We propose a machine learning framework for parameter estimation of single mode Gaussian quantum states. Under a Bayesian framework, our approach estimates parameters of suitable prior distributions from measured data. For phase-space displacement an
d squeezing parameter estimation, this is achieved by introducing Expectation-Maximization (EM) based algorithms, while for phase parameter estimation an empirical Bayes method is applied. The estimated prior distribution parameters along with the observed data are used for finding the optimal Bayesian estimate of the unknown displacement, squeezing and phase parameters. Our simulation results show that the proposed algorithms have estimation performance that is very close to that of Genie Aided Bayesian estimators, that assume perfect knowledge of the prior parameters. Our proposed methods can be utilized by experimentalists to find the optimum Bayesian estimate of parameters of Gaussian quantum states by using only the observed measurements without requiring any knowledge about the prior distribution parameters.
We propose a multiple-input multiple-output (MIMO) quantum key distribution (QKD) scheme for terahertz (THz) frequency applications operating at room temperature. Motivated by classical MIMO communications, a transmit-receive beamforming scheme is pr
oposed that converts the rank-$r$ MIMO channel between Alice and Bob into $r$ parallel lossy quantum channels. Compared with existing single-antenna QKD schemes, we demonstrate that the MIMO QKD scheme leads to performance improvements by increasing the secret key rate and extending the transmission distance. Our simulation results show that multiple antennas are necessary to overcome the high free-space path loss at THz frequencies. We demonstrate a non-monotonic relation between performance and frequency, and reveal that positive key rates are achievable in the $10-30$ THz frequency range. The proposed scheme can be used for both indoor and outdoor QKD applications for beyond fifth-generation ultra-secure wireless communications systems.
We consider multi-antenna wireless systems aided by large intelligent surfaces (LIS). LIS presents a new physical layer technology for improving coverage and energy efficiency by intelligently controlling the propagation environment. In practice howe
ver, achieving the anticipated gains of LIS requires accurate channel estimation. Recent attempts to solve this problem have considered the least-squares (LS) approach, which is simple but also sub-optimal. The optimal channel estimator, based on the minimum mean-squared-error (MMSE) criterion, is challenging to obtain and is non-linear due to the non-Gaussianity of the effective channel seen at the receiver. Here we present approaches to approximate the optimal MMSE channel estimator. As a first approach, we analytically develop the best linear estimator, the LMMSE, together with a corresponding majorization-minimization based algorithm designed to optimize the LIS phase shift matrix during the training phase. This estimator is shown to yield improved accuracy over the LS approach by exploiting second-order statistical properties of the wireless channel and the noise. To further improve performance and better approximate the globally-optimal MMSE channel estimator, we propose data-driven non-linear solutions based on deep learning. Specifically, by posing the MMSE channel estimation problem as an image denoising problem, we propose two convolutional neural network (CNN) based methods to perform the denoising and approximate the optimal MMSE channel estimation solution. Our numerical results show that these CNN-based estimators give superior performance compared with linear estimation approaches. They also have low computational complexity requirements, thereby motivating their potential use in future LIS-aided wireless communication systems.
Large intelligent surfaces (LIS) present a promising new technology for enhancing the performance of wireless communication systems. Realizing the gains of LIS requires accurate channel knowledge, and in practice the channel estimation overhead can b
e large due to the passive nature of LIS. Here, we study the achievable rate of a LIS-assisted single-input single-output communication system, accounting for the pilot overhead of a least-squares channel estimator. We demonstrate that there exists an optimal $K^{*}$, which maximizes achievable rate by balancing the power gains offered by LIS and the channel estimation overhead. We present analytical approximations for $K^{*}$, based on maximizing an analytical upper bound on average achievable rate that we derive, and study the dependencies of $K^*$ on statistical channel and system parameters.
We study a multiple-input single-output (MISO) communication system assisted by a reconfigurable intelligent surface (RIS). A base station (BS) having multiple antennas is assumed to be communicating to a single-antenna user equipment (UE), with the
help of a RIS. We assume that the system operates in an environment with line-of-sight (LoS) between the BS and RIS, whereas the RIS-UE link experiences Rayleigh fading. We present a closed form expression for the optimal active and passive beamforming vectors at the BS and RIS respectively. Then, by characterizing the statistical properties of the received SNR at the UE, we apply them to derive analytical approximations for different system performance measures, including the outage probability, average achievable rate and average symbol error probability (SEP). Our results, in general, demonstrate that the gain due to RIS can be substantial, and can be significantly greater than the gains reaped by using multiple BS antennas.
Let $mathbf{W}_1$ and $mathbf{W}_2$ be independent $ntimes n$ complex central Wishart matrices with $m_1$ and $m_2$ degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices $(mathbf
{W}_1+mathbf{W}_2)^{-1}mathbf{W}_1$, which are analogous to those of F matrices ${bf W}_1 {bf W}_2^{-1}$ and those of the Jacobi unitary ensemble (JUE). Defining $alpha_1=m_1-n$ and $alpha_2=m_2-n$, we derive new exact distribution formulas in terms of $(alpha_1+alpha_2)$-dimensional matrix determinants, with elements involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-$n$ analysis with $alpha_1$ and $alpha_2$ fixed (i.e., under the so-called hard-edge scaling limit); the analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as $n to infty$ in terms of $alpha_1$- and $alpha_2$-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-$n$ corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painleve differential equations, or hypergeometric functions of matrix arguments.
Sample correlation matrices are employed ubiquitously in statistics. However, quite surprisingly, little is known about their asymptotic spectral properties for high-dimensional data, particularly beyond the case of null models for which the data is
assumed independent. Here, considering the popular class of spiked models, we apply random matrix theory to derive asymptotic first-order and distributional results for both the leading eigenvalues and eigenvectors of sample correlation matrices. These results are obtained under high-dimensional settings for which the number of samples n and variables p approach infinity, with p/n tending to a constant. To first order, the spectral properties of sample correlation matrices are seen to coincide with those of sample covariance matrices; however their asymptotic distributions can differ significantly, with fluctuations of both the sample eigenvalues and eigenvectors often being remarkably smaller than those of their sample covariance counterparts.
Robust estimators of large covariance matrices are considered, comprising regularized (linear shrinkage) modifications of Maronnas classical M-estimators. These estimators provide robustness to outliers, while simultaneously being well-defined when t
he number of samples does not exceed the number of variables. By applying tools from random matrix theory, we characterize the asymptotic performance of such estimators when the numbers of samples and variables grow large together. In particular, our results show that, when outliers are absent, many estimators of the regularized-Maronna type share the same asymptotic performance, and for these estimators we present a data-driven method for choosing the asymptotically optimal regularization parameter with respect to a quadratic loss. Robustness in the presence of outliers is then studied: in the non-regularized case, a large-dimensional robustness metric is proposed, and explicitly computed for two particular types of estimators, exhibiting interesting differences depending on the underlying contamination model. The impact of outliers in regularized estimators is then studied, with interesting differences with respect to the non-regularized case, leading to new practical insights on the choice of particular estimators.
We study the design of portfolios under a minimum risk criterion. The performance of the optimized portfolio relies on the accuracy of the estimated covariance matrix of the portfolio asset returns. For large portfolios, the number of available marke
t returns is often of similar order to the number of assets, so that the sample covariance matrix performs poorly as a covariance estimator. Additionally, financial market data often contain outliers which, if not correctly handled, may further corrupt the covariance estimation. We address these shortcomings by studying the performance of a hybrid covariance matrix estimator based on Tylers robust M-estimator and on Ledoit-Wolfs shrinkage estimator while assuming samples with heavy-tailed distribution. Employing recent results from random matrix theory, we develop a consistent estimator of (a scaled version of) the realized portfolio risk, which is minimized by optimizing online the shrinkage intensity. Our portfolio optimization method is shown via simulations to outperform existing methods both for synthetic and real market data.
A large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, refer
red to as outliers. Building upon recent random matrix advances in the area of robust statistics, we specifically show that the so-called Maronna M-estimator of scatter asymptotically behaves similar to well-known random matrices when the population and sample sizes grow together to infinity. The introduction of outliers leads the robust estimator to behave asymptotically as the weighted sum of the sample outer products, with a constant weight for all legitimate samples and different weights for the outliers. A fine analysis of this structure reveals importantly that the propensity of the M-estimator to attenuate (or enhance) the impact of outliers is mostly dictated by the alignment of the outliers with the inverse population covariance matrix of the legitimate samples. Thus, robust M-estimators can bring substantial benefits over more simplistic estimators such as the per-sample normalized version of the sample covariance matrix, which is not capable of differentiating the outlying samples. The analysis shows that, within the class of Maronnas estimators of scatter, the Huber estimator is most favorable for rejecting outliers. On the contrary, estimators more similar to Tylers scale invariant estimator (often preferred in the literature) run the risk of inadvertently enhancing some outliers.
