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Revista Mexicana de Física

Entanglement and control operations in Heisenberg 3-D interactions of two qubits

Francisco Delgado

Published May 19, 2010• 2 citations
OpenAlex

Abstract

Entanglement generated by the Heisenberg model has been studied by several authors in order to understand its relation to the magnetic properties of materials, using mainly particular cases in one or two dimensions for two or more particles. In this work, the evolution of the Heisenberg model is solved for two particles including an inhomogeneous magnetic field in three dimensions, giving a detailed study of the entanglement properties derived from this interaction. Some relations between entanglement and energy or spin are verified, based on known relations for these observables. Finally, some possible quantum control operations are suggested to drive bipartite qubits with an external magnetic field, controlling their evolution into a periodical behavior. These operations become useful to preserve the system properties as well as to transfer information between two parts which can be exploited in engineering applications (e.g. quantum computation or quantum information).

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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.

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Physical Review A•Nov 8, 2006•132 citations
Entanglement generated by the Heisenberg model has been studied by several authors in order to understand its relation to the magnetic properties of materials, using mainly particular cases in one or two dimensions for two or more particles. In this work, the evolution of the Heisenberg model is solved for two particles including an inhomogeneous magnetic field in three dimensions, giving a detailed study of the entanglement properties derived from this interaction. Some relations between entanglement and energy or spin are verified, based on known relations for these observables. Finally, some possible quantum control operations are suggested to drive bipartite qubits with an external magnetic field, controlling their evolution into a periodical behavior. These operations become useful to preserve the system properties as well as to transfer information between two parts which can be exploited in engineering applications (e.g. quantum computation or quantum information).
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.
Control of quantum entanglement has been considered as elemental physical resource for quantum applications in Quantum Information and Quantum Computation. Control of entangled states on a couple of atoms, ions or quantum dots are milestones in almost all quantum applications towards a scalable spin-based quantum computers or quantum devices. For magnetic systems, Ising model is an interaction which generates and modifies entanglement properties of quantum systems based on matter. In addition, when this interaction includes driven magnetic fields, it can be controlled to sustain, characterize or modify entanglement and other quantum properties. In this work, recent results about evolution in a general anisotropic three dimensional Ising model including an inhomogeneous magnetic field is considered to obtain some general quantum control effects for their sustainability, programmed evolution or transformation: Evolution loops and Exchange operations. This control is achievable through a set of physical parameters, whose prescriptions are reported. The use of a non local basis in the model to express time evolution lets take advantage to describe and control the system, in particular with those issues associated with entanglement and operations mentioned before. Finally, some analysis about equivalent gates based on our development is made including an example with teleportation, using one of the gates constructed.
Quantum correlations and entanglement are fundamental resources for quantum information and quantum communication processes. Developments in these fields normally assume stable resources, not susceptible of distortion. That is not always the case, Heisenberg interactions between qubits can produce distortion on entangled pairs generated for engineering purposes (e. g. quantum computation or quantum cryptography). The presence of parasite magnetic fields modifies the expected properties and behavior for which the pair was intended. Quantum measurement and control help to discriminate the original state in order to correct it or reconstruct it using some procedures which do not alter their quantum nature. Different kinds of quantum entangled pairs driven by a Heisenberg Hamiltonian with an additional inhomogeneous magnetic field become distorted. They can be reconstructed by adding an external magnetic field with fidelity close to one. In addition, each state can be efficiently discriminated. Combining both processes, first reconstruction without discrimination and after discrimination with adequate non-local measurements, it is possible to (a) improve the discrimination, and (b) reprepare faithfully the original state. The complete process gives fidelities better than 0.9. Some results about a class of equivalence for the required measurements are found, allowing to select the experimentally most adequate.
The effect of the control on bipartite entanglement is discussed from a geometric viewpoint for a nuclear magnetic resonance (NMR) system as a model of the n-qubit control system. The Hamiltonian of the model is the sum of the drift and control Hamiltonians, each of which describes the interaction between pairs of qubits and between one of qubits and an external magnetic field, respectively. According to a bipartite partition, the Schroedinger equation for the NMR system is put in the matrix form. This paper gives a solution to the Schroedinger equation with the assumptions of small coupling among qubits and of constant controls. The solution is put in the form of power series in small parameters. In particular, in the case of the two-qubit NMR system, the drift and control Hamiltonians are shown to be coupled to work for entanglement promotion, by examining solutions to the Schroedinger equation in detail. The concurrence, a measure of entanglement, is evaluated along the solution for a small time interval in order to observe that the control effect appears, not at the first-order terms in t, but at the higher-order terms in t. The evaluated concurrence also suggests which control makes the two-qubit more entangled or less.
Entanglement is considered a basic physical resource for modern quantum applications as Quantum Information and Quantum Computation. Interactions based on specific physical systems able to generate and sustain entanglement are subject to deep research to get understanding and control on it. Atoms, ions or quantum dots are considered key pieces in quantum applications because they are elements in the development toward a scalable spin-based quantum computer through universal and basic quantum operations. Ising model is a type of interaction generating entanglement in quantum systems based on matter. In this work, a general bipartite anisotropic Ising model including an inhomogeneous magnetic field is analyzed in a non-local basis. This model summarizes several particular models presented in literature. When evolution is expressed in the Bell basis, it shows a regular block structure suggesting a [Formula: see text] decomposition. Then, their algebraic properties are analyzed in terms of a set of physical parameters which define their group structure. In particular, finite products of pulses in this interaction are analyzed in terms of [Formula: see text] covering. Thus, evolution denotes remarkable properties, in particular those related potentially with entanglement and control, which give a fruitful arena for further quantum developments and generalization.
We study the bipartite entanglement dynamics in an anisotropic two-qubit Heisenberg XYZXYZXYZ model under the influence of population relaxation and in the presence of various types of magnetic fields. We find that the maximal value of the concurrence has an intrinsic rigidity for the different external magnetic fields. Moreover, our results demonstrate that it is possible to produce entangled states and to control or to modulate the concurrence within the intrinsic maximal value with the help of external time-varying fields despite the existence of dissipation.