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arXiv:quant-ph/9903062 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Stochastic resonance for quantum channels

Julian Juhi-Lian Ting

Published Mar 1, 1999• 23 citations
arXivHTMLPDF

Abstract

The concept of stochastic resonance in nonlinear dynamics is applied to interpret the capacity of noisy quantum channels. The two-Pauli channel is used to illustrate the idea. The fidelity of the channel is also considered. Noise enhancement is found for the channel fidelity but not for the channel capacity.
DOI
10.1103/physreve.59.2801

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This chapter introduces the notion of noisy quantum channels , and the different types of “quantum noise” that affect qubit messages passed through such channels. The main types of noisy channel reviewed here are the depolarizing , bit-flip , phase-flip , and bit-phase-flip channels. Then the quantum channel capacity χ is defined through the Holevo–Schumacher–Westmoreland (HSW) theorem. Such a theorem can conceptually be viewed as the elegant quantum counterpart of Shannon's (noisy) channel coding theorem , which was described in Chapter 13. Here, I shall not venture into the complex proof of the HSW theorem but only provide a background illustrating the similarity with its classical counterpart. The resemblance with the channel capacity χ and the Holevo bound , as described in Chapter 21, and with the classical mutual information H ( X ; Y ), as described in Chapter 5, are both discussed. For advanced reference, a hint is provided as to the meaning of the still not fully explored concept of quantum coherent information . Several examples of quantum channel capacity, derived from direct applications of the HSW theorem, along with the solution of the maximization problem, are provided. Noisy quantum channels The notion of “noisiness” in a classical communication channel was first introduced in Chapter 12, when describing channel entropy . Such a channel can be viewed schematically as a probabilistic relation between two random sources, X for the originator, and Y for the recipient.
We determine conditions for the presence of stochastic resonance in a lossy bosonic channel with a nonlinear, threshold decoding. The stochastic resonance effect occurs if and only if the detection threshold is outside of a 'forbidden interval'. We show that it takes place in different settings: when transmitting classical messages through a lossy bosonic channel, when transmitting over an entanglement-assisted lossy bosonic channel and when discriminating channels with different loss parameters. Moreover, we consider a setting in which stochastic resonance occurs in the transmission of a qubit over a lossy bosonic channel with a particular encoding and decoding. In all cases, we assume the addition of Gaussian noise to the signal and show that it does not matter who, between sender and receiver, introduces such a noise. Remarkably, different results are obtained when considering a setting for private communication. In this case, the symmetry between sender and receiver is broken and the 'forbidden interval' may vanish, leading to the occurrence of stochastic resonance effects for any value of the detection threshold.
We study encodings that give the best-known thresholds for the nonzero capacity of quantum channels, i.e., the upper bound for correctable noise, using an entropic approach to calculation of the threshold values. Our results show that Pauli noise is correctable up to the hashing bound. For a depolarizing channel, this approach allows one to achieve a nonzero capacity for a fidelity (probability of no error) of f=0.80870f=0.808\phantom{\rule{0.2em}{0ex}}70f=0.80870.
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.
We determine conditions for the presence of stochastic resonance in a lossy bosonic channel with a nonlinear, threshold decoding. The stochastic resonance effect occurs if and only if the detection threshold is outside of a 'forbidden interval'. We show that it takes place in different settings: when transmitting classical messages through a lossy bosonic channel, when transmitting over an entanglement-assisted lossy bosonic channel and when discriminating channels with different loss parameters. Moreover, we consider a setting in which stochastic resonance occurs in the transmission of a qubit over a lossy bosonic channel with a particular encoding and decoding. In all cases, we assume the addition of Gaussian noise to the signal and show that it does not matter who, between sender and receiver, introduces such a noise. Remarkably, different results are obtained when considering a setting for private communication. In this case, the symmetry between sender and receiver is broken and the 'forbidden interval' may vanish, leading to the occurrence of stochastic resonance effects for any value of the detection threshold. © 2013 IOP Publishing Ltd.