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Chinese Physics B

Control of purity and entanglement for two spatially two-separated qubits via phase damping

A.‐S. F. Obada, H. A. Hessian, A.‐B. A. Mohamed, Mostafa Hashem

Published Oct 1, 2012• 2 citations
Publisher

Abstract

Control of purity and entanglement of two two-qubits dispersively coupled to a field with a reservoir are investigated. Initially the qubits are entangled, while the field is either in a coherent state or a statistical mixture of two coherent states. For an alternative entanglement measure we calculate the negativity of the eigenvalues of the partially transposed density matrix. A measure related to the mutual entropy, namely the index of entropy, is employed to measure the entanglement. Its results agree well with the negativity. It is found that the entanglement and purity have strong sensitivity to phase damping. The asymptotic behaviour of the states of the field, the two two-qubits, and the total system fall into mixed states.

DOI
10.1088/1674-1056/21/10/100310

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Control of purity and entanglement of two two-qubits dispersively coupled to a field with a reservoir are investigated. Initially the qubits are entangled, while the field is either in a coherent state or a statistical mixture of two coherent states. For an alternative entanglement measure we calculate the negativity of the eigenvalues of the partially transposed density matrix. A measure related to the mutual entropy, namely the index of entropy, is employed to measure the entanglement. Its results agree well with the negativity. It is found that the entanglement and purity have strong sensitivity to phase damping. The asymptotic behaviour of the states of the field, the two two-qubits, and the total system fall into mixed states.
Numerical simulations and calculation of both the total entropy and the sub-entropies are used to investigate the purity loss in nonlinear interaction of a qubit with coherent field. While, the mutual entropy is used as a measure of the total correlations and the negativity as a measure to the entanglement. We show that the purity and entanglement behave different for standard energy dissipation as compared to an energy preserving phase noise coupling as introduced by Milburn. It is found that a very strong sensitivity of the degradation both total correlation with entanglement to the amount of phase damping and the chosen coupling. In particular for a coherent initial field state the maximum obtainable correlation is strongly reduced even for very weak noise.
An analytical description of a superconducting (SC) phase qubit coupled to a torsional resonator, which is damped by a dispersive reservoir, is presented based on the master equation. Therefore, the effect of the qubit phase damping on the dynamical behavior of the entanglement, purity loss and qubit inversion are investigated. It is found that the collapse and revival phenomena of qubit inversion are very sensitive not only to the damping parameter but also to the frequency detuning and the qubit distribution angle of the initial state. It is interesting to note that the purity of the state of the SC-qubit, which is measured by von Neumann entropy, can be completely lost due to the dispersive reservoir parameter. Because of the existence of dispersive reservoir, the von Neumann entropy cannot be a measure for the entanglement in open system. So, the negative eigenvalue of the partially transposed density matrix of qubit-resonator system is used to quantify the entanglement. For certain parameter sets, it is possible to control the degree and the dynamics of entanglement between the qubit and the torsional resonator.
The atomic decay for a two level atom interacting with a single mode of electromagnetic field is considered. In particular for a coherent state or statistical mixture (SM) of two opposite coherent states as initial field states, the exact solution of the master equation is found. Effect of the atomic damping on the partial entropies of the atom or the field and the total entropy as a measures of the purity loss is investigated. The degree of entanglement by the negativity and the mutual information and the atomic coherence through the master equation is studied.
Amplitude damping changes entangled pure states into usually less-entangled mixed states. We show, however, that even local amplitude damping of one or two qubits can result in mixed states more entangled than pure states if one compares the relative entropy of entanglement (REE) for a given degree of the Bell--Clauser-Horne-Shimony-Holt inequality violation (referred to as nonlocality). By applying Monte Carlo simulations, we find the maximally entangled mixed states and show that they are likely to be optimal by checking the Karush-Kuhn-Tucker conditions, which generalize the method of Lagrange multipliers for this nonlinear optimization problem. We show that the REE for mixed states can exceed that of pure states if the nonlocality is in the range (0,0.82) and the maximal difference between these REEs is 0.4. A former comparison [Phys. Rev. A 78, 052308 (2008)] of the REE for a given negativity showed analogous property but the corresponding maximal difference in the REEs is one order smaller (i.e., 0.039) and the negativity range is (0,0.53) only. For appropriate comparison, we normalized the nonlocality measure to be equal to the standard entanglement measures, including the negativity, for arbitrary two-qubit pure states. We also analyze the influence of the phase-damping channel on the entanglement of the initially pure states. We show that the minimum of the REE for a given nonlocality can be achieved by this channel, contrary to the amplitude-damping channel.
We study pure phase damping of two qubits due to fluctuating fields. As frequently employed, decoherence is thus described in terms of random unitary (RU) dynamics, that is, a convex mixture of unitary transformations. Based on a separation of the dynamics into an average Hamiltonian and a noise channel, we are able to analytically determine the evolution of both entanglement and purity. This enables us to characterize the dynamics in a concurrence-purity (CP) diagram: We find that RU phase-damping dynamics sets constraints on accessible regions in the CP plane. We show that initial state and dynamics contribute to final entanglement independently.
We study the exact entanglement and entropy dynamics of two qubits interacting with a common zero-temperature non-Markovian reservoir. It is a commonly held view that entanglement loss due to environmental decoherence is accompanied by loss of purity of the state of the system. We demonstrate that such an intuitive picture does not always apply: the deterioration of entanglement and purity does not necessarily come together; i.e. revivals of entanglement can be accompanied by deterioration of purity. To complete our investigation on entanglement–mixedness interplay we consider the case of initial mixed states and study how the entanglement dynamics and its revivals are related to both the initial purity and the initial entanglement.