dorsal/arxiv
View SchemaEntropic Geometry from Logic
| Authors | Bob Coecke |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0212065 |
| URL | https://arxiv.org/abs/quant-ph/0212065 |
| Journal | ENTCS - MFPS 2003 |
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_A\cong(\Delta^n,\sqsubseteq) when A is the n-element powerset and D_A\cong(\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.)
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"abstract": "We produce a probabilistic space from logic, both classical and quantum,\nwhich is in addition partially ordered in such a way that entropy is monotone.\nIn particular do we establish the following equation:\n Quantitative Probability = Logic + Partiality of Knowledge + Entropy.\n That is: 1. A finitary probability space \\Delta^n (=all probability measures\non {1,...,n}) can be fully and faithfully represented by the pair consisting of\nthe abstraction D^n (=the object up to isomorphism) of a partially ordered set\n(\\Delta^n,\\sqsubseteq), and, Shannon entropy; 2. D^n itself can be obtained via\na systematic purely order-theoretic procedure (which embodies introduction of\npartiality of knowledge) on an (algebraic) logic. This procedure applies to any\nposet A; D_A\\cong(\\Delta^n,\\sqsubseteq) when A is the n-element powerset and\nD_A\\cong(\\Omega^n,\\sqsubseteq), the domain of mixed quantum states, when A is\nthe lattice of subspaces of a Hilbert space.\n (We refer to http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.html\nfor a domain-theoretic context providing the notions of approximation and\ncontent.)",
"arxiv_id": "quant-ph/0212065",
"authors": [
"Bob Coecke"
],
"categories": [
"quant-ph",
"gr-qc",
"math-ph",
"math.LO",
"math.MP",
"math.PR"
],
"journal_ref": "ENTCS - MFPS 2003",
"title": "Entropic Geometry from Logic",
"url": "https://arxiv.org/abs/quant-ph/0212065"
},
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