Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e. the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology.

We extend the quantum discord to continuous variable systems and evaluate Gaussian quantum discord C(̺) for bipartite Gaussian states. In particular, for squeezed thermal states (STS), we explicitly maximize the extractable information over Gaussian measurements: C(̺) is minimized by a generalized measurement rather than a projective one. Almost all STS have nonzero Gaussian discord: they may be either separable or entangled if the discord is below the threshold C(̺) = 1, whereas they are all entangled above the threshold. We elucidate the general role of state parameters in determining the discord and discuss its evolution in noisy channels.PACS numbers: 03.67.-a, 03.65.Ta Quantum correlations have been the subject of intensive studies in the last two decades, mainly due to the general belief that they are a fundamental resource for quantum information processing tasks. The first rigorous attempt to address the classification of quantum correlation from has been put forward by Werner [1], who put on firm basis the elusive concept of quantum entanglement. A state of a bipartite system is called entangled if it cannot be written as follows:where ̺ Ak and ̺ Bk are generic density matrices describing the states of the two subsystems. The definition above has an immediate operational interpretation: separable states can be prepared by local operations and classical communication between the two parties, whereas entangled states cannot. One might have thought that such classical information exchange could not bring any quantum character to the correlations in the state. In this sense separability has often been regarded as a synonymous of classicality. However, it has been shown that this is not the case [2,3]. A measure of correlations -quantum discord-has been defined as the mismatch between two quantum analogues of classically equivalent expression of the mutual information. For pure entangled states quantum discord coincides with the entropy of entanglement. However, quantum discord can be different from zero also for some (mixed) separable state. In other words, classical communication can give rise to quantum correlations. This can be understood by considering that the states ̺ Ak and ̺ Bk above may be physically non distinguishable, i.e. non-orthogonal and thus not all the information about them can then be locally retrieved. This phenomenon has no classical counterpart, thus accounting for the quantumness of the correlations in separable state with positive discord. Quantum discord has been shown to be a property hold by almost all quantum states [4] and recently attracted considerable attention [5][6][7][8][9][10]. In particular, the vanishing of quantum discord between two systems has been shown to be a requirement for the complete positivity of the reduced subsystem dynamics [11].While the discord is a fundamental notion allowing for the description of the quantumness of the correlations present in the state of a quantum system, its evaluation requires an optimization procedure over the set of all measurem...

We present a universal technique for quantum state estimation based on the maximum-likelihood method. This approach provides a positive definite estimate for the density matrix from a sequence of measurements performed on identically prepared copies of the system. The method is versatile and can be applied to multimode radiation fields as well as to spin systems. The incorporation of physical constraints, which is natural in the maximum-likelihood strategy, leads to a substantial reduction of statistical errors. Numerical implementation of the method is based on a particular form of the Gauss decomposition for positive definite Hermitian matrices.Comment: 4 pages, 3 figures (5 eps files). Submitted to Phys. Rev. A as a Rapid Communicatio