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IEEE Transactions on Information Theory

Quantum electrodynamics of a communication channel

C. Y. She

Published Jan 1, 1968• 10 citations
Publisher

Abstract

A simple one-dimensional model is taken to illustrate the quantum effects of a narrowband communication system. The system is quantized, and its dynamics are discussed in terms of the Heisenberg equations for field operators. The detection process of a coherent joint whose compatibility with quantum physics was justified elsewhere, is discussed. The detector, which is capable of measuring the quadrature-modulating components at Nyquist rate with minimum possible uncertainty allowed by quantum theory and which is most suitable for a continuous channel, is used to determine the channel capacity. It is shown that the noise of the channel, due to both thermal and quantum fluctuations, is additive and Gaussian. The classical concepts and expressions for the average mutual information and channel capacity are still valid, provided that the quantum noise of one quantum is included properly. The results obtained approach straightforwardly Shannon's in the classical limit.

DOI
10.1109/tit.1968.1054095

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IEEE Journal on Selected Areas in Communications•Jan 23, 2020•15 citations
A simple one-dimensional model is taken to illustrate the quantum effects of a narrowband communication system. The system is quantized, and its dynamics are discussed in terms of the Heisenberg equations for field operators. The detection process of a coherent joint whose compatibility with quantum physics was justified elsewhere, is discussed. The detector, which is capable of measuring the quadrature-modulating components at Nyquist rate with minimum possible uncertainty allowed by quantum theory and which is most suitable for a continuous channel, is used to determine the channel capacity. It is shown that the noise of the channel, due to both thermal and quantum fluctuations, is additive and Gaussian. The classical concepts and expressions for the average mutual information and channel capacity are still valid, provided that the quantum noise of one quantum is included properly. The results obtained approach straightforwardly Shannon's in the classical limit.
The problem of including quantum effects within existing methods of communication system analysis is investigated.A mathematical model is postulated for describing narrow bandwidth systems.Utilizing the Glauber P representation, the model permits analysis of both real field detection (measurement of electric and magnetic field magnitudes) and photon detection.The resulting equation which describes photon detection is shown to be consistent with previous work which has been proven experimentally.Unique aspects of the postulated model are its ability to describe both random and deterministic modulation functions in conventional terms, and its ability to demonstrate the inapplicability of the "photon channel" model for describing present-day systems.An analysis of quantum effects at the transmitter shows that modulation of the radiated field must take place by one of two distinctly different processes.Wave modulation is defined as a process which conveys information in the amplitude and phase of the electromagnetic field.Photon modulation is defined as a process which conveys information in the exact number of photons per pulse of the transmitted field (i.e., the "photon channel" of Stern, Gordon, ani othfrs).A wave modulated signal is shown to describe present-day modulation schemes and, moreover, to yield random photon fluctuations.It is concluded a wave modulated signal is not capable of describing a photon modulated signal.In contrast, a photon modulated signal is shown to possess a completely random phase fluctuation thus demonstrating the inability of the "photon channel" to describe presently used wave modulation schemes.For wave modulation the Glauber P function is shown to be identical (with a change of variable) to the classical joint probability density of the quadrature components of a narrowband sinusoid used in conventional analysis.Equations are derived which "transform" classical probability distributions into distributions which include quantum effects.The "partitioning noise" studied by Hagfors and Bowen is found to constitute a source of noise only in the "photon channel."The "partitioning effect" is shown to merely preserve the Poisson character of photon counts in wave modulated fields and therefore does not introduce a noise in present-day systems.For a wave modulated system with real field detection it is shown that photon noise can be included in conventional analysis by adding to the normal input noise, a noise density of hf c /2 Watts/Hertz where h is Planck's constant and f is the system carrier frequency.Use of photon heterodyne detection requires the addition of an input noise density of hf c Watts/Hertz.The photon noise is exactly additive Gaussian for real field detection and approximately additive Gaussian for photon heterodyne detection.Detection by counting the received photons is shown to yield neither Gaussian nor Poissonian statistics.However, with an "ideal measurement process," when the effective received noise density n c is much less than hf c , the variance of the counts are found to approach that of a Poisson distribution.Conversely, when nc is much larger than hf c , the .variance approaches that derivable from a classical analysis.A channel capacity equation for a wave modulated system is found which is equal to those derived by Lachs and Jelsma.The equation is shown to give capacities less than that derived by Gordon.
Dynamics of quantum information channel is studied with the use of a boson detector model. Explicit results are given for quantum dense coding process on the basis of an equivalence relation between a classical capacity and von Neumann mutual information. Reduced density matrices for an entangled spin degree of freedom of an incoming particle are exactly solved to give complete time evolution under influence of fluctuations due to a detector. Quantum mechanical channel matrix and the classical capacity are determined as functions of time and temperature. Decoherence properties of the entangled states under quantum and thermal noises are fully discussed.
Communication theory problems incorporating quantum effects for optical-frequency applications are discussed. Under suitable conditions, a unique quantum channel model corresponding to a given classical space-time varying linear random channel is established. A procedure is described by which a proper density-operator representation applicable to any receiver configuration can be constructed directly from the channel output field. Some examples illustrating the application of our methods to the development of optical quantum channel representations are given. Optimizations of communication system performance under different criteria are considered. In particular, certain necessary and sufficient conditions on the optimal detector in M-ary quantum signal detection are derived. Some examples are presented. Parameter estimation and channel capacity are discussed briefly.
Abstract The aim of the paper is the derivation of formulas for the capacities of a classical–quantum communication channel with Gaussian noise, which are analogs of the classical capacities and arise in many problems of quantum information theory, on a simple, heuristic and physically intuitively clear level. The classical capacity of a classical communication channel with limited frequency band and Gaussian information states diverges as noise tends to zero. In a consistent quantum consideration, the divergence is removed, which is a consequence of the fundamental physical principle of indistinguishability of particles—bosons in information states.
Quantum information processing leverages the principles of quantum mechanics, utilizing qubits, to improve computational and communicative tasks. In this realm, the quantum channel’s capacity is pivotal in determining the efficiency and accuracy of quantum information handling, with its performance being significantly influenced by channel noise. Our study aims to establish a holistic hybrid quantum noise model to determine the quantum channel capacity. In this paper, we formulated a mathematical expression for this capacity and conducted simulations for both Gaussian and non-Gaussian inputs. A hybrid noise model is constructed by convolution of Poisson-distributed quantum noise with classical additive white Gaussian noise. We characterized the quantum-classical noise and the received signal using Gaussian Mixture Models. The maximum amount of quantum information that can be reliably transmitted over a quantum channel (per use of the channel) is determined by its capacity, and entropy and related quantities like mutual information play a role in calculating this capacity. Our formulation of quantum channel capacity is derived from the mutual information shared between the transmitter and receiver, encompassing the entropies of the signals. The quantum channel presents a higher capacity-to-signal-to-noise ratio for Gaussian inputs than non-Gaussian ones.
Based upon the Fluctuation-Dissipation Theorem of Nyquist-Callen-Welton and the spectrum balancing relationship of Kubo-Martin-Schwinger for operator-valued stochastic noise processes, a two-parameter Quantum Noise-Energy Spectral Density (QN-ESD) is constructed and Fourier-Laplace transformed to obtain its associated Quantum Noise-Autocorrelation Function (QN-ACF). By means of a Taylor series expansion, the quantum thermal noise correlations are linked to Number Theory's Reimann-Zeta function. From this perspective, a panoply of spectral components and their QN correlations are characterized for three disjoint regions located across the Electromagnetic Spectrum (EMS). The second-moment characterizations and frequency boundaries for each region are specified as a function of two frequency dependent design parameters, viz., a normalized temperature dependent frequency and a quantum receiver cutoff frequency. These boundaries define where the quadrature noise correlation projections of the QN-ESD become noticeably asymmetric-in-frequency. For the operator-valued white quantum noise region, various communication system performance metrics are characterized and their limits are presented and compared via interconnecting parameter relations to their classical communications (CC) counterparts. These metrics include Shannon's non-additive, but regularized, zero-error capacity in bits/photon-pair, bits/photon, bits/channel-use and bits/sec, the spectral efficiency in bits/sec/Hz and the error-correcting code rate. For long coded messages, Shannon's capacity quadrature approaches 2π√5/ ln(2) ≈ 20 bits/photon-pair or (10 bits/photon) while for single-photon per channel use transmission it approaches 3 bits/photon-pair. Results are graphically illustrated for the quantum noise spectral density, quantum noise correlations, Shannon's Î-Q̂ capacities and spectral efficiency.