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.
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.
In this thesis we study the information transmission through Gaussian quantum channels. Gaussian quantum channels model physical communication links, such as free space communication or optical fibers and therefore, may be considered as the most relevant quantum channels. One of the central characteristics of any communication channel is its capacity. In this work we are interested in the classical capacity, which is the maximal number of bits that can be reliably transmitted per channel use. An important lower bound on the classical capacity is given by the Gaussian capacity, which is the maximal transmission rate with the restriction that only Gaussian encodings are allowed: input messages are encoded in so-called Gaussian states for which the mean field amplitudes are Gaussian distributed. We focus in this work mainly on the Gaussian capacity for the following reasons. First, Gaussian encodings are easily accessible experimentally. Second, the difficulty of studying the classical capacity, which arises due to an optimization problem in an infinite dimensional Hilbert space, is greatly reduced when considering only Gaussian input encodings. Third, the Gaussian capacity is conjectured to coincide with the classical capacity, even though this longstanding conjecture is unsolved until today. We start with the investigation of the capacities of the single-mode Gaussian channel. We show that the most general case can be reduced to a simple, fiducial Gaussian channel which depends only on three parameters: its transmissivity (or gain), the added noise variance and the squeezing of the noise. Above a certain input energy threshold, the optimal input variances are given by a quantum water-filling solution, which implies that the optimal modulated output state is a thermal state. This is a quantum extension (or generalization) of the well-known classical water-filling solution for parallel Gaussian channels. Below the energy threshold the solution is given by a transcendental equation and only the less noisy quadrature is modulated. We characterize in detail the dependence of the Gaussian capacity on its channel parameters. In particular, we show that the Gaussian capacity is a non-monotonous function of the noise squeezing and analytically specify the regions where it exhibits one maximum, a maximum and a minimum, a saddle point or no extrema. Then, we investigate the case of n-mode channels with noise correlations (i.e. memory), where we focus in particular on the classical additive noise channel. We consider memory models for which the noise correlations can be unraveled by a passive symplectic transformation. Therefore, we can simplify the problem to the study of the Gaussian capacity in an uncorrelated basis, which corresponds to the Gaussian capacity of n single-mode channels with a common input energy constraint. Above an input energy threshold the solutions is given by a global quantum water-filling solution, which implies that all modulated single-mode output states are thermal states with the same temperature. Below the threshold the channels are divided into three sets: i) those that are excluded from information transmission, ii) those for which only the less noisy quadrature is modulated, and iii) those for which the quantum water-filling solution is satisfied. As an example we consider a Gauss-Markov correlated noise, which in the uncorrelated basis corresponds to a collection of single-mode classical additive noise channels. When rotating the collection of optimal single-mode input states back to the original, correlated basis the optimal multi-mode input state becomes a highly entangled state. We then compare the performance of the optimal input state with a simple coherent state encoding and conclude that one gains up to 10% by using the optimal encoding. Since the preparation of the optimal input state may be very challenging we consider sub-optimal Gaussian-matrix product states (GMPS) as input states as well. GMPS have a known experimental setup and, though being heavily entangled, can be generated sequentially. We demonstrate that for the Markovian correlated noise as well as for a non-Markovian noise model in a wide range of channel parameters, a nearest-neighbor correlated GMPS achieves more than 99.9% of the Gaussian capacity. At last, we introduce a new noise model for which the GMPS is the exact optimal input state. Since GMPS are known to be ground states of quadratic Hamiltonians this suggests a starting point to develop links between optimization problems of quantum communication and many body physics.
Quantum communication is at the heart of the quantum information theory. We study two types of quantum communication protocols over noisy transmission channels, i.e. super dense coding and cryptography protocols. In the first part of this thesis, for various scenarios, it is discussed how the super dense coding capacity is influenced by noisy quantum channels. The case of memoryless channels as well as those channels with memory which are modelled by uncorrelated and correlated noise, respectively, are considered. Explicitly Pauli channels over arbitrary dimensions are treated and the super dense coding capacity for some resource states is derived. For the qubit depolarizing channel, when noise is uncorrelated, the super dense coding capacity with respect to the input state is also optimized. This illustrates a threshold value of the noise parameter below which the super dense coding capacity is optimized by a maximally entangled initial state, while above the threshold value the super dense coding capacity is optimized by a product state. For Pauli channels, with correlated noise, the case of non unitary encoding is studied and the super dense coding capacity is derived. The first part of this thesis is concluded with an example for multipartite super dense coding.
In the second part of the thesis, the problem of optimal eavesdropping on noisy states in quantum key distribution is investigated. The case of the six state protocol, when the signal states are mixed with white noise, is considered. This situation may arise either when Alice deliberately adds noise to the signal states before they leave her lab, or, in a realistic scenario when the eavesdropper (referred to as Eve) is not assumed to replace the noisy quantum channel by a noiseless one. For individual attacks, we find Eve's optimal mutual information with Alice as a function of the quantum bit error rate. Finally, the results illustrate that adding noise on the quantum level can make quantum key distribution more robust against eavesdropping.
This paper presents photonic communications and data storage capacitates for classical and quantum communications over a quantum channel. These capacities represent a generalization of Shannon’s classical channel capacity and coding theorem in two ways. First, it extends classical results for bit communication transport to all frequencies in the electromagnetic spectrum. Second, it extends the results to quantum bit (qubit) transport as well as a hybrid of classical and quantum communications. Nature’s limits on the rate at which classical and/or quantum information can be sent error-free over a quantum channel using classical and/or quantum error-correcting codes are presented as a function of the thermal background light level and Einstein zero-point energy. Graphical results are given as well as numerical results regarding communication rate limits using Planck’s natural frequency and time-interval units!
In classical digital communications two main families of “M-ary” modulation schemes are generally distinguished: bandwidth limited and power limited. The canonical comparison of these modulation methods is based on normalized data rate (R/W) (bits per second per hertz of bandwidth) and the signal to noise rate per bit required to achieve a given error probability for different M. In a classical picture, the two families reside in two separate semi-plains R/W < 1 and R/W > 1, i.e. energy efficiency and bandwidth efficiency cannot be optimized at the same time. However, we find an alphabet family that can be paired with a quantum receiver to simultaneously optimize bandwidth and power efficiency of a communication channel. Particularly we found that coherent frequency shift keying (CFSK) gives rise to a family of communication protocols that are bandwidth limited in nature, but whose bandwidth usage can be optimized so that R/W>1 for a range of alphabet lengths M, while power sensitivity beats that of power-limited protocols. We will report our theoretical findings and experimental progress towards implementation of this protocol family.
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.
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.