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arXiv:2003.01942 · Scientific Reports

Holevo Capacity of Discrete Weyl Channels

Junaid ur Rehman, Youngmin Jeong, Jeong San Kim, Hyundong Shin

Published Nov 23, 2018• 21 citations
arXivHTMLPDF

Abstract

Holevo capacity is the maximum rate at which a quantum channel can reliably transmit classical information without entanglement. However, calculating the Holevo capacity of arbitrary quantum channels is a nontrivial and computationally expensive task since it requires the numerical optimization over all possible input quantum states. In this paper, we consider discrete Weyl channels (DWCs) and exploit their symmetry properties to model DWC as a classical symmetric channel. We characterize lower and upper bounds on the Holevo capacity of DWCs using simple computational formulae. Then, we provide a sufficient and necessary condition where the upper and lower bounds coincide. The framework in this paper enables us to characterize the exact Holevo capacity for most of the known special cases of DWCs.

DOI
10.1038/s41598-018-35777-7

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Holevo capacity is the maximum rate at which a quantum channel can reliably transmit classical information without entanglement. However, calculating the Holevo capacity of arbitrary quantum channels is a nontrivial and computationally expensive task since it requires the numerical optimization over all possible input quantum states. In this paper, we consider discrete Weyl channels (DWCs) and exploit their symmetry properties to model DWC as a classical symmetric channel. We characterize lower and upper bounds on the Holevo capacity of DWCs using simple computational formulae. Then, we provide a sufficient and necessary condition where the upper and lower bounds coincide. The framework in this paper enables us to characterize the exact Holevo capacity for most of the known special cases of DWCs.
We propose a tight lower bound on the Holevo capacity of discrete Weyl channels (DWCs), which gives the exact Holevo capacity of known special cases of DWCs. Furthermore, for a physically available but otherwise unknown (channel parameters) DWC, this bound can be directly estimated by utilizing d+1d+1d+1 measurement configurations without entanglement and joint measurements. We show that the performance of directly estimating the proposed bound is comparable with first performing the entanglement-based optimal parameter estimation of the DWC and then computing the proposed lower bound.
Abstract Finding the classical capacity of a quantum channel is not easy, yet we are able to analytically calculate this capacity for new channels. We analyze the bounds of the Holevo capacity and classical capacity for the generalized Pauli channels. In particular, by generalizing earlier results for the Weyl channels, we obtain the lower and upper bounds of the Holevo capacity and show that, if these bounds coincide, the Holevo capacity is weakly additive. We also prove the weak additivity of the lower bound. Two new examples of the generalized Pauli channels with known classical capacity are presented. Finally, we relate the change rate of the classical capacity to the P-divisibility of Pauli channels.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
We discuss discrete Weyl channels (DWCs) with correlated noise in consecutive uses. Some special cases of DWCs have been shown to have the so-called transition behavior, where the optimal signal states, which maximize the Holevo information between the transmitter and the receiver, sharply transition from product states to maximally entangled states as the degree of memory between the consecutive channel uses increases from a threshold value. We first show the general existence of this transition behavior for a class of DWCs where the product states are optimal for the memoryless case. We also provide two methods to estimate the degree of memory exhibited by the correlated noise. These estimators rely on generating classical data by measuring the channel output which shows the same memory characteristics as the consecutive uses of the channel. We specifically consider the Markovian noise but one of our proposed estimators can be straightforwardly generalized to an arbitrary type of noise as it provides the direct access to the sequence in which noise affects the consecutive input states.
We propose a framework to variationally obtain detectable capacity bounds for quantum channels. The proposal of this framework is motivated by the difficulty of estimating the von Neumann entropy of an unknown quantum state. Instead of estimating the von Neumann entropy at the channel output, we propose to estimate the state purity---which can be measured by a single measurement setting---followed by bounding the von Neumann entropy from above and below. This procedure leads to new upper and lower bounds on various communication rates of quantum channels for some fixed input states. Then, by utilizing the variational method to find optimal input states we obtain lower bounds on (i) the quantum capacity of arbitrary channels, (ii) the entanglement-assisted classical capacity of arbitrary channels, and (iii) the classical capacity of covariant channels. Corresponding to these lower bounds, we also obtain upper bounds on (i) NNN-shot coherent information, (ii) the entanglement-assisted classical capacity, and (iii) the NNN-shot Holevo capacity of arbitrary quantum channels. All these bounds can be estimated by a single measurement setting without needing a full process tomography or any further a priori knowledge, e.g., preferred basis of the channel.
We study the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Rényi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Rényi information.