Contents 10 Quantum Shannon Theory 1 10.1 Shannon for Dummies 2 10.1.1 Shannon entropy and data compression 2 10.1.2 Joint typicality, conditional entropy, and mutual information 6 10.1.3 Distributed source coding 8 10.1.4 The noisy channel coding theorem 9 10.2 Von Neumann Entropy 16 10.2.1 Mathematical properties of H(ρ) 18 10.2.2 Mixing, measurement, and entropy 20 10.2.3 Strong subadditivit...

In the search for robust, accurate, and highly efficient financial option valuation techniques, we here present the SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. The nature of the local Shannon wavelets basis enables us to adaptively determine the proper s...

The aim of this thesis is to formulate and prove quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the Shannon-McMillan-Breiman theorem is one of the fundamental limit theorems for classical discrete dynamical systems. It can be interpreted as a special case of the individual ergodic theorem. In this work, we consider spin latti...

In the search for robust, accurate, and highly efficient financial option valuation techniques, we here present the SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. The nature of the local Shannon wavelets basis enables us to adaptively determine the proper s...

We introduce a novel parametric family of symmetric information-theoretic distances based on Jensen’s inequality for a convex functional generator. In particular, this family unifies the celebrated Jeffreys divergence with the Jensen-Shannon divergence when the Shannon entropy generator is chosen. We then design a generic algorithm to compute the unique centroid defined as the minimum average d...

The calculation of a statistical measure of complexity and the Fisher-Shannon information in nuclei is carried out in this work. We use the nuclear shell model in order to obtain the fractional occupation probabilities of nuclear orbitals. The increasing of both magnitudes, the statistical complexity and the Fisher-Shannon information, with the number of nucleons is observed. The shell structur...

We tackle the problem of estimating the Shannon capacity of cycles of odd length. We present some strategies which allow us to nd tight bounds on the Shannon capacity of cycles of various odd lengths, and suggest that the diiculty of obtaining a general result may be related to diierent behaviours of the capacity, depending on the \structure" of the odd integer representing the cycle length. We...

Guesswork forms the mathematical framework for quantifying computational security subject to brute-force determination by query. In this paper, we consider guesswork subject to a per-symbol Shannon entropy budget. We introduce inscrutability rate to quantify the asymptotic difficulty of guessing U out of V secret strings drawn from the string-source and prove that the inscrutability rate of any...

Constraints on the entropy function are of fundamental importance in information theory. For a long time, the polymatroidal axioms, or equivalently the nonnegativity of the Shannon information measures, are the only known constraints. Inequalities that are implied by nonnegativity of the Shannon information measures are categorically referred to as Shannontype inequalities. If the number of ran...