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arXiv:quant-ph/0206199

Distant entanglement with nearest neighbor interactions

Gavin K. Brennen

Published Jun 28, 2002• 1 citation
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Abstract

Preparing many body entangled states efficiently using available interactions is a challenging task. One solution may be to couple a system collectively with a probe that leaves residual entanglement in the system. We investigate the entanglement produced between two possibly distant qubits 1 and 2 that interact locally with a third qubit 3 under unitary evolution generated by pairwise Hamiltonians. For the case where the Hamiltonians commute, relevant to certain quantum nondemolition measurements, the entanglement between qubits 1 and 2 is calculated explicitly for several classes of initial states and compared with the case of noncommuting interaction Hamiltonians. This analysis can be helpful to identify preferable physical system interactions for entangled state synthesis.

DOI
10.48550/arxiv.quant-ph/0206199

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Preparing many body entangled states efficiently using available interactions is a challenging task. One solution may be to couple a system collectively with a probe that leaves residual entanglement in the system. We investigate the entanglement produced between two possibly distant qubits 1 and 2 that interact locally with a third qubit 3 under unitary evolution generated by pairwise Hamiltonians. For the case where the Hamiltonians commute, relevant to certain quantum nondemolition measurements, the entanglement between qubits 1 and 2 is calculated explicitly for several classes of initial states and compared with the case of noncommuting interaction Hamiltonians. This analysis can be helpful to identify preferable physical system interactions for entangled state synthesis.
The evolution of entanglement in a 3-spin chain with nearest-neighbor Heisenberg-XY interactions for different initial states is investigated here. In an NMR experimental implementation, we generate multipartite entangled states starting from initial separable pseudo-pure states by simulating nearest-neighbor XY interactions in a 3-spin linear chain of nuclear spin qubits. For simulating XY interactions, we follow algebraic method of Zhang et al. [Phys. Rev. A72 (2005) 012331]. Bell state between end qubits has been generated by using only the unitary evolution of the XY Hamiltonian. For generating W-state and GHZ-state a single qubit rotation is applied on second and all the three qubits, respectively after the unitary evolution of the XY Hamiltonian.
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.
We propose a scheme to generate entanglement between two distant qubits (two-level atom) which are separately trapped in their own (in general) non-Markovian dissipative cavities by utilizing entangling swapping, considering the case in which the qubits can move along their cavity axes rather than a static state of motion. We first examine the role of movement of the qubit by studying the entropy evolution for each subsystem. The average entropy over the initial states of the qubit is calculated. Then by performing a Bell state measurement on the fields leaving the cavities, we swap the entanglement between qubit-field in each cavity into qubit-qubit and field-field subsystems. The entangling power is used to measure the average amount of swapped entanglement over all possible pure initial states. Our results are presented in two weak and strong coupling regimes, illustrating the positive role of movement of the qubits on the swapped entanglement. It is revealed that by considering certain conditions for the initial state of qubits, it is possible to achieve a maximally long-leaving stationary entanglement (Bell state) which is entirely independent of the environmental variables as well as the velocity of qubits. This happens when the two qubits have the same velocities.
This study establishes a theorem that provides sufficient criteria for identifying terms in any time-independent Hamiltonian that have no influence on local dynamics, local observables, or any local phenomena, without any approximation. The usefulness of this theorem is demonstrated by predicting the behavior of three systems. The first system consists of multiple qubit chains with Ising interactions. The second system is formed by a single qubit chain, differentiated by the substitution of Dzyalonshinskii-Moriya (DM) interaction for the last Ising interaction. The predictions were verified by analytically deriving the reduced dynamics of both systems. A third system was also considered, namely, a short qubit chain with a transverse magnetic field on the intermediary environment qubit. This transverse magnetic field signals that the commutation relation required by our theorem no longer holds. The third system's local dynamics were also derived analytically, demonstrating that all Hamiltonian constituents contributed to the reduced dynamics. The physical consequences of our theorem's inapplicability were also explored using this third system by analyzing the entanglement dynamics between the qubits that do not directly interact. The results showed that the entanglement is caused by an \textit{emergent coupling} between the two non-directly interacting subsystems. This \textit{emergent coupling} arises from the non-commutativity of the intermediate interactions connecting the two subsystems. When searching for these \textit{emergent couplings}, our theorem can eliminate intermediate interactions that will not produce them.
Entanglement is a defining feature of many-body quantum systems and is an essential requirement for quantum computing. It is therefore useful to study physical processes which generate entanglement within a large system, as they maybe replicated for applications involving the said requirements in quantum information processing. A possible avenue to maximize entanglement generation is to rely on the phenomena of information scrambling, i.e. transport of initially localized information throughout the system. Here the rationale is that the spread of information carries with it an inherent capacity of entanglement generation. Scrambling greatly depends upon the dynamical nature of the system Hamiltonian, and the interplay between entanglement generation and information scrambling maybe investigated taking a chain of interacting spins on a one dimensional lattice. This system is analogous to an array of qubits and this relative simplicity implies that the resulting unitary dynamics can be efficiently simulated using present-day cloud based NISQ devices. In our present work, we consider such a spin model which is made up of nearest and next nearest neighbor XXZ Model, along with an introduced coupling term λ. This coupling term serves as a tuning parameter which modifies the dynamical nature of the system from the integrable to the quantum chaotic regime. In order to quantify the entanglement generated within the system we use the more general multipartite metric which computes the average entanglement across all system bipartitions to obtain a global picture of the entanglement structure within the entire system. Further to make our study initial state independent, we consider an ensemble of random initial states and evaluate the multipartite entanglement generated for each of these states under time evolution. We find clear differences in the entanglement measures as the system undergoes through different regimes. We also analyze the fluctuations present within our entanglement metric and find a distinct pattern as we head from the integrable to the chaotic regime.
We investigate the selective generation of Bell-pair entanglement in three qubit networks using a standard quantum collision model. By varying the network geometry, the node contacted by the ancilla, and the interaction Hamiltonians, we identify configurations in which high pairwise concurrence and Bell state fidelity are generated between different target qubits, including nonadjacent qubits and qubits not directly coupled to the ancilla. For comparison, we also consider the continuous time unitary evolution of the complete ancilla network system, sampled at the same discrete time intervals. This reference calculation clarifies how discarding each outgoing ancilla and removing its correlations from subsequent dynamics modifies the attainable entanglement and preparation time. Our results provide a proof of principle demonstration of geometry dependent Bell pair generation and routing in small qubit networks.