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Physical Review E

Monte Carlo simulation of a noisy quantum channel with memory

Ismail Yunus Akhalwaya, Mervlyn Moodley, Francesco Petruccione

Published Oct 15, 2015• 0 citations
Publisher

Abstract

The classical capacity of quantum channels is well understood for channels with uncorrelated noise. For the case of correlated noise, however, there are still open questions. We calculate the classical capacity of a forgetful channel constructed by Markov switching between two depolarizing channels. Techniques have previously been applied to approximate the output entropy of this channel and thus its capacity. In this paper, we use a Metropolis-Hastings Monte Carlo approach to numerically calculate the entropy. The algorithm is implemented in parallel and its performance is studied and optimized. The effects of memory on the capacity are explored and previous results are confirmed to higher precision.

DOI
10.1103/physreve.92.043304

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The classical capacity of quantum channels is well understood for channels with uncorrelated noise. For the case of correlated noise, however, there are still open questions. We calculate the classical capacity of a forgetful channel constructed by Markov switching between two depolarizing channels. Techniques have previously been applied to approximate the output entropy of this channel and thus its capacity. In this paper, we use a Metropolis-Hastings Monte Carlo approach to numerically calculate the entropy. The algorithm is implemented in parallel and its performance is studied and optimized. The effects of memory on the capacity are explored and previous results are confirmed to higher precision.
The classical product state capacity of a noisy quantum channel with memory is investigated. A forgetful noise-memory channel is constructed by Markov switching between two depolarizing channels which introduces non-Markovian noise correlations between successive channel uses. The computation of the capacity is reduced to an entropy computation for a function of a Markov process. A reformulation in terms of algebraic measures then enables its calculation. The effects of the hidden-Markovian memory on the capacity are explored. An increase in noise correlations is found to increase the capacity.
The classical product state capacity of a noisy quantum channel with memory is investigated. A forgetful noise‐memory channel is constructed by Markov switching between two depolarizing channels which introduces non‐Markovian noise correlations between successive channel uses. This function of a Markov process can be reformulated as an algebraic measure. This framework provides an expression for the asymptotic entropy rate and thus enables the calculation of the classical capacity. The effects of the hidden‐Markovian memory on the capacity are explored. An increase in noise‐correlations is found to increase the capacity.
This paper was presented in poster form at and in the proceedings of the QCMC 2008. It is a summary of a fuller paper to appear separately. The classical product state capacity of a noisy quantum channel with memory is investigated. A forgetful noise-memory channel is constructed by Markov switching between two depolarizing channels which introduces non-Markovian noise correlations between successive channel uses. This function of a Markov process can be reformulated as an algebraic measure. This framework provides an expression for the asymptotic entropy rate and thus enables the calculation of the classical capacity. The effects of the hidden-Markovian memory on the capacity are explored. An increase in noise-correlations is found to increase the capacity. 1
We study the classical and entanglement-assisted capacity of a forgetful quantum memory channel that randomly switches between two qubit depolarizing channels. We show that when the input consists of two qubits then depending on channel parameters either the maximally entangled input states or product input states achieve the two-use classical capacity. We conjecture that as the number of input qubits is increased the classical capacity approaches the product state capacity for all values of the parameters. We also derive an expression for the entanglement-assisted classical capacity of this quantum memory channel in terms of the entropy rate of a Markov chain.
We present an algorithm for calculation of the Gaussian classical capacity of a quantum bosonic memory channel with additive Gaussian noise. The algorithm, restricted to Gaussian input states, is applicable to all channels with noise correlations obeying certain conditions and works in the full input energy domain, beyond previous treatments of this problem. As an illustration, we study the optimal input states and capacity of a quantum memory channel with Gauss-Markov noise [J. Sch\"afer, Phys. Rev. A 80, 062313 (2009)]. We evaluate the enhancement of the transmission rate when using these optimal entangled input states by comparison with a product coherent-state encoding and find out that such a simple coherent-state encoding achieves not less than 90% of the capacity.
Quantum mechanics contains a fresh and mysterious view of reality.Besides the philosophical intrigue, it has also produced and continues to inspire tantalizing new technological innovations.In any technological system, the designers must contend with the problem of noise.This thesis studies classical noise in two different quantum settings.The first is the classical capacity of a quantum channel with memory.Adding forgetfulmemory, attempts to push the boundaries of our understanding of how best to transmit information in the presence of correlated noise.We study the noise within two different frameworks; Algebraic Measure theory and Monte Carlo simulations.Both tools are used to calculate the capacity of the channel as correlations in the noise are increased.The second classical-quantum system investigated is atomic clocks.Using power spectral density methods we study aliasing noise induced by periodic-correction which includes the Dick Effect.We propose a novel multi-window scheme that extends the standard method of noise correction and exhibits better anti-aliasing properties.A uniting thread that emerges is that correlations can be put to good use.In the classical capacity setting, correlations occur between uses of the quantum channel.We show that stronger correlations increase the classical capacity.The benefits of correlation are even seen at a meta-level within the framework of Monte Carlo simulations.Correlations are designed into the algorithm which have nothing to do with real-world correlations, but are abstract correlations created by a Markov chain employed in the algorithm to help efficiently sample from a distribution of exponential size.Finally, in the atomic clock setting, correlations in the measured noise are used to help predict and cancel noise on a short time-scale while trying to limit aliasing.Channel capacity and precise time-keeping are distinct topics and require very different approaches to study.However, common to both topics is their application to communication and other tasks, the need to overcome noise and the benefits of exploiting correlations in the noise.