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Canadian Journal of Physics

Non-Markovinity of single qubit channels: analytical and numerical methods

Jian-Song Zhang, Aixi Chen

Published Oct 10, 2013• 0 citations
Publisher

Abstract

We propose methods to calculate non-Markovianity of four typical single qubit channels including flip (bit-flip, phase-flip, and bit-phase flip channels), depolarizing, phase damping, and amplitude damping channels analytically. Explicit expressions of non-Markovianity for some single qubit channels are obtained. For general channels we propose the Euler parametrization representation of quantum states to calculate non-Markovianity numerically.

DOI
10.1139/cjp-2013-0276

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We propose methods to calculate non-Markovianity of four typical single qubit channels including flip (bit-flip, phase-flip, and bit-phase flip channels), depolarizing, phase damping, and amplitude damping channels analytically. Explicit expressions of non-Markovianity for some single qubit channels are obtained. For general channels we propose the Euler parametrization representation of quantum states to calculate non-Markovianity numerically.
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Abstract In this paper, we investigate the cohering and decohering power of the one-qubit Markovian channels with respect to coherence measures based on the l 1 - norm, the Rényi α - relative entropy and the Tsallis α - relative entropy of coherence, respectively. The amplitude damping channel, phase damping channel, depolarizing channel, and flip channels are analytically calculated. It shows that the decohering power of the amplitude damping channel on the x , y , and z basis is equal to each other. The same phenomenon can be seen for the phase damping channel and the flip channels. The cohering power for the phase damping channel and the flip channels on the x , y basis also equals to that on the z basis. However, the cohering and decohering power of the depolarizing channel is independent to the reference basises. And the cohering power of the amplitude damping channel on the x , y basis is different to that on the z basis.
We propose an effective method which can simplify the optimization of the increase of the trace distance over all pairs of initial states in calculating the non-Markovianity N\mathcal{N}N for single-channel open systems. For the amplitude-damping channel, we can unify the results of Breuer et al. [Phys. Rev. Lett. 103, 210401 (2009)] in the large-detuning case and the results of Xu et al. [Phys. Rev. A 81, 044105 (2010)] in the resonant case; furthermore, for the general off-resonant cases, we can obtain a very tight lower bound of N\mathcal{N}N. As another application of our method, we also discuss N\mathcal{N}Nfor the non-Markovian depolarizing channel.
We study the dynamics of coherence-induced state ordering under incoherent channels, particularly four specific Markovian channels: amplitude damping channel, phase damping channel, depolarizing channel and bit flit channel for single-qubit states. We show that the amplitude damping channel, phase damping channel, and depolarizing channel do not change the coherence-induced state ordering by l1 norm of coherence, relative entropy of coherence, geometric measure of coherence, and Tsallis relative α-entropies, while the bit flit channel does change for some special cases.