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Physica Scripta

Tracking control of two qubit entanglement using piecewise time-independent method

Guilherme Jurkevicz Delben, Marcus W. Beims

Published Dec 4, 2020• 6 citations
Publisher

Abstract

Abstract The tracking control of quantum correlations between two coupled qubits is analyzed, where only one of the qubits is coupled to a Markovian environment. Such a system is a generalized model that can be used, for example, to study the problem of a nuclear spin coupled to the electron spin exposed to the environment. Using an external control field, we increase the time for which the system remains coherent and entangled. The control field is an external potential applied to the system, which contains two adjustable parameters, the intensity and the phase of the field. Furthermore, we present quantum control protocols with different targets. First, for population control of both qubits, second for the coherence control of both two-level systems, and finally for the control of entanglement. Due to analytical for X states the target function can directly be identified with the entanglement. Furthermore, we have shown that when smaller dissipation rates are considered, the control method generates larger stable entanglement between the qubits.

DOI
10.1088/1402-4896/abd0bc

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arXiv:quant-ph/0702147•Journal of Physics B Atomic Molecular and Optical Physics•Apr 18, 2007•100 citations
Abstract The tracking control of quantum correlations between two coupled qubits is analyzed, where only one of the qubits is coupled to a Markovian environment. Such a system is a generalized model that can be used, for example, to study the problem of a nuclear spin coupled to the electron spin exposed to the environment. Using an external control field, we increase the time for which the system remains coherent and entangled. The control field is an external potential applied to the system, which contains two adjustable parameters, the intensity and the phase of the field. Furthermore, we present quantum control protocols with different targets. First, for population control of both qubits, second for the coherence control of both two-level systems, and finally for the control of entanglement. Due to analytical for X states the target function can directly be identified with the entanglement. Furthermore, we have shown that when smaller dissipation rates are considered, the control method generates larger stable entanglement between the qubits.
We investigate the quantum control of two qubits interacting with a Markovian environment and evolving as an XXX state. The control is implemented via a simple applied field, whose profile is determined by means of the piecewise time-independent quantum control method. The goal is to make either the population or the system concurrence to follow a specified tracking control trajectory. By considering the system under the influence of three kinds of noise---phase damping, amplitude damping, or a combination of both---we unveil the conditions (like energy balance) under which the quantum control is properly achieved and fidelity (monitored through the trace distance) is kept satisfactorily high, even if the qubits interact differently with the dissipative medium. We further find that the effects of phase damping, a type of noise which notoriously destroys coherence and entanglement, can be mitigated if during the control the system is also exposed to amplitude damping. Of potential interest for the usage of entanglement as resource for quantum information, our results indicate that distinct entanglement measures and coherence can be maintained relatively high (even at long times) as a consequence of the tracking quantum control process.
An essential action in quantum information processing is the manipulation (control) of a single qubit, ideally a closed two-level system. However, in realistic applications, quantum processes are often under the influence of the external environment, e.g., presenting some degree of dissipation and decoherence. In this paper we address the emerging difficulties in the (tracking) quantum control of a two-level system under the influence of both Markovian and non-Markovian noise. We employ a same framework to treat both situations, a Lindblad-type equation, but considering that for the former (latter) case, the decay rate $\mathrm{\ensuremath{\Gamma}}$ is time independent (dependent). We discuss the conditions leading to a breakdown of the quantum control and eventual ways to overcome the problem, like employing a fast control scheme or controlling the off-diagonal terms of the system density matrix. Surprisingly, for Markovian noise such breakdown time decreases with $\mathrm{\ensuremath{\Gamma}}$ not as an exponential but as a power law. This indicates that the quantum control should be possible for a coupling between the system and the environment stronger than previously expected. Moreover, we find that for non-Markovian noise, the breakdown time is longer when there is backflow, i.e., $\mathrm{\ensuremath{\Gamma}}(t)$ can be negative. The present theoretical results point to certain favorable scenarios to operate qubits even in a noisy medium.
In this paper, we consider two examples of an entanglement in two-qubit systems and an example of entanglement in quantum field theory (QFT). In the beginning, we study the entanglement of two spin states by a magnetic field. A nonzero entanglement appears for interacting spins. When the coupling between the spins is constant, we study the entanglement by several types of time-dependent magnetic fields. In the case of a constant difference between [Formula: see text] components of magnetic fields acting on each spin, we find several time-dependent coupling functions [Formula: see text] that also allow us to analyze analytically and numerically the entanglement measure. Considering two photons moving in an electron medium, we demonstrate that they can be entangled in a controlled way by applying an external magnetic field. The magnetic field affecting electrons of the medium affects photons and, thus, causes an entanglement of the photon beams. The third example is related to the effect of production of electron–positron pairs from the vacuum by a strong external electric field. Here, we have used a general nonperturbative expression for the density operator of the system under consideration. Applying a reduction procedure to this density operator, we construct mixed states of electron and positron subsystems. Calculating the von Neumann entropy of such states, we obtain the loss of information due to the reduction and, at the same time, the entanglement measure of electron and positron subsystems. This entanglement can be considered as an example of an entanglement in QFT.
QUANTUM CONTROL AND GENERATION OF QUANTUM ENTANGLEMENT In this thesis, the generation of entanglement is studied in a controlled environment. The model system of interest includes a cavity field interacting with a pair of atoms. The cavity field is heavily damped and it is pumped in order to maintain a steady state field population. Thus, we can eliminate the cavity field adiabatically and obtain the master equation describing only the qubits evolution in time. At first, this system is analyzed in the steady state, without making any measurement on the photons leaking through the cavity walls. In this way, the ideal physical parameter set for maximum entanglement in this model is investigated. In the second step, we assume a direct measurement on the leaking cavity photons, and observe the evolution of entanglement in a quantum trajectories approach. We simulate quantum trajectories approach by applying Monte Carlo method. The amount of entanglement is obtained as a function of time and number of photon detections.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via the optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.