Entropic geometry from logic
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Publication:276466
DOI10.1016/S1571-0661(03)50003-2zbMATH Open1338.81033arXivquant-ph/0212065MaRDI QIDQ276466
Publication date: 3 May 2016
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 order-theoretic 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 n-element 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/rr-02-07.html for a domain-theoretic context providing the notions of approximation and content.)
Full work available at URL: https://arxiv.org/abs/quant-ph/0212065
Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) (81P10) Measures of information, entropy (94A17) Quantum state spaces, operational and probabilistic concepts (81P16)
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