We perform a reconstruction of the polarization sector of the density matrix of an intense polarization squeezed beam starting from a complete set of Stokes measurements. By using an appropriate quasidistribution, we map this onto the Poincaré space providing a full quantum mechanical characterization of the measured polarization state.

We establish a simple method to assess the quantum Fisher information required for resolving two incoherent point sources with an imaging system. The resulting Cramér-Rao bound shows that the standard Rayleigh limit can be surpassed by suitable coherent measurements. We explicitly find these optimal strategies and present a realization for Gaussian and slit apertures. This involves a projection onto the optimal bases that is accomplished with digital holographic techniques and is compared with a CCD position measurement. Our experimental results unequivocally confirm unprecedented sub-Rayleigh precision.

We introduce a unitary operator representing the exponential of the phase difference between two modes of the electromagnetic field. The eigenvalue spectrum has a discrete character that is fully analyzed. We relate this operator with a suitable polar decomposition of the Stokes parameters of the field, obtaining a natural classical limit. The cases of weakly and highly excited states are considered, discussing to what extent it is possible to talk about the phase for a single-mode field. This operator is applied to some interesting two-mode fields.

We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost important in realistic scenarios, such as observational astronomy.The time-honored Rayleigh criterion [1] specifies the minimum separation between two incoherent optical sources using a linear imaging system. As a matter of fact, it is the size of the point spread function [2] that determines the resolution: two points closer than the PSF width will be difficult to resolve due to the substantial overlap of their images.Thus far, this Rayleigh criterion has been considered as a fundamental limit. Resolution can only be improved either by reducing the wavelength or by building higher numericalaperture optics, thereby making the PSF narrower. Nonetheless, outstanding methods have been developed lately that can break the Rayleigh limit under special circumstances [3][4][5][6][7][8][9][10][11][12]. Though promising, these techniques are involved and require careful control of the source, which is not always possible, especially in astronomical applications.Despite being very intuitive, the common derivation of the Rayleigh limit is heuristic and it is deeply rooted in classical optical technology [13]. Recently, inspired by ideas of quantum information, Tsang and coworkers [14][15][16][17] have revisited this problem using the Fisher information and the associated Cramér-Rao lower bound (CRLB) to quantify how well the separation between two point sources can be estimated. When only the intensity at the image is measured (the basis of all the conventional techniques), the Fisher information falls to zero as the separation between the sources decreases and the CRLB diverges accordingly; this is known as the Rayleigh curse [14]. However, when the Fisher information of the complete field is calculated, it stays constant and so does the CRLB, revealing that the Rayleigh limit is not essential to the problem.These remarkable predictions prompted a series of experimental implementations [18][19][20] and further generalizations [21][22][23][24][25], including the related question of source localization [26][27][28]. All this previous work has focused on the estimation of the separation, taking for granted a highly symmetric configuration with identical sources. In this Letter, we approach the issue in a more realistic scenario, where both sources may have unequal intensities. This involves the si...

A complete set of d + 1 mutually unbiased bases exists in a Hilbert spaces of dimension d, whenever d is a power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements. One of these classes is diagonal, and can be mapped to cyclic operators by means of the finite Fourier transform, which allows one to understand complementarity in a similar way as for the position-momentum pair in standard quantum mechanics. The relevant examples of two and three qubits and two qutrits are discussed in detail.

Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension. In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime. Fidelities of reconstruction within the range of 0.81-0.96 are obtained for both pure and mixed states. These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown

We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as superlattices, photonic crystals, and optical resonators. In all these cases, the transfer matrix has the same algebraic properties as the Lorentz group in a (2 + 1)-dimensional spacetime, as well as the group of unimodular real matrices underlying the structure of the abcd law, which explains many subtle details. We elaborate on the geometrical interpretation of the transfer-matrix action as a mapping on the unit disk and apply a simple trace criterion to classify the systems into three types with very different geometrical and physical properties. This approach is applied to some practical examples and, in particular, an alternative framework to deal with periodic (and quasiperiodic) systems is proposed.

For a system of N qubits, spanning a Hilbert space of dimension d = 2 N , it is known that there exists d + 1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive the four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of sixteen sets of bases,and show some of them, and their interrelations, as examples. The extension of the method to the general case of N qubits is outlined.