Quantum illumination (QI) is the task of querying a scene using a transmitter probe whose quantum state is entangled with a reference beam retained in ideal storage, followed by optimally detecting the target-returned light together with the stored reference, to make decisions on characteristics of targets at stand-off range, at precision that exceeds what is achievable with a classical transmitter of the same brightness and otherwise identical conditions. Using tools from perturbation theory, we show that in the limit of low transmitter brightness, high loss, and high thermal background, there is a factor of four improvement in the Chernoff exponent of the error probability in discriminating any number of apriori-known reflective targets when using a Gaussian-state entangled QI probe, over using classical coherent-state illumination (CI). While this advantage was known for detecting the presence or absence of a target, it had not been proven for the generalized task of discriminating between arbitrary target libraries. In proving our result, we derive simple general analytic expressions for the lowest-order asymptotic expansions of the quantum Chernoff exponents for QI and CI in terms of the signal brightness, loss, thermal noise, and the modal expansion coefficients of the target-reflected light's radiant exitance profiles when separated by a spatial mode sorter after entering the entrance pupil of the receiver's aperture.

Quantum cryptography provides absolute security against an all-powerful eavesdropper (Eve). However, in practice Eve's resources may be restricted to a limited aperture size so that she cannot collect all paraxial light without alerting the communicating parties (Alice and Bob). In this paper we study a quantum wiretap channel in which the connection from Alice to Eve is lossy, so that some of the transmitted quantum information is inaccessible to both Bob and Eve. For a pureloss channel under such restricted eavesdropping, we show that the key rates achievable with a two-mode squeezed vacuum state, heterodyne detection, and public classical communication assistance-given by the Hashing inequality-can exceed the secret key distillation capacity of the channel against an omnipotent eavesdropper. We report upper bounds on the key rates under the restricted eavesdropping model based on the relative entropy of entanglement, which closely match the achievable rates. For the pure-loss channel under restricted eavesdropping, we compare the secret-key rates of continuous-variable (CV) quantum key distribution (QKD) based on Gaussian-modulated coherent states and heterodyne detection with the discrete variable (DV) decoystate BB84 QKD protocol based on polarization qubits encoded in weak coherent laser pulses.

Depth profile reconstruction of a scene at low light levels using an active imaging setup has wide-ranging applications in remote sensing. In such low-light imaging scenarios, single-photon detectors are employed to time-resolve individual photon detections. However, even with single-photon detectors, current frameworks are limited to using hundreds of photon detections at each pixel to mitigate Poisson noise inherent in light detection. In this paper, we discuss two pixelwise imaging frameworks that allow accurate reconstruction of depth profiles using small numbers of photon detections. The first framework addresses the problem of depth reconstruction of an opaque target, in which it is assumed that each pixel contains exactly one reflector. The second framework addresses the problem of reconstructing multiple-depth pixels. In each scenario, our framework achieves photon efficiency by combining accurate statistics for individual photon detections with a longitudinal sparsity constraint tailored to the imaging problem. We demonstrate the photon efficiencies of our frameworks by comparing them with conventional imagers that use more naïve models based on high light-level assumptions.

Many partially-successful attempts have been made to find the most natural discrete-variable version of Shannon's entropy power inequality (EPI). We develop an axiomatic framework from which we deduce the natural form of a discrete-variable EPI and an associated entropic monotonicity in a discrete-variable central limit theorem. In this discrete EPI, the geometric distribution, which has the maximum entropy among all discrete distributions with a given mean, assumes a role analogous to the Gaussian distribution in Shannon's EPI. The entropy power of X is defined as the mean of a geometric random variable with entropy H(X). The crux of our construction is a discrete-variable version of Lieb's scaled addition X ???ηY of two random variables X and Y with η ∈ (0, 1). We discuss the relationship of our discrete EPI with recent work of Yu and Johnson who developed an EPI for a restricted class of random variables that have ultra-log-concave (ULC) distributions. Even though we leave open the proof of the aforesaid natural form of the discrete EPI, we show that this discrete EPI holds true for variables with arbitrary discrete distributions when the entropy power is redefined as eH(X)in analogy with the continuous version. Finally, we show that our conjectured discrete EPI is a special case of the yet-unproven Entropy Photon-number Inequality (EPnI), which assumes a role analogous to Shannon's EPI in capacity proofs for Gaussian bosonic (quantum) channels.

A single-pixel camera is able to computationally form spatially resolved images using one photodetector and a spatial light modulator. The images it produces in low-light-level operation are imperfect, even when the number of measurements exceeds the number of pixels, because its photodetection measurements are corrupted by Poisson noise. Conventional performance analysis for single-pixel imaging generates estimates of mean-square error (MSE) from Monte Carlo simulations, which require long computational times. In this letter, we use random matrix theory to develop a closed-form approximation to the MSE of the widely used least-squares inversion method for Poisson noise-limited single-pixel imaging. We present numerical experiments that validate our approximation and a motivating example showing how our framework can be used to answer practical optical design questions for a single-pixel camera.

Light detection and ranging systems reconstruct scene depth from time-of-flight measurements. For low light-level depth imaging applications, such as remote sensing and robot vision, these systems use single-photon detectors that resolve individual photon arrivals. Even so, they must detect a large number of photons to mitigate Poisson shot noise and reject anomalous photon detections from background light. We introduce a novel framework for accurate depth imaging using a small number of detected photons in the presence of an unknown amount of background light that may vary spatially. It employs a Poisson observation model for the photon detections plus a union-of-subspaces constraint on the discrete-time flux from the scene at any single pixel. Together, they enable a greedy signal-pursuit algorithm to rapidly and simultaneously converge on accurate estimates of scene depth and background flux, without any assumptions on spatial correlations of the depth or background flux. Using experimental single-photon data, we demonstrate that our proposed framework recovers depth features with 1.7 cm absolute error, using 15 photons per image pixel and an illumination pulse with 6.7-cm scaled root-mean-square length. We also show that our framework outperforms the conventional pixelwise log-matched filtering, which is a computationally-efficient approximation to the maximum-likelihood solution, by a factor of 6.1 in absolute depth error.

Capturing depth and reflectivity images at low light levels from active illumination of a scene has wide-ranging applications. Conventionally, even with single-photon detectors, hundreds of photon detections are needed at each pixel to mitigate Poisson noise. We introduce a robust method for estimating depth and reflectivity using on the order of 1 detected photon per pixel averaged over the scene. Our computational imager combines physically accurate single-photon counting statistics with exploitation of the spatial correlations present in real-world reflectivity and 3D structure. Experiments conducted in the presence of strong background light demonstrate that our computational imager is able to accurately recover scene depth and reflectivity, while traditional maximum likelihood-based imaging methods lead to estimates that are highly noisy. Our framework increases photon efficiency 100-fold over traditional processing and thus will be useful for rapid, low-power, and noise-tolerant active optical imaging.

Range estimation at low light-levels is accomplished using pulsed illumination of the target and time-of-flight measurement of backscattered light using single-photon detectors. Photon arrival statistics for this problem are time-inhomogeneous Poisson point processes where the rate function is determined by the illumination waveform. Given the flexibility to choose from different illumination waveforms, an important design question is - how does the range estimation performance depend on the pulse shape? The maximum-likelihood (ML) range estimation problem is nonlinear and thus it is difficult to analytically compare the estimation performance from different illumination waveforms. In this paper, we present an information-theoretic framework for evaluating ML range estimation performance. We derive relationships between the entropy of the photon arrival observations and the Cramér-Rao lower bound (CRLB) on the range estimate by extending De Brujin's identity and isoperimetric properties for non-Gaussian distributions.

We develop a theory of quantum M-ary phase shift keying in which quantum states of optical modes are modulated at the transmitter by applying one of M uniformly-spaced phase shifts. We allow full freedom in choosing modulation states with any number of signal, i.e., transmitted, and ancilla modes, subject only to an average energy, i.e., photon number, constraint in either the signal modes alone or in the signal and ancilla modes together. For lossless operation and unrestricted POVM measurements at the receiver, we find the explicit form of the modulation state that minimizes the average error probability under an energy constraint of N photons. Multiple signal modes, mixed states, and entanglement with an ancilla are shown to be unnecessary for optimum performance. We show that communication with zero error is possible if and only if N ≥ (M - 1)/2.

When classical information is sent over a channel with quantum-state modulation alphabet, such as the free-space optical (FSO) channel, attaining the ultimate (Holevo) limit to channel capacity requires the receiver to make joint measurements over long codeword blocks. In recent work, we showed a receiver for a pure-state channel that can attain the ultimate capacity by applying a single-shot optical (unitary) transformation on the received codeword state followed by simultaneous (but separable) projective measurements on the single-modulation-symbol state spaces. In this paper, we study the ultimate tradeoff between photon efficiency and spectral efficiency for the FSO channel. Based on our general results for the pure-state quantum channel, we show some of the first concrete examples of codes and laboratory-realizable joint-detection optical receivers that can achieve fundamentally higher (superadditive) channel capacity than receivers that physically detect each modulation symbol one at a time, as is done by all conventional (coherent or direct-detection) optical receivers.

The asymptotic capacity at low input powers of an average-power limited or an average- and peak-power limited discrete-time Poisson channel is considered. For a Poisson channel whose dark current is zero or decays to zero linearly with its average input powerε, capacity scales likeεlog 1/εfor smallε. For a Poisson channel whose dark current is a nonzero constant, capacity scales, to within a constant, likeεlog log 1/ε for smallε.

Determining the ultimate classical information carrying capacity of electromagnetic waves requires quantum-mechanical analysis to properly account for the bosonic nature of these waves. Recent work has established capacity theorems for bosonic single-user and broadcast channels, under the presumption of two minimum output entropy conjectures. Despite considerable accumulated evidence that supports the validity of these conjectures, they have yet to be proven. In this paper, it is shown that the second conjecture suffices to prove the classical capacity of the bosonic wiretap channel, which in turn would also prove the quantum capacity of the lossy bosonic channel. The preceding minimum output entropy conjectures are then shown to be simple consequences of an entropy photon-number inequality (EPnl), which is a conjectured quantum-mechanical analog of the entropy power inequality (EPI) from classical information theory.

We show that when coherent-state encoding is employed in conjunction with coherent detection, the Bosonic broadcast channel is equivalent to a classical degraded Gaussian broadcast channel whose capacity region is dual to that of the classical Gaussian multiple-access channel. We further show that if a minimum output-entropy conjecture holds true, then the ultimate classical information capacity of the Bosonic broadcast channel can be achieved by a coherent-state encoding. We provide some evidence in support of the conjecture.

Atmospheric optical communication systems that use optical preamplifiers and diversity reception are addressed. The particular diversity techniques that are investigated include aperture averaging, linear combining, and adaptive optics. On-off keying and binary pulse position modulation are considered. The effects of atmospheric turbulence, background light, source extinction ratio, amplified spontaneous emission, and receiver thermal noise are studied in the context of a semiclassical photon-counting approach. Numerical results are presented using a conditional Gaussian approximation method. By this method, we can measure the power penalty incurred under various operating conditions as well as the link margin improvement due to optical preamplification and diversity reception.

Most atmospheric optical links are set up to operate in the far-field power transfer regime, in which diffraction spread is the dominant effect on the beam, resulting in very weak power coupling between the transmitter and the receiver. However, it is also possible to establish geometries such that the link operates in the near-field regime, where, in the absence of turbulence, it is possible to focus the beam onto the receiver with almost perfect power coupling. Work on the performance of near-field atmospheric optical communication systems is scarce in existing literature, perhaps due to increased complexity in prescribed models. We analyze error probability bounds for optical communication links operating in the nearfield regime, utilizing on-off keying (OOK) and a coherent or direct detection receiver. In addition, we obtain bounds on the ergodic capacity of a near-field atmospheric optical link with local oscillator shot-noise dominated coherent detection, or shot-noise limited direct detection.

The capacity C for transmitting classical information is investigated for noisy bosonic channel models. An exact result is obtained for the pure-loss case. Upper and lower bounds are established for channels with active noise sources

Atmospheric optical communication systems that use optical preamplifiers and diversity reception to combat the effects of atmospheric turbulence are addressed. The effects of atmospheric turbulence, background light, source extinction ratio, amplified spontaneous emission, and receiver thermal noise are considered in a context of a semiclassical photon-counting approach. The particular diversity techniques that are investigated include aperture averaging, linear combining, and adaptive optics. Numerical results, obtained using a conditional Gaussian approximation, quantify the link margin improvement due to optical preamplification and diversity reception.

We consider the ergodic capacity and capacity-versus-outage probability of direct-detection optical communication through a turbulent atmosphere using multiple transmit and receive apertures. We assume shot-noise-limited operation in which detector outputs are doubly stochastic Poisson processes whose rates are proportional to the sum of the transmitted powers, scaled by lognormal random fades, plus a background noise. In the high and low signal-to-background ratio regimes, we show that the ergodic capacity of this fading channel equals or exceeds that for a channel with deterministic path gains. Furthermore, knowledge of these path gains is not necessary to achieve capacity when the signal-to-background ratio is high. In the low signal-to-background ratio regime, path-gain knowledge provides minimal capacity improvement when using a moderate number of transmit apertures. We also develop expressions for the capacity-versus-outage probability in the high and low signal-to-background ratio regimes, by means of a moment-matching approximation to the distribution for the sum of lognormal random variables. Monte Carlo simulations show that these capacity-versus-outage approximations are quite accurate for moderate numbers of apertures.

Maximum-likelihood range imaging is considered for pulsed-imager operation of a coherent laser radar. The expectation-maximization (EM) algorithm is used to develop an explicit procedure for maximum-likelihood fitting of a multiresolution (wavelet) basis-at a sequence of increasingly fine resolutions-to laser radar range data. Specialization to the Haar-wavelet basis yields a procedure that is both computationally efficient and numerically robust. Basic analytical properties of the estimation algorithm and its performance are presented, along with results based on simulated and real laser radar range data. It is shown that the weights associated with the expectation-maximization iterations provide a reliable indicator for terminating the coarse-to-fine resolution progression. At the weight-determined stopping point, estimation performance approaches the ultimate limit set by the complete-data bound.

An experimental 10-Mb/s token-ring local area network (LAN) that uses atmospheric optical communication links is described. Objective and subjective performance results are reported for operation over a 170-m-long outdoor path. The data support the following hypotheses. Atmospheric optical communications may find LAN applications as bridges between buildings containing cable subnetworks or as temporary quick connects for new outlying hosts for which cable runs are initially unavailable. LAN protocols for networks incorporating atmospheric optical communications should be modified to best insulate users from the occasional error bursts and bad-weather outages suffered by such links.>

Under severe rain conditions there may be significant amounts of multiply-scattered millimeter wave radiation present in the vicinity of the receiving antenna for a satellite downlink. This paper explores angle diversity reception of the scattered radiation as an alternative to spatial (multiple site) diversity for ameliorating rain attenuation. Results from scalar plane-wave transport theory for the attenuation, angle-of-arrival spread, multipath spread, and Doppler spread of the scattered field are summarized. When combined with a diversity reception communication analysis, these solutions imply that a large performance gain over nondiversity reception can be achieved if adaptive coherent combining can successfully be employed. Although this gain does not require multiple receiver sites, it does entail a transmission bandwidth reduction.

A recent result on the operator representation for photoemissive detection is used to derive the operator measured by Kennedy's near-optimum receiver for binary coherent-state quantum communication.

In Part I of this three-part study it was shown that the use of two-photon coherent state (TCS) radiation may yield siginificant performance gains in free-space optical communicatinn if the receiver makes a quantum measurement of a single field quadrature. In Part II it was shown that homodyne detection achieves the same signal-to-noise ratio as the quantum field quadrature measurement, thus providing a receiver which realizes the linear modulation TCS performance gain found in Part I. Furthermore, it was shown in Part il that ff homodyne detection does exactly correspond to the field quadrature measurement, then a large binary communication performance gain is afforded by homodyne detection of antipodal TCS signals. The full equivalence of honmdyne detection and single-quadrature field measurement, as well as that of heterodyne detection and two-quadrature field measurement, is established. Furthermore, a heterodyne configuration which uses a TCS image-band oscillator in addition to the usual coherent state local oscillator is studied. This coafiguration termed TCS heterodyne detection is shown to realize all the quantum measurements described by arbitrary TCS. The foregoing results are obtained by means of a representation theorem which shows that photoemissive detection realizes the photon flux density measurement.

In Part I of this three-part study, it was shown that novel quantum states, called two-photon coherent states (TCS), have significant potential for improving free-space optical communications. Because TCS radiation does not possess a classical analog, i.e., its diagonalP-representation is highly singular, the semiclassical conditional Poisson process model for direct detection is not applicable to TCS reception. In this paper, photoemissive detection of arbitrary quantized radiation fields is studied with incorporation of the nontrivial effects of detector quantum efficiency. General theorems are derived permitting the application of classical point process results to the detection and estimation of signals in arbitrary quantum states. These general theorems are applied to determining the performance of TCS optical communication systems that employ direct, heterodyne, or homodyne detection in binary decision as well as in linear modulation problems. It is shown that the use of TCS radiation with direct detection or heterodyne detection results in minimal performance increments over comparable coherent-state systems. Homodyne detection, however, can achieve the full TCS signal-to-noise ratio improvement predicted in Part I of this study. The increase in homodyne signal-to-noise ratio obtained by use of TCS radiation yields significant performance gains in both linear modulation and antipodal signal detection.

Recent theoretical work has shown that novel quantum states, called two-photon coherent states (TCS), have significant potential for improving free-space optical communications. The first part of a three-part study of the communication theory of TCS radiation is presented. The issues of quantum-field propagation and optimum quantum-state generation are addressed. In particular, the quantum analog of the classical paraxial diffraction theory for quasimonochromatic scalar waves is developed. This result, which describes the propagation of arbitrary quantum states as a boundary-value problem suitable for communication system analysis, is used to treat a number of quantum transmitter optimization problems. It is shown that, under near-field propagation conditions, a TCS transmitter maximizes field-measurement signal-to-noise ratio among all transmitter quantum states; the performance of the TCS system exceeds that for a conventional (coherent state) transmitter by a factor ofN_{s} + 1, whereN_{s}is the average number of signal photons (transmitter energy constraint). Under far-field propagation conditions, it is shown that use of a TCS local oscillator in the receiver can, in principle, attenuate field-measurement quantum noise by a factor equal to the diffraction loss of the channel, if appropriate spatial mode mixing can be achieved. These communcation results are derived by assuming that field-quadrature quantum measurement is performed. In part II of this study, photoemissive reception of TCS radiation will be considered; it will be shown therein that homodyne detection of TCS fields can realize the field-quadrature signal - to-noise ratio performance of part I. In part III, the relationships between photoemissive detection and general quantum measurements will be explored. In particular, a synthesis procedure will be obtained for realizing all the measurements described by arbitrary TCS.