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Physical Review A

Ground-state entanglement in a system with many-body interactions

Xinhua Peng, Jingfu Zhang, Jiangfeng Du, Dieter Suter

Published Apr 27, 2010• 28 citations
Publisher

Abstract

Entanglement refers to the inability of many-body quantum mechanical systems to be separated into independent subsystems. This has been well investigated in systems consisting of two entangled subsystems. In larger systems, more complex types of entanglement exist. In a system consisting of three subsystems (e.g., qubits), it is possible that all three subsystems are entangled with each other in a way that cannot be reduced to bipartite entanglement, and it is known that two different, inequivalent forms of tripartite entanglement exist such as the GHZ and $W$ states (GHZ denotes ``Greenberger-Horne-Zeilinger''). Here, we investigate a particularly interesting system with competing one-, two-, and three-body interactions. Its ground state can be a product state, a GHZ state, or a $W$ state, depending on the type and strength of the spin-spin couplings. By varying an external control parameter, the system can be made to undergo quantum transitions between the various ground-state-entanglement phases. We implement the system in an NMR quantum simulator and use adiabatic evolution of the effective Hamiltonian to drive the system through the quantum transitions. In the experimental and numerical simulations, we check the suitability of different observables for making the quantum transitions visible and for characterizing the different phases.

DOI
10.1103/physreva.81.042327

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Entanglement refers to the inability of many-body quantum mechanical systems to be separated into independent subsystems. This has been well investigated in systems consisting of two entangled subsystems. In larger systems, more complex types of entanglement exist. In a system consisting of three subsystems (e.g., qubits), it is possible that all three subsystems are entangled with each other in a way that cannot be reduced to bipartite entanglement, and it is known that two different, inequivalent forms of tripartite entanglement exist such as the GHZ and WWW states (GHZ denotes ``Greenberger-Horne-Zeilinger''). Here, we investigate a particularly interesting system with competing one-, two-, and three-body interactions. Its ground state can be a product state, a GHZ state, or a WWW state, depending on the type and strength of the spin-spin couplings. By varying an external control parameter, the system can be made to undergo quantum transitions between the various ground-state-entanglement phases. We implement the system in an NMR quantum simulator and use adiabatic evolution of the effective Hamiltonian to drive the system through the quantum transitions. In the experimental and numerical simulations, we check the suitability of different observables for making the quantum transitions visible and for characterizing the different phases.
Quantum phase transitions occur at zero temperature, when the ground state of a Hamiltonian undergoes a qualitative change as a function of a control parameter. We consider a particularly interesting system with competing one-, two-, and three-body interactions. Depending on the relative strength of these interactions, the ground state of the system can be a product state, or it can exhibit genuine tripartite entanglement. We experimentally simulate such a system in a NMR quantum simulator and observe the different ground states. By adiabatically changing the strength of one coupling constant, we push the system from one ground state to a qualitatively different ground state. We show that these ground states can be distinguished and the transitions between them observed by measuring correlations between the spins or the expectation values of suitable entanglement witnesses.
We investigate the entanglement in the ground state of systems comprising two and three qubits with random interactions. Since the Hamiltonians also contain deterministic one-body terms, by varying the interaction strength, one can continuously interpolate between deterministic separable eigenstates and fully random entangled eigenstates, with nontrivial intermediate behavior. Entanglement strongly depends on the underlying topology of the interaction among the qubits. Since W states correspond to a zero-measure set as compared to the set of Greenberger-Horne-Zeilinger (GHZ) states, in all investigated cases the ground states are of the latter type. However, for a certain class of interactions (nonseparable collective potential) high GHZ entanglement is produced, while for fully separable pairwise interactions the marginal GHZ ground states concentrate in the vicinity of W states.
Quantum phase transitions occur when the ground state of a Hamiltonian undergoes qualitative changes with a control parameter changing. In this paper we consider a particular system--an Isng-type spin ring with competing many-body interactions. Depending on the relative strength interactions, the ground state of the system is either a product state or entangled state. We implement the system in a cavity-assisted neutral atomic simulator and study the non-locality and entanglement of the simulated ground state of an Ising-type three-spin ring with the control parameter changing. The simplicity of the setup and its robustness to noise give it a great practicality within the framework of current experimental technology.
We theoretically study the macroscopic quantum entanglement in superconducting flux qubits. A phase-coupling scheme is proposed to offer enough strength of interactions between qubits. It is shown that due to the two-qubit tunneling processes both the ground state and excited states of coupled two flux qubits can be Bell type states, maximally entangled, in experimentally accessible regimes. The parameter regimes for the Bell states are discussed in terms of external magnetic flux and Josephson coupling energies. We also investigate two types of genuine three-qubit entanglement, known as the Greenberger-Horne- Zeilinger(GHZ) and W states. While an excited state can be the W state, the GHZ state is formed at the ground state of the coupled three flux qubits. The GHZ and W states are shown to be robust against external flux fluctuations for feasible experimental realizations.
We study a Rabi type Hamiltonian system in which a qubit and a d d -level quantum system (qudit) are coupled through a common resonator. In the weak and strong coupling limits the spectrum is analysed through suitable perturbative schemes. The analysis show that the presence of the multilevels of the qudit effectively enhance the qubit-qudit interaction. The ground state of the strongly coupled system is found to be of Greenberger-Horne-Zeilinger (GHZ) type. Therefore, despite the qubit-qudit strong coupling, the nature of the specific tripartite entanglement of the GHZ state suppresses the bipartite entanglement. We analyze the system dynamics under quenching and adiabatic switching of the qubit-resonator and qudit-resonator couplings. In the quench case, we found that the non-adiabatic generation of photons in the resonator is enhanced by the number of levels in the qudit. The adiabatic control represents a possible route for preparation of GHZ states. Our analysis provides relevant information for future studies on coherent state transfer in qubit-qudit systems.
The dynamics of entanglement is investigated on the basis of exactly solvable models of multiple quantum (MQ) NMR spin dynamics. It is shown that the time evolution of MQ coherences of systems of coupled nuclear spins in solids is directly connected with dynamics of the quantum entanglement. We studied analytically the dynamics of entangled states for two- and three-spin systems coupled by the dipole-dipole interaction. In this case the dynamics of the quantum entanglement is uniquely determined by the time evolution of MQ coherences of the second order. The real part of the density matrix describing MQ dynamics in solids is responsible for MQ coherences of the zeroth order while its imaginary part is responsible for the second order. Thus, one can conclude that the dynamics of the entanglement is connected with transitions from the real part of the density matrix to the imaginary one, and vice versa. A pure state which generalizes the Greenberger-Horne-Zeilinger (GHZ) and W states is found. Different measures of the entanglement of this state are analyzed for tripartite systems.