Point Estimation of States of Finite Quantum Systems

The estimation of the density matrix of a $k$-level quantum system is studied when the parametrization is given by the real and imaginary part of the entries and they are estimated by independent measurements. It is established that the properties of the estimation procedure depend very much on the invertibility of the true state. In particular, in case of a pure state the estimation is less efficient. Moreover, several estimation schemes are compared for the unknown state of a qubit when one copy is measured at a time. It is shown that the average mean quadratic error matrix is the smallest if the applied observables are complementary. The results are illustrated by computer simulations.Comment: 16 pages, 5 figure

Covariance and Fisher information in quantum mechanics

Variance and Fisher information are ingredients of the Cramer-Rao inequality. We regard Fisher information as a Riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and Fisher information. In this approach we show that there is a kind of dual one-to-one correspondence between the candidates of the two concepts. We emphasis that Fisher informations are obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold.Comment: LATE

On Mutual Information in Multipartite Quantum States and Equality in Strong Subadditivity of Entropy

The challenge of equality in the strong subadditivity inequality of entropy is approached via a general additivity of correlation information in terms of nonoverlapping clusters of subsystems in multipartite states (density operators). A family of tripartite states satisfying equality is derived.Comment: 8 pages; Latex2e and Revtex

Asymptotics of Quantum Relative Entropy From Representation Theoretical Viewpoint

In this paper it was proved that the quantum relative entropy $D(\sigma \| \rho)$ can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of POVMs. The sequence of POVMs depends on $\rho$, but is independent of the choice of $\sigma$.Comment: LaTeX2e. 8 pages. The title was changed from "Asymptotic Attainment for Quantum Relative Entropy

Hamilton-Jacobi Theory and Information Geometry

Recently, a method to dynamically define a divergence function $D$ for a given statistical manifold $(\mathcal{M}\,,g\,,T)$ by means of the Hamilton-Jacobi theory associated with a suitable Lagrangian function $\mathfrak{L}$ on $T\mathcal{M}$ has been proposed. Here we will review this construction and lay the basis for an inverse problem where we assume the divergence function $D$ to be known and we look for a Lagrangian function $\mathfrak{L}$ for which $D$ is a complete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes.Comment: 8 page

A new quantum version of f-divergence

This paper proposes and studies new quantum version of $f$-divergences, a class of convex functionals of a pair of probability distributions including Kullback-Leibler divergence, Rnyi-type relative entropy and so on. There are several quantum versions so far, including the one by Petz. We introduce another quantum version ($\mathrm{D}_{f}^{\max}$, below), defined as the solution to an optimization problem, or the minimum classical $f$- divergence necessary to generate a given pair of quantum states. It turns out to be the largest quantum $f$-divergence. The closed formula of $\mathrm{D}_{f}^{\max}$ is given either if $f$ is operator convex, or if one of the state is a pure state. Also, concise representation of $\mathrm{D}_{f}^{\max}$ as a pointwise supremum of linear functionals is given and used for the clarification of various properties of the quality. Using the closed formula of $\mathrm{D}_{f}^{\max}$, we show: Suppose $f$ is operator convex. Then the\ maximum $f\,$- divergence of the probability distributions of a measurement under the state $\rho$ and $\sigma$ is strictly less than $\mathrm{D}_{f}^{\max}\left( \rho\Vert\sigma\right)$. This statement may seem intuitively trivial, but when $f$ is not operator convex, this is not always true. A counter example is $f\left( \lambda\right) =\left\vert 1-\lambda\right\vert$, which corresponds to total variation distance. We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.Comment: The proof of dual representation of the former version was misstated. An alternative proof is presente

Mapping ecosystem functions and services in Eastern Europe using global-scale data sets

To assess future interactions between the environment and human well-being, spatially explicit ecosystem service models are needed. Currently available models mainly focus on provisioning services and do not distinguish changes in the functioning of the ecosystem (Ecosystem Functions – ESFs) and human use of such functions (Ecosystem Services – ESSs). This limits the insight on the impact of global change on human well-being. We present a set of models for assessing ESFs and ESSs. We mapped a diverse set of provisioning, regulating and cultural services, focusing on services that depend on the landscape structure. Services were mapped using global-scale data sets. We evaluated the models for a sample area comprising Eastern Europe. ESFs are mainly available in natural areas, while hotspots of ESS supply are found in areas with heterogeneous land cover. Here, natural land cover where ESFs are available is mixed with areas where the ESSs are utilized. We conclude that spatial patterns of several ESFs and ESSs can be mapped at global scale using existing global-scale data sets. As land-cover change has different impacts on different aspects of the interaction between humans and the environment, it is essential to clearly distinguish between ESFs and ESSs in integrated assessment studies

Quantum hypothesis testing with group symmetry

The asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system. We consider various asymptotic error exponents in connection with the problems of the Chernoff bound, the Hoeffding bound and Stein's lemma, and derive bounds on these quantities in terms of their corresponding statistical distance measures. A special emphasis is put on the comparison of the performances of group-invariant and unrestricted measurements.Comment: 33 page

An entropic uncertainty principle for positive operator valued measures

Extending a recent result by Frank and Lieb, we show an entropic uncertainty principle for mixed states in a Hilbert space relatively to pairs of positive operator valued measures that are independent in some sense. This yields spatial-spectral uncertainty principles and log-Sobolev inequalities for invariant operators on homogeneous spaces, which are sharp in the compact case.Comment: 14 pages. v2: a technical assumption removed in main resul