynamics to convert a product-state basis into a special maximally entangled basis is temporally enhanced by the application of a suitable magnetic field. Similar basis transformations are found to be feasible in the case of isotropic XYXYXY couplings with a magnetic field.
Simple physical interactions between spin-1∕21∕21∕2 particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-1∕21∕21∕2 particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-1∕21∕21∕2 particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an NNN-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the XXX-state). This XXX-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset AAA of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of N=2n+1N=2n+1N=2n+1 qubits is maximally entangled with the remaining qubits (set BBB) in the chain. In addition, we prove that by performing a local operation just on the subset BBB, one can transform the XXX-state into a direct product of kkk singlets shared by the parties AAA and BBB. This property of the XXX-state can be utilized for new secure multipartite communication protocols.
In this work, we investigate the bipartite entanglement of decohered mixed states generated from maximally entangled cluster states of [Formula: see text] qubits physical system. We introduce the disconnected cluster states for an ensemble of [Formula: see text] non-interacting qubits and we give the corresponding separable density matrices. The maximally entangled states can be generated from disconnected cluster states, by assuming that the dynamics of the multi-qubit system is governed by a quadratic Hamiltonian of Ising type. When exposed to a local noisy interaction with the environment, the multi-qubit system evolves from its initial pure maximally entangled state to a decohered mixed state. The decohered mixed states generated from bipartite, tripartite and multipartite maximally entangled cluster states are explicitly expressed and their bipartite entanglements are investigated.
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Entanglement generated by Ising model has been studied for several authors in order to understand the relation between it and magnetic properties of materials, principally using one or two dimensional models for two or more particles. In this work, Ising model evolution is solved in three dimensions for two parts including an inhomogeneous magnetic field, giving an detailed study of the entanglement properties derived from interaction. Some relations between entanglement and energy or spin are developed specifically to stablish candidates for entanglement witness. Finally, some basic quantum control operations are prescripted for this model as to preserve the properties of the system as for transfer information between the two parts. These last schemes of control are useful when initial state is completely known, to gain domain on the system. It prevents to introduce more sophisticated control schemes which normally are necessary when initial state is at least partially unknown.
Abstract The classical and quantum dynamics of two particles constrained on S1S^1S1 S1 is discussed via Dirac’s approach. We show that when state is maximally entangled between two subsystems, the product of dispersion in the measurement reduces. We also quantify the upper bound on the external field B⃗\vec {B}B B→ such that B⃗≥B⃗upper\vec {B}\ge \vec {B}_{upper }B≥Bupper​ B→≥B→upper implies no reduction in the product of dispersion pertaining to one subsystem. Further, we report on the cut-off value of the external field B⃗cutoff\vec {B}_{cutoff }Bcutoff​ B→cutoff , above which the bipartite entanglement is lost and there exists a direct relationship between uncertainty of the composite system and the external field. We note that, in this framework it is possible to tune the external field for entanglement/unentanglement of a bipartite system. Finally, we show that the additional terms arising in the quantum Hamiltonian, due to the requirement of Hermiticity of operators, produce a shift in the energy of the system.