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arXiv:quant-ph/0407166 · Physical Review A

Depolarization channels with zero-bandwidth noises

Andrzej Dragan, Krzysztof Wódkiewicz

Published Jan 18, 2005• 15 citations
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Abstract

A simple model describing depolarization channels with zero-bandwidth environment is presented and exactly solved. The environment is modeled by Lorentzian, telegraphic, and Gaussian zero-bandwidth noises. Such channels can go beyond the standard Markov dynamics and therefore can illustrate the influence of memory effects of the noisy communication channel on the transmitted information. To quantify the disturbance of quantum states the entanglement fidelity between arbitrary input and output states is investigated.

DOI
10.1103/physreva.71.012322

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A simple model describing depolarization channels with zero-bandwidth environment is presented and exactly solved. The environment is modeled by Lorentzian, telegraphic, and Gaussian zero-bandwidth noises. Such channels can go beyond the standard Markov dynamics and therefore can illustrate the influence of memory effects of the noisy communication channel on the transmitted information. To quantify the disturbance of quantum states the entanglement fidelity between arbitrary input and output states is investigated.
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Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρρρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.
The fidelity and the entanglement fidelity are two important quantities in describing the transmission of quantum information through (possibly noisy) quantum channels. In this paper these two quantities are calculated for different kinds of state which are transmitted in noisy channels. A general equation is obtained for these fidelities. Some examples are given to illustrate the general result and possible applications of these results to quantum cryptography and teleportation via noisy channels are discussed.
The amount of classical information that is reliably transmitted over two uses of general Pauli channels with memory, modeled as a correlated noise between a single pair of uses, is investigated. The maximum of the mutual information between the input and the output is proven to be achieved by a class of product states that is explicitly given in terms of the relevant channel parameters below some memory threshold, and by maximally entangled states above this threshold. In particular, this proves a conjecture on the depolarizing channel by Macchiavello and Palma [Phys. Rev. A 65, 050301(R) (2002)]. Furthermore, it also shows that no other scenario can occur for Pauli channels as for example the existence of an intermediate optimal degree of entanglement reported for some Gaussian channels with memory.