A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the qq-zz Family

The so-called $q$-z-\textit{R\'enyi Relative Entropies} provide a huge two-parameter family of relative entropies which includes almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant $(0,2)$ symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined differential forms on a suitable unfolding space without the need to introduce a specific set of coordinates. To make the reader acquainted with the intrinsic formalism introduced, we first perform the computation for the qubit case, and then, we extend the computation of the metric-like tensors to a generic $n$-level system. By suitably varying the parameters $q$ and $z$, we are able to recover well-known examples of quantum metric tensors that, in our treatment, appear written in terms of globally defined geometrical objects that do not depend on the coordinates system used. In particular, we obtain a coordinate-free expression for the von Neumann-Umegaki metric, for the Bures metric and for the Wigner-Yanase metric in the arbitrary $n$-level case.Comment: 50 pages, 1 figur

Dynamical aspects in the Quantizer-Dequantizer formalism

The use of the quantizer-dequantizer formalism to describe the evolution of a quantum system is reconsidered. We show that it is possible to embed a manifold in the space of quantum states of a given auxiliary system by means of an appropriate quantizer-dequantizer system. If this manifold of states is invariant with respect to some unitary evolution, the quantizer-dequantizer system provides a classical-like realization of such dynamics, which in general is non linear. Integrability properties are also discussed. Weyl systems and generalized coherente states are used as a simple illustration of these ideas.Comment: 15 page

Schwinger's Picture of Quantum Mechanics IV: Composition and independence

The groupoids description of Schwinger's picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger's groupoid of the system. The groupoids picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, as well as the notion of free independence, find a natural realm in the groupoids formalism. The ideas described in this paper will be illustrated by using the EPRB experiment. It will be observed that, in addition to the notion of the non-separability provided by the entangled state of the system, there is an intrinsic `non-separability' associated to the impossibility of identifying the entangled particles as subsystems of the total system.Comment: 32 pages. Comments are welcome

Covariant Jacobi Brackets for Test Particles

We show that the space of observables of test particles carries a natural Jacobi structure which is manifestly invariant under the action of the Poincar\'{e} group. Poisson algebras may be obtained by imposing further requirements. A generalization of Peierls procedure is used to extend this Jacobi bracket on the space of time-like geodesics on Minkowski space-time.Comment: 13 pages Submitted to MPL

Geometry from divergence functions and complex structures

Motivated by the geometrical structures of quantum mechanics, we introduce an almost-complex structure $J$ on the product $M\times M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a metric-like tensor on $M\times M$ from a divergence function. These tensors may be pulled back to $M$, and we compute them in the case of an N-dimensional symplex with respect to the Kullback-Leibler relative entropy, and in the case of (a suitable unfolding space of) the manifold of faithful density operators with respect to the von Neumann-Umegaki relative entropy.Comment: 19 pages, comments are welcome

Covariant Variational Evolution and Jacobi Brackets: Fields

The analysis of the covariant brackets on the space of functions on the solutions to a variational problem in the framework of contact geometry initiated in the companion letter Ref.19 is extended to the case of the multisymplectic formulation of the free Klein-Gordon theory and of the free Schr\"{o}dinger equation.Comment: 16 page

First assessment of age and sex structures of elephants by using dung size analysis in a West African savannah

1. An age and sex structure study of the West African savannah elephant (Loxodonta africana africana) population of the Nazinga Game ranch (Burkina Faso) was carried out using direct sightings of individuals near water points, where they group during hot periods of the day, and droppings circumference measurements of dung pile along line transects, during the dry seasons of 2007 and 2008. The age structure, from direct observation data, was estimated by classifying the individuals into 9 age-class-categories based on body size and eruption (length) of tusks.2. The frequency distribution of number of individuals into the age-class-categories showed high similarity with the distribution of circumference measurements of dung-piles that the individuals were seen depositing. Fifty dung-piles measured soon after they were deposited (Class A) and after they dried (to class C) showed a significant difference between the circumference measurements in relation to the time elapsed between the measurements.3. The frequency distribution of dung circumference classes of the observed elephants was similar to that of the dung-piles measured along the transects in 2008. Therefore, dung-piles measurements were used to estimate the Nazinga Game Ranch savanna elephant population’s age and sex structures at the end of dry seasons of 2007 and 2008. We advocated that the Nazinga elephant population consisted mainly of sub-adults.4. The sex ratio was estimated to be in favour of females (1/2). The age-class-specific sex ratio was uneven for calves and young individuals, while being in favour of females with adults. Individuals of less than 1 year represented 6% of the population during the study period

CT-based tumour response criteria compared after combined treatment for liver metastases of colorectal cancer

open6noPurpose: The aim of this analysis is to compare different tumour response criteria (TRC) after chemotherapy combined with bevacizumab in liver metastases from colorectal cancer (mCRC) to ascertain the best early prognostic indicator of response. Methods and Materials: 103 target liver metastases from 65 mCRC patients treated with chemoterapy plus bevacizumab were examined at the Istituto Oncologico Veneto IOV-IRCSS (March 2008-January 2013). All patients had baseline CT and at least one follow-up scan. Tumour response was retrospectively analyzed by two radiologists using RECIST1.1, modified Choi, and Chun morphologic criteria. Tumour response, classified as good (complete or partial response) or poor (stable or progressive disease), was compared with progression-free survival (PFS) at first follow-up (t1) and time of best response. Interobserver agreement and concordance between TRC were measured. Results: At t1, 32.31% showed a good response according to RECIST1.1 (median PFS 11.1), 84.62% according to Choi (median PFS 10.8). These percentages rose to 49.23% (median PFS 12.1) and 87.69% (median PFS 10.8), respectively, at the time of best response. According to Chun, 67.69% showed a good response at the time of best response (median PFS 10.8). The Choi criteria detected a higher proportion of good responders at t1, showing a better correlation with PFS; all methods correlated with PFS at the time of best response. Conclusion: The Choi criteria proved more consistent in the early detection of response in mCRC treated with chemotherapy plus bevacizumab, underscoring the importance of using these criteria in the early assessment of response to combined treatment.openopenVarotto, A.; Di Grazia, L.; Aliberti, C.; Bergamo, F.; Nardin, M.; Pomerri, F.Varotto, A.; Di Grazia, L.; Aliberti, C.; Bergamo, F.; Nardin, M.; Pomerri, Fabi