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Entropic Geometry from Logic  Bob Coecke
;  Date: 
10 Dec 2002  Journal:  ENTCS  MFPS 2003  Subject:  Quantum Physics; Probability; Logic; Mathematical Physics  quantph grqc mathph math.LO math.MP math.PR  Abstract:  We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic + Partiality of Knowledge + Entropy. That is: 1. A finitary probability space Delta^n (=all probability measures on {1,...,n}) can be fully and faithfully represented by the pair consisting of the abstraction D^n (=the object up to isomorphism) of a partially ordered set (Delta^n,sqsubseteq), and, Shannon entropy; 2. D^n itself can be obtained via a systematic purely ordertheoretic procedure (which embodies introduction of partiality of knowledge) on an (algebraic) logic. This procedure applies to any poset A; D_Acong(Delta^n,sqsubseteq) when A is the nelement powerset and D_Acong(Omega^n,sqsubseteq), the domain of mixed quantum states, when A is the lattice of subspaces of a Hilbert space. (We refer to http://web.comlab.ox.ac.uk/oucl/publications/tr/rr0207.html for a domaintheoretic context providing the notions of approximation and content.)  Source:  arXiv, quantph/0212065  Services:  Forum  Review  PDF  Favorites 


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