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arXiv:0810.5716 · Physica Scripta

A semiclassical model for a memory dephasing channel

Antonio D’Arrigo, Giuliano Benenti, G. Falci

Published Jul 1, 2009• 1 citation
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Abstract

We study a dephasing channel with memory, described by a Hamiltonian model in which the system-environment interaction is described by a stochastic process. We propose a useful way to describe the channel uses correlations. Moreover, we give a general expression for the coherences decay factors as a function of the number of channel uses and of the stochastic process power spectrum. We also study the impact of memory on the three qubit code, showing that correlations among channel uses affect very little the code performance.

DOI
10.1088/0031-8949/2009/t135/014052

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We study a dephasing channel with memory, described by a Hamiltonian model in which the system-environment interaction is described by a stochastic process. We propose a useful way to describe the channel uses correlations. Moreover, we give a general expression for the coherences decay factors as a function of the number of channel uses and of the stochastic process power spectrum. We also study the impact of memory on the three qubit code, showing that correlations among channel uses affect very little the code performance.
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In this article we treat the subject of chaotic environments with few degrees of freedom in quantum communication by investigating a conservative dynamical map as a model of a dephasing quantum channel. When the channel's dynamics is chaotic, we investigate the model's semi-classical limit and show that the entropy exchange grows at a constant rate which depends on a single parameter (the interaction strength), analogous to stochastic models of dephasing channels. We analyze memory effects in the channel and present strong physical arguments to support that the present model is forgetful in the chaotic regime while memory effects in general cannot be ignored when channel dynamics is regular. In order to render the nonchaotic channel forgetful, it becomes necessary to apply a reset to the channel and this reset can efficiently be modeled by application of a chaotic map. We may then refer to encoding theorems (valid in the case of forgetful channels) to present evidence of a transition from noiseless to noisy channel due to the environment's transition from regular to chaotic dynamics.
We show that the amount of coherent quantum information that can be reliably transmitted down a dephasing channel with memory is maximized by separable input states. In particular, we model the channel as a Markov chain or a multimode environment of oscillators. While in the first model, the maximization is achieved for the maximally mixed input state, in the latter it is convenient to exploit the presence of a decoherence-protected subspace generated by memory effects. We explicitly compute the quantum channel capacity for the first model while numerical simulations suggest a lower bound for the latter. In both cases memory effects enhance the coherent information. We present results valid for arbitrary input size.
We investigate measurement uncertainty of two incompatible observables, dense coding capacity, teleportation fidelity, and quantum Fisher information for two qubits traversing sequentially a dephasing channel modeled by the multimode bosonic field. Using a memory factor determined by the time separation between two channel uses and the transmit time of each qubit, it is shown that the memory effect of this channel on these four quantum characteristics is state-dependent. It is beneficial for some initial states and detrimental for the other states. There is also circumstance in which the memory effects with different origins have different influences on these quantum characteristics. These results may enrich our comprehension on the interplay between memory effect of a channel and decoherence process of a multi-qubit system.
The quantum capacity of a channel is often used as a single measure to characterize the ability of a channel to transmit quantum information coherently. The capacity determines the maximal asymptotic rate at which we can code reliably over a channel. Here we argue that this asymptotic treatment is insufficient to the point of being irrelevant in the quantum setting when decoherence limits our ability to manipulate large quantum systems. For all practical purposes we should instead focus on the tradeoff between three parameters: the rate of the code, the number of coherent uses of the channel, and the fidelity of the transmission. The aim is then to specify the region determined by allowed combinations of these parameters. Towards this goal, we find an approximate characterization of the region of allowed triplets for the qubit dephasing channel and for the erasure channel with classical post-processing. In each case the region is parametrized by a second channel parameter, the dispersion of the quantum capacity. In the process we also develop general bounds on the achievable region that are valid for all quantum channels.