Probability tables

The idea of writing a table of probabilistic data for a quantum or classical system, and of decomposing this table in a compact way, leads to a shortcut for Hardy's formalism, and gives new perspectives on foundational issues.Comment: LaTeX, 17 pages, 1 figure. Contribution to the conference ``Quantum Theory: Reconsideration of Foundations 2'' (Vaexjoe, Sweden, 2003). To appear in the Proceedings (the notation in this version has been slightly modified, and the references updated

On the relation between plausibility logic and the maximum-entropy principle: a numerical study

What is the relationship between plausibility logic and the principle of maximum entropy? When does the principle give unreasonable or wrong results? When is it appropriate to use the rule `expectation = average'? Can plausibility logic give the same answers as the principle, and better answers if those of the principle are unreasonable? To try to answer these questions, this study offers a numerical collection of plausibility distributions given by the maximum-entropy principle and by plausibility logic for a set of fifteen simple problems: throwing dice.Comment: 24 pages of main text and references, 8 pages of tables, 7 pages of additional reference

On distinguishability, orthogonality, and violations of the second law: contradictory assumptions, contrasting pieces of knowledge

Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from _two_ different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.Comment: LaTeX2e/RevTeX4, 18 pages, 6 figures. V2: Important revisio

Numerical Bayesian state assignment for a three-level quantum system. I. Absolute-frequency data; constant and Gaussian-like priors

This paper offers examples of concrete numerical applications of Bayesian quantum-state-assignment methods to a three-level quantum system. The statistical operator assigned on the evidence of various measurement data and kinds of prior knowledge is computed partly analytically, partly through numerical integration (in eight dimensions) on a computer. The measurement data consist in absolute frequencies of the outcomes of N identical von Neumann projective measurements performed on N identically prepared three-level systems. Various small values of N as well as the large-N limit are considered. Two kinds of prior knowledge are used: one represented by a plausibility distribution constant in respect of the convex structure of the set of statistical operators; the other represented by a Gaussian-like distribution centred on a pure statistical operator, and thus reflecting a situation in which one has useful prior knowledge about the likely preparation of the system. In a companion paper the case of measurement data consisting in average values, and an additional prior studied by Slater, are considered.Comment: 23 pages, 14 figures. V2: Added an important note concerning cylindrical algebraic decomposition and thanks to P B Slater, corrected some typos, added reference

The Laplace-Jaynes approach to induction

An approach to induction is presented, based on the idea of analysing the context of a given problem into `circumstances'. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti's representation theorem and on the notion of infinite exchangeability. In particular, it gives an alternative interpretation of those formulae that apparently involve `unknown probabilities' or `propensities'. Various advantages and applications of the presented approach are discussed, especially in comparison to that based on exchangeability. Generalisations are also discussed.Comment: 38 pages, 1 figure. V2: altered discussion on some points, corrected typos, added reference

Numerical Bayesian quantum-state assignment for a three-level quantum system. II. Average-value data with a constant, a Gaussian-like, and a Slater prior

This paper offers examples of concrete numerical applications of Bayesian quantum-state assignment methods to a three-level quantum system. The statistical operator assigned on the evidence of various measurement data and kinds of prior knowledge is computed partly analytically, partly through numerical integration (in eight dimensions) on a computer. The measurement data consist in the average of outcome values of N identical von Neumann projective measurements performed on N identically prepared three-level systems. In particular the large-N limit will be considered. Three kinds of prior knowledge are used: one represented by a plausibility distribution constant in respect of the convex structure of the set of statistical operators; another one represented by a prior studied by Slater, which has been proposed as the natural measure on the set of statistical operators; the last prior is represented by a Gaussian-like distribution centred on a pure statistical operator, and thus reflecting a situation in which one has useful prior knowledge about the likely preparation of the system. The assigned statistical operators obtained with the first two kinds of priors are compared with the one obtained by Jaynes' maximum entropy method for the same measurement situation. In the companion paper the case of measurement data consisting in absolute frequencies is considered.Comment: 10 pages, 4 figures. V2: added "Post scriptum" under Conclusions, slightly changed Acknowledgements, and corrected some spelling error

A Creative Review on Integer Additive Set-Valued Graphs

For a non-empty ground set $X$, finite or infinite, the {\em set-valuation} or {\em set-labeling} of a given graph $G$ is an injective function $f:V(G) \to \mathcal{P}(X)$, where $\mathcal{P}(X)$ is the power set of the set $X$. A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \to \mathcal{P}(X)$ such that the function $f^{\ast}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\ast}(uv) = f(u){\ast} f(v)$ for every $uv{\in} E(G)$ is also injective, where $\ast$ is a binary operation on sets. An integer additive set-indexer is defined as an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}_0})$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective, where $\mathbb{N}_0$ is the set of all non-negative integers. In this paper, we critically and creatively review the concepts and properties of integer additive set-valued graphs.Comment: 14 pages, submitted. arXiv admin note: text overlap with arXiv:1312.7672, arXiv:1312.767