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arXiv:1703.03000 · Physical Review A

Preparing Greenberger-Horne-Zeilinger and W states on a long-range Ising spin model by global controls

Jiahui Chen, Hui Zhou, Chang‐Kui Duan, Xinhua Peng

Published Mar 31, 2017• 27 citations
arXivHTMLPDF

Abstract

Entanglement, a unique quantum resource with no classical counterpart, remains at the heart of quantum information. The Greenberger-Horne-Zeilinger (GHZ) and $W$ states are two inequivalent classes of multipartite entangled states which cannot be transformed into each other by means of local operations and classic communication. In this paper, we present the methods to prepare the GHZ and $W$ states via global controls on a long-range Ising spin model. For the GHZ state, general solutions are analytically obtained for an arbitrary-size spin system, while for the $W$ state, we find a standard way to prepare the $W$ state that is analytically illustrated in three- and four-spin systems and numerically demonstrated for larger-size systems. The number of parameters required in the numerical search increases only linearly with the size of the system.

DOI
10.1103/physreva.95.032340

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Entanglement, a unique quantum resource with no classical counterpart, remains at the heart of quantum information. The Greenberger-Horne-Zeilinger (GHZ) and WWW states are two inequivalent classes of multipartite entangled states which cannot be transformed into each other by means of local operations and classic communication. In this paper, we present the methods to prepare the GHZ and WWW states via global controls on a long-range Ising spin model. For the GHZ state, general solutions are analytically obtained for an arbitrary-size spin system, while for the WWW state, we find a standard way to prepare the WWW state that is analytically illustrated in three- and four-spin systems and numerically demonstrated for larger-size systems. The number of parameters required in the numerical search increases only linearly with the size of the system.
We study how to prepare two important types of entangled states (i.e., GHZ and WWW states) via global controls in closed linear Ising spin chains with nearest-neighbor couplings. To achieve the goal, we use a numerical optimization algorithm to search an optimal set of control parameters, and then analytically obtain the time-optimal solutions for a three-spin chain by a geometrical explanation on the Bloch sphere. For experimental demonstration, we also consider the most robust sequence under the realistic imperfections such as the errors of global controls and the time evolutions as well as in the numerical optimization algorithm, and experimentally implement the numerically derived robust pulse sequences in a three-spin system where the closed linear Ising spin chain is simulated by nuclear-magnetic-resonance technique. This experiment demonstrates that our scheme is feasible in realistic settings under the experimental imperfections.
Greenberger-Horne-Zeilinger (GHZ) state is a typical maximally entangled state which is pursued in both fundamental research and emerging quantum technologies. Preparing large-scale GHZ states in lattice spin systems is particularly appealing for quantum advantages, but conventional schemes face great challenges in scalability. Here we propose a universal and scalable scheme to generate large-scale GHZ-like states, which share similar entanglement and metrological properties with standard GHZ states, in lattice spin systems through global Floquet engineering. Our scheme requires only global operations and shows great advantage for large particle number. It is applicable to systems with arbitrary interaction ranges, offering a practical pathway for large-scale implementation of many-body entangled states in various systems.
We present two schemes for realizing the remote preparation of a Greenberger–Horne–Zeilinger (GHZ) state. The first scheme is to remotely prepare a general N -particle GHZ state with two steps. One is to prepare a qubit state by using finite classical bits from sender to receiver via a two-particle entangled state, and the other is that the receiver introduces N — 1 additional particles and performs N — 1 controlled-not (C-Not) operations. The second scheme is to remotely prepare an N -atom GHZ state via a two-atom entangled state in cavity quantum electrodynamics (QED). The two schemes require only a two-particle entangled state used as a quantum channel, so we reduce the requirement for entanglement.
We discuss how to prepare an Ising chain in a GHZ state using a single global control field only. This model does not require the spins to be individually addressable and is applicable to quantum systems such as cold atoms in optical lattices, some liquid- or solid-state NMR experiments, and many nanoscale quantum structures. We show that GHZ states can always be reached asymptotically from certain easy-to-prepare initial states using adiabatic passage, and under certain conditions finite-time reachability can be ensured. To provide a reference useful for future experimental implementations, three different control strategies to achieve the objective---adiabatic passage, Lyapunov control, and optimal control---are compared, and their advantages and disadvantages discussed, in particular in the presence of realistic imperfections such as imperfect initial state preparation, system inhomogeneity, and dephasing.
In this paper, we study the entanglement classification of the two-dimensional subspace spanned by nnn-qubit Greenberger-Horne-Zeilinger (GHZ) and WWW states under stochastic local operations and classical communication (SLOCC). We show that for $n\ensuremath{\ge}4$ such a system can have infinite SLOCC inequivalent classes. Specifically, two states, ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+x{|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+y{|W\ensuremath{\rangle}}_{n}$, from the subspace spanned by nnn-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if (xy)n=1{(\frac{x}{y})}^{n}=1(yx​)n=1.
A proof of Bell's theorem without inequalities valid for both inequivalent classes of three-qubit entangled states under local operations assisted by classical communication, namely Greenberger-Horne-Zeilinger (GHZ) and W,W,W, is described. This proof leads to a Bell inequality that allows more conclusive tests of Bell's theorem for three-qubit systems. Another Bell inequality involving both tri- and bipartite correlations is introduced which illustrates the different violations of local realism exhibited by the GHZ and WWW states.