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

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

Mapping and modelling the effects of land use and land management change on ecosystem services from local ecosystems and landscapes to global biomes

Herstel en duurzaam beheer van biodiversiteit en ecosysteemdiensten worden steeds meer geïntegreerd in nationaal en internationaal beleid. In dit proefschrift wordt een methodologie ontwikkeld voor de kwantificering van effecten van landmanagement op de ruimtelijke verspreiding van ecosysteemdiensten, zodat de door landmanagement veroorzaakte trade-offs tussen ecosysteemdiensten bepaald kunnen worden voor zowel lokale ecosystemen en landschappen als regionale en mondiale biomen. Een groot aantal ecosysteemdiensten zijn bestudeerd. De karterings- en modelleringsmethoden zijn toegepast en gecombineerd met scenario-analyse in de Nederlandse en Zuid-Afrikaanse studies. Voor Nederland is het landschap van Het Groene Woud bestudeerd

Continuity and Stability of Partial Entropic Sums

Extensions of Fannes' inequality with partial sums of the Tsallis entropy are obtained for both the classical and quantum cases. The definition of kth partial sum under the prescribed order of terms is given. Basic properties of introduced entropic measures and some applications are discussed. The derived estimates provide a complete characterization of the continuity and stability properties in the refined scale. The results are also reformulated in terms of Uhlmann's partial fidelities.Comment: 9 pages, no figures. Some explanatory and technical improvements are made. The bibliography is extended. Detected errors and typos are correcte

Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs

We prove multiplicativity of maximal output $p$ norm of classical noise channels and thermal noise channels of arbitrary modes for all $p>1$ under the assumption that the input signal states are Gaussian states. As a direct consequence, we also show the additivity of the minimal output entropy and that of the energy-constrained Holevo capacity for those Gaussian channels under Gaussian inputs. To the best of our knowledge, newly discovered majorization relation on symplectic eigenvalues, which is also of independent interest, plays a central role in the proof.Comment: 9 pages, no figures. Published Versio

A volume inequality for quantum Fisher information and the uncertainty principle

Let $A_1,...,A_N$ be complex self-adjoint matrices and let $\rho$ be a density matrix. The Robertson uncertainty principle $$det(Cov_\rho(A_h,A_j)) \geq det(- \frac{i}{2} Tr(\rho [A_h,A_j]))$$ gives a bound for the quantum generalized covariance in terms of the commutators $[A_h,A_j]$. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case $N=2m+1$. Let $f$ be an arbitrary normalized symmetric operator monotone function and let $_{\rho,f}$ be the associated quantum Fisher information. In this paper we conjecture the inequality $$det (Cov_\rho(A_h,A_j)) \geq det (\frac{f(0)}{2} _{\rho,f})$$ that gives a non-trivial bound for any natural number $N$ using the commutators $i[\rho, A_h]$. The inequality has been proved in the cases $N=1,2$ by the joint efforts of many authors. In this paper we prove the case N=3 for real matrices

Quantum Chi-Squared and Goodness of Fit Testing

The density matrix in quantum mechanics parameterizes the statistical properties of the system under observation, just like a classical probability distribution does for classical systems. The expectation value of observables cannot be measured directly, it can only be approximated by applying classical statistical methods to the frequencies by which certain measurement outcomes (clicks) are obtained. In this paper, we make a detailed study of the statistical fluctuations obtained during an experiment in which a hypothesis is tested, i.e. the hypothesis that a certain setup produces a given quantum state. Although the classical and quantum problem are very much related to each other, the quantum problem is much richer due to the additional optimization over the measurement basis. Just as in the case of classical hypothesis testing, the confidence in quantum hypothesis testing scales exponentially in the number of copies. In this paper, we will argue 1) that the physically relevant data of quantum experiments is only contained in the frequencies of the measurement outcomes, and that the statistical fluctuations of the experiment are essential, so that the correct formulation of the conclusions of a quantum experiment should be given in terms of hypothesis tests, 2) that the (classical) $\chi^2$ test for distinguishing two quantum states gives rise to the quantum $\chi^2$ divergence when optimized over the measurement basis, 3) present a max-min characterization for the optimal measurement basis for quantum goodness of fit testing, find the quantum measurement which leads both to the maximal Pitman and Bahadur efficiency, and determine the associated divergence rates.Comment: 22 Pages, with a new section on parameter estimatio