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arXiv:2203.00182

Generation of Maximally Entangled States by Lyapunov Control Based on Entanglement Measure

Yun-Yan Lee, Dong, Daoyi, Yang, Ciann-Dong

Published Mar 1, 2022• 0 citations
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Abstract

Maximally entangled states (MES) are highly valued in quantum information processing. In quantum control, the creation of MES is typically treated as a state transfer problem with a predefined MES as the target. However, this approach is limited by the requirement to predetermine the MES structure. This paper introduces an improved quantum Lyapunov control approach that relies on the quantum entanglement measure to construct the Lyapunov function, instead of using the distance between quantum states. This strategy enables the preparation of any MES, regardless of whether its structure is known beforehand, using a single control scheme. The proposed entanglement control technique is unaffected by the number of entangled subsystems since it targets the entanglement measure as a scalar. Initially applied to bipartite pure states, this method demonstrates its capability to generate Bell states and their equivalents. Subsequent applications to bipartite mixed states and multipartite systems illustrate that the technique can produce MES with unspecified structures.

DOI
10.48550/arxiv.2203.00182

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Maximally entangled states (MES) are highly valued in quantum information processing. In quantum control, the creation of MES is typically treated as a state transfer problem with a predefined MES as the target. However, this approach is limited by the requirement to predetermine the MES structure. This paper introduces an improved quantum Lyapunov control approach that relies on the quantum entanglement measure to construct the Lyapunov function, instead of using the distance between quantum states. This strategy enables the preparation of any MES, regardless of whether its structure is known beforehand, using a single control scheme. The proposed entanglement control technique is unaffected by the number of entangled subsystems since it targets the entanglement measure as a scalar. Initially applied to bipartite pure states, this method demonstrates its capability to generate Bell states and their equivalents. Subsequent applications to bipartite mixed states and multipartite systems illustrate that the technique can produce MES with unspecified structures.
Entangled states are fundamental to quantum information processing. However, many existing quantum control methods rely on predefined target states, limiting their flexibility in accommodating diverse entanglement structures. This article introduces a sliding mode control framework that utilizes an entanglement measure as the sliding surface, enabling the generation of entangled states without specifying a fixed target. By adjusting the desired entanglement level, the proposed method can generate a wide range of states, including both bipartite and multipartite configurations, as well as pure and mixed states. Since the entanglement measure is scalar-valued, the resulting control law is inherently independent of the number of subsystems-an important advantage of the proposed approach. Among various entangled states, maximally entangled states (MESs) are of particular interest. Lyapunov stability of the control scheme is established, and numerical simulations confirm its effectiveness in robustly generating MESs in both bipartite and multipartite systems.
Quantum control concerns driving quantum systems to the desired properties or states. This objective is hindered by the non-deterministic nature of quantum systems and the symmetric topology of the quantum state space. This paper introduces a coupling Lyapunov functions approach to counter for the stochastic nature of quantum measurements and break the symmetric topology of resulting filter state space. As an example, we then apply it to synthesize non-smooth control almost surely globally stabilizing the desired Bell state-the maximally entangled two-qubit state with various applications. It turns out that this approach enables the short time convergence of the system state, which is essential in the control of entanglement. More interestingly, its combination with the generalized concept of SWM (simultaneous weak measurement)-induced quantum state reduction allows us to deterministically generate the maximally entangled Bell state with little non-local effects. We use numerical illustration to validate the proposed scheme.
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In this paper, we propose a scheme for the entanglement creations and the quantum gate implementations of spin qubits via Lyapunov control. We first show that three types of Bell states can be generated in one step with the traditional Lyapunov control. After that, the traditional Lyapunov control is generalized with the design of two Lyapunov functions that characterize the distance between two unitary operators. Thus, the Lyapunov control can be also used in constructing desired unitary operations. The performance of the entanglement creations and the quantum gate implementations are both estimated via numerical simulations. The results show the scheme holds robustness against the errors of the fields and couplings. As the generalization of Lyapunov control can not only be used in spin systems, the scheme may benefit the quantum information processing in different quantum systems.
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