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arXiv:0907.0052 · Physical Review A

Genuine tripartite entanglement in quantum brachistochrone evolution of a three-qubit system

Bao‐Kui Zhao, Fu‐Guo Deng, Zhang Feng-Shou, Hong-Yu Zhou

Published Nov 11, 2009• 13 citations
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Abstract

We explore the connection between quantum brachistochrone (time-optimal) evolution of a three-qubit system and its residual entanglement called three-tangle. The result shows that the entanglement between two qubits is not required for some brachistochrone evolutions of a three-qubit system. However, the evolution between two distinct states cannot be implemented without its three-tangle, except for the trivial cases in which less than three qubits attend evolution. Although both the probability density function of the time-averaged three-tangle and that of the time-averaged squared concurrence between two subsystems become more and more uniform with the decrease in angles of separation between an initial state and a final state, the features of their most probable values exhibit a different trend.

DOI
10.1103/physreva.80.052106

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We explore the connection between quantum brachistochrone (time-optimal) evolution of a three-qubit system and its residual entanglement called three-tangle. The result shows that the entanglement between two qubits is not required for some brachistochrone evolutions of a three-qubit system. However, the evolution between two distinct states cannot be implemented without its three-tangle, except for the trivial cases in which less than three qubits attend evolution. Although both the probability density function of the time-averaged three-tangle and that of the time-averaged squared concurrence between two subsystems become more and more uniform with the decrease in angles of separation between an initial state and a final state, the features of their most probable values exhibit a different trend.
Abstract We analytically investigate the role of entanglement in time-optimal state evolution as an application of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly coupled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entanglement plays a role for W and Greenberger–Horne–Zeilinger ( GHz ) initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.
Entanglement plays a fundamental role in the brachistochrone evolution of composite quantum systems. In the case of composite systems with distinguishable subsystems quantum brachistochrone evolutions cannot be implemented without entanglement, excepting trivial cases in which only one of the subsystems evolves. Here we explore the connection between entanglement and time-optimal quantum evolution for systems of two identical particles, elucidating its dependence on the type of statistics obeyed by the particles.
Quantum and classical correlations are investigated during quantum brachistochrone evolution (QBE) in this paper. We found some typical properties of the pair of quantum states sampled randomly by use of the Harr measure in this special kind of quantum evolution. This kind of evolution of a three-qubit system between two distinct states cannot be implemented without classical correlations (including bipartite J 2 and tripartite J 3 ) and quantum correlations (including bipartite D 2 and tripartite D 3 ). We also found that some QBEs between two distinct GHZ states do not follow the typical behaviour, and that this kind of evolution can be implemented without bipartite quantum correlations. Although the probability density function of the time-averaged bipartite classical correlation, time-averaged bipartite quantum correlations and time-averaged genuine tripartite correlations become more and more uniform with the decrease of angles of separation between an initial state and a final state, the features of their most probable values exhibit a different trend.
We consider a pure 3-qubits system interacting through a XY-Hamiltonian with antiferromagnetic constant J. We employ the 3-tangle as an efficient measure of the entanglement between such a 3-qubit system. The time evolution of such a 3-tangle is studied. In order to do the above, the 3-tangle associated to the pure 3-qubits state|ψ(t)⟩ = c0(t)|000⟩+c1(t)|001⟩+c2(t)|010⟩+c3(t)|011⟩+ c4(t)|100⟩+c5(t)|101⟩+c6(t)|110⟩+c7(t)|111⟩ is calculated as a function of the initial coefficients {ci(t = 0)} (i = 0,1,...,7), the time t and the antiferromagnetic constant J. We find that the 3-tangle of the 3-qubits system is periodic with period t = 4π/J. Furthermore, we also find that the 3-tangle as a function of the time t and J has maximal and minimum values. The maximal values of the 3-tangle can be employed in Quantum Information Protocols (QIP) that use entanglement as a basic resource. The pattern found for the 3-tangle of the system of three qubits interacting through a XY Hamiltonian as a function of J and the time t resembles to a quantized physical quantity.
The three-tangle is a measure of three-way entanglement in a system of three qubits. For a pure state, it can be understood as the residual entanglement not accounted for by pairwise entanglements between individual qubits. Here we define and evaluate the analogous quantity for three rebits (that is, binary systems in the real-amplitude variant of quantum theory). We find that the resulting formula is the same as in the complex case, except that an overall absolute value sign is missing. As a result, the rebit three-tangle can be negative, expressing the possibility of non-monogamous entanglement in real-amplitude quantum theory (for entanglement based on the convex-roof construction). We then relate the entanglement among three rebits to the entanglement of two qubits, by re-expressing the two-qubit state as a three-rebit state in the ubit model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.
Entanglement is closely related to some fundamental features of the dynamics of composite quantum systems: quantum entanglement enhances the "speed" of evolution of certain quantum states, as measured by the time required to reach an orthogonal state. The concept of "speed" of quantum evolution constitutes an important ingredient in any attempt to determine the fundamental limits that basic physical laws impose on how fast a physical system can process or transmit information. Here we explore the relationship between entanglement and the speed of quantum evolution in the context of the quantum brachistochrone problem. Given an initial and a final state of a composite system we consider the amount of entanglement associated with the brachistochrone evolution between those states, showing that entanglement is an essential resource to achieve the alluded time-optimal quantum evolution.