Direct measurement of the Wigner function by photon counting

We report a direct measurement of the Wigner function characterizing the quantum state of a light mode. The experimental scheme is based on the representation of the Wigner function as an expectation value of a displaced photon number parity operator. This allowed us to scan the phase space point-by-point, and obtain the complete Wigner function without using any numerical reconstruction algorithms.Comment: 4 pages, REVTe

A measure of the non-Gaussian character of a quantum state

We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a reference Gaussian state. We analyze in details the properties of the proposed measure and exploit it to evaluate the non-Gaussianity of some relevant single- and multi-mode quantum states. The evolution of non-Gaussianity is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows to define a corresponding measure for the non-Gaussian character of a quantum operation.Comment: revised and enlarged version, 7 pages, 4 figure

Iterative maximum-likelihood reconstruction in quantum homodyne tomography

I propose an iterative expectation maximization algorithm for reconstructing a quantum optical ensemble from a set of balanced homodyne measurements performed on an optical state. The algorithm applies directly to the acquired data, bypassing the intermediate step of calculating marginal distributions. The advantages of the new method are made manifest by comparing it with the traditional inverse Radon transformation technique

Optimization of Bell's Inequality Violation For Continuous Variable Systems

Two mode squeezed vacuum states allow Bell's inequality violation (BIQV) for all non-vanishing squeezing parameter $(\zeta)$. Maximal violation occurs at $\zeta \to \infty$ when the parity of either component averages to zero. For a given entangled {\it two spin} system BIQV is optimized via orientations of the operators entering the Bell operator (cf. S. L. Braunstein, A. Mann and M. Revzen: Phys. Rev. Lett. {\bf68}, 3259 (1992)). We show that for finite $\zeta$ in continuous variable systems (and in general whenever the dimensionality of the subsystems is greater than 2) additional parameters are present for optimizing BIQV. Thus the expectation value of the Bell operator depends, in addition to the orientation parameters, on configuration parameters. Optimization of these configurational parameters leads to a unique maximal BIQV that depends only on $\zeta.$ The configurational parameter variation is used to show that BIQV relation to entanglement is, even for pure state, not monotonic.Comment: An example added; shows that the amount of Bell's inequality violation as a measure of entanglement is doubtfu

Maximum likelihood estimation of photon number distribution from homodyne statistics

We present a method for reconstructing the photon number distribution from the homodyne statistics based on maximization of the likelihood function derived from the exact statistical description of a homodyne experiment. This method incorporates in a natural way the physical constraints on the reconstructed quantities, and the compensation for the nonunit detection efficiency.Comment: 3 pages REVTeX. Final version, to appear in Phys. Rev. A as a Brief Repor

Effect of noise and enhancement of nonlocality in on/off photodetection

Nonlocality of two-mode states of light is addressed by means of CHSH inequality based on displaced on/off photodetection. Effects due to non-unit quantum efficiency and nonzero dark counts are taken into account. Nonlocality of both balanced and unbalanced superpositions of few photon-number states, as well as that of multiphoton twin beams, is investigated. We find that unbalanced superpositions show larger nonlocality than balanced one when noise affects the photodetection process. De-Gaussification by means of (inconclusive) photon subtraction is shown to enhance nonlocality of twin beams in the low energy regime. We also show that when the measurement is described by a POVM, rather than a set of projectors, the maximum achievable value of the Bell parameter in the CHSH inequality is decreased, and is no longer given by the Cirel'son bound.Comment: 21 Figure

Nonlocality of Two-Mode Squeezing with Internal Noise

We examine the quantum states produced through parametric amplification with internal quantum noise. The internal diffusion arises by coupling both modes of light to a reservoir for the duration of the interaction time. The Wigner function for the diffused two-mode squeezed state is calculated. The nonlocality, separability, and purity of these quantum states of light are discussed. In addition, we conclude by studying the nonlocality of two other continuous variable states: the Werner state and the phase-diffused state for two light modes.Comment: 7 pages, 5 figures, submitted to PR

Parameters estimation in quantum optics

We address several estimation problems in quantum optics by means of the maximum-likelihood principle. We consider Gaussian state estimation and the determination of the coupling parameters of quadratic Hamiltonians. Moreover, we analyze different schemes of phase-shift estimation. Finally, the absolute estimation of the quantum efficiency of both linear and avalanche photodetectors is studied. In all the considered applications, the Gaussian bound on statistical errors is attained with a few thousand data.Comment: 11 pages. 6 figures. Accepted on Phys. Rev.