▓ Local Randomness for True Random Number generators ▓ Non-local Randomness for distribution of Cryptographic Keys ▓ Towards faster, longer distances and cheaper QKD engines ▓ Quantum Repeaters ▓ Device-Independent Quantum Information Processing

Quantum communication

Information geometry is a new mathematical discipline which applies the methodology of differential geometry to statistics. Therefore, families of exponential distributions are considered as embedded manifolds, called statistical manifolds. This includes so important families like the multivariate normal or the gamma distributions. Fisher information — well known in information theory — becomes a metric on statistical manifolds. The Fisher information metric enables a hyperbolic structure on the multivariate normal distributions. Information geometry offers new methods for hypothesis testings, estimation theory or stochastic filtering. These can be used in engineering areas like signal processing or video processing or finance.

http://ieeexplore.ieee.org/iel5/6423297/6450609/06450636.pdf

Information geometry and its applications

General Instructions Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. If an electronic calculator is used, write its serial number in the box at the top right hand corner of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the horizontal box on the front cover of its corresponding answer book. Hand in THREE answer books even if they have not all been used. You are reminded that the Examiners attach great importance to legibility, accuracy and clarity of expression.

Foundations of Quantum Mechanics

The Jensen–Shannon divergence is a renowned bounded symmetrization of the unbounded Kullback–Leibler divergence which measures the total Kullback–Leibler divergence to the average mixture distribution. However, the Jensen–Shannon divergence between Gaussian distributions is not available in closed form. To bypass this problem, we present a generalization of the Jensen–Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using parameter mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen–Shannon divergence between probability densities of the same exponential family; and (ii) the geometric JS-symmetrization of the reverse Kullback–Leibler divergence between probability densities of the same exponential family. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen–Shannon divergence between scale Cauchy distributions. Applications to clustering with respect to these novel Jensen–Shannon divergences are touched upon.

https://www.mdpi.com/1099-4300/21/5/485/pdf

On the Jensen-Shannon Symmetrization of Distances Relying on Abstract Means.

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Elementary differential geometry

Motivated by a geometric decomposition of the vector field associated with the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact Hamiltonian systems for the description of dissipative-like dynamical systems in the context of (non-necessarily exact) contact manifolds. In particular, we show how this class of dynamical systems naturally emerges in the context of Lagrangian Mechanics and in the case of nonlinear evolutions on the space of pure states of a finite-level quantum system.

Contact manifolds and dissipation, classical and quantum

A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger’s algebra of selective measurements and helps to un...

Schwinger’s picture of quantum mechanics I: Groupoids

Schwinger’s algebra of selective measurements has a natural interpretation in the formalism of groupoids. Its kinematical foundations, as well as the structure of the algebra of observables of the ...

Schwinger's picture of quantum mechanics III : The statistical interpretation

The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appea...

Schwinger’s picture of quantum mechanics II: Algebras and observables

In this paper, we review Schwinger’s formulation of Quantum Mechanics and argue that the mathematical structure behind Schwinger’s “Symbolism of Atomic Measurements” is that of a groupoid. In this ...

A gentle introduction to Schwinger’s formulation of quantum mechanics: The groupoid picture

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Florio M. Ciaglia

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