Conditional states and joint distributions on MV-algebras

summary:In this paper we construct conditional states on semi-simple MV-algebras. We show that these conditional states are not given uniquely. By using them we construct the joint probability distributions and discuss the properties of these distributions. We show that the independence is not symmetric

Binary hidden Markov models and varieties

The technological applications of hidden Markov models have been extremely diverse and successful, including natural language processing, gesture recognition, gene sequencing, and Kalman filtering of physical measurements. HMMs are highly non-linear statistical models, and just as linear models are amenable to linear algebraic techniques, non-linear models are amenable to commutative algebra and algebraic geometry. This paper closely examines HMMs in which all the hidden random variables are binary. Its main contributions are (1) a birational parametrization for every such HMM, with an explicit inverse for recovering the hidden parameters in terms of observables, (2) a semialgebraic model membership test for every such HMM, and (3) minimal defining equations for the 4-node fully binary model, comprising 21 quadrics and 29 cubics, which were computed using Grobner bases in the cumulant coordinates of Sturmfels and Zwiernik. The new model parameters in (1) are rationally identifiable in the sense of Sullivant, Garcia-Puente, and Spielvogel, and each model's Zariski closure is therefore a rational projective variety of dimension 5. Grobner basis computations for the model and its graph are found to be considerably faster using these parameters. In the case of two hidden states, item (2) supersedes a previous algorithm of Schonhuth which is only generically defined, and the defining equations (3) yield new invariants for HMMs of all lengths $\geq 4$. Such invariants have been used successfully in model selection problems in phylogenetics, and one can hope for similar applications in the case of HMMs

Algebraic conformal quantum field theory in perspective

Conformal quantum field theory is reviewed in the perspective of Axiomatic, notably Algebraic QFT. This theory is particularly developped in two spacetime dimensions, where many rigorous constructions are possible, as well as some complete classifications. The structural insights, analytical methods and constructive tools are expected to be useful also for four-dimensional QFT.Comment: Review paper, 40 pages. v2: minor changes and references added, so as to match published versio

Nonlinear spectral analysis: A local Gaussian approach

The spectral distribution $f(\omega)$ of a stationary time series $\{Y_t\}_{t\in\mathbb{Z}}$ can be used to investigate whether or not periodic structures are present in $\{Y_t\}_{t\in\mathbb{Z}}$, but $f(\omega)$ has some limitations due to its dependence on the autocovariances $\gamma(h)$. For example, $f(\omega)$ can not distinguish white i.i.d. noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that $f(\omega)$ can be an inadequate tool when $\{Y_t\}_{t\in\mathbb{Z}}$ contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the local Gaussian autocorrelations introduced in Tj{\o}stheim and Hufthammer (2013), and these local measures of dependence can be used to construct the local Gaussian spectral density presented in this paper. A key feature of the new local spectral density is that it coincides with $f(\omega)$ for Gaussian time series, which implies that it can be used to detect non-Gaussian traits in the time series under investigation. In particular, if $f(\omega)$ is flat, then peaks and troughs of the new local spectral density can indicate nonlinear traits, which potentially might discover local periodic phenomena that remain undetected in an ordinary spectral analysis.Comment: Version 4: Major revision from version 3, with new theory/figures. 135 pages (main part 32 + appendices 103), 11 + 16 figure

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold. The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X -> 1+1 carry such effect module structure, and can be used as predicates. Predicates are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays an important role. It appears here in the form of an adjunction, where we use maps 1 -> X as states. For such a state s and a predicate p, the validity probability s |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of `instruments', using L\"uders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C*-algebras, side-effect appear exclusively in the non-commutative case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems

Invertibility of random matrices: unitary and orthogonal perturbations

We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is close to orthogonal. As an application, these results completely eliminate a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur and Zeitouni.Comment: 46 pages. A more general result on orthogonal perturbations of complex matrices added. It rectified an inaccuracy in application to Single Ring Theorem for orthogonal matrice

Different quantum f-divergences and the reversibility of quantum operations

The concept of classical $f$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard $f$-divergences), Matsumoto's maximal $f$-divergences, measured $f$-divergences, and sandwiched and $\alpha$-$z$-R\'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum $f$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum $f$-divergence by a quantum operation. In particular, we compare the standard and the maximal $f$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum $f$-divergences are strictly different for non-commuting operators unless $f$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard $f$-divergences. We also study the monotonicity of the $\alpha$-$z$-R\'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\'enyi divergence invariant, and determine domains of the parameters $\alpha,z$ where monotonicity holds, and where the preservation of the $\alpha$-$z$-R\'enyi divergence implies the reversibility of the quantum operation.Comment: 70 pages. v4: New Proposition 3.8 and Appendix D on the continuity properties of the standard f-divergences. The 2-positivity assumption removed from Theorem 3.34. The achievability of the measured f-divergence is shown in Proposition 4.17, and Theorem 4.18 is updated accordingl