dorsal/arxiv
View SchemaApproximate Consistency and Prediction Algorithms in Quantum Mechanics
| Authors | Jim McElwaine |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9707053 |
| URL | https://arxiv.org/abs/quant-ph/9707053 |
| Journal | Ph.D. Thesis, Cambridge (1996) |
Abstract
This dissertation investigates questions arising in the consistent histories formulation of the quantum mechanics of closed systems. Various criteria for approximate consistency are analysed. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several new bounds on the violation of probability sum rules. The quantum Zeno effect is also analysed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed. The possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set for pure initial density matrices is investigated. The problems that arise are explained, and different possible algorithms discussed. Their properties are analysed with the aid of simple models. A set of computer programs is described which apply the algorithms to more complicated examples. Another algorithm is proposed that selects the consistent set (formed using Schmidt projections) with the highest Shannon information. This is applied to a simple model and shown to produce physically sensible histories. The theory is capable of unconditional probabilistic prediction for closed quantum systems, and is strong enough to be falsifiable. Ideas on applying the theory to more complicated examples are discussed.
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"abstract": "This dissertation investigates questions arising in the consistent histories\nformulation of the quantum mechanics of closed systems. Various criteria for\napproximate consistency are analysed. The connection between the\nDowker-Halliwell criterion and sphere packing problems is shown and used to\nprove several new bounds on the violation of probability sum rules. The quantum\nZeno effect is also analysed within the consistent histories formalism and used\nto demonstrate some of the difficulties involved in discussing approximate\nconsistency. The complications associated with null histories and infinite sets\nare briefly discussed.\n The possibility of using the properties of the Schmidt decomposition to\ndefine an algorithm which selects a single, physically natural, consistent set\nfor pure initial density matrices is investigated. The problems that arise are\nexplained, and different possible algorithms discussed. Their properties are\nanalysed with the aid of simple models. A set of computer programs is described\nwhich apply the algorithms to more complicated examples.\n Another algorithm is proposed that selects the consistent set (formed using\nSchmidt projections) with the highest Shannon information. This is applied to a\nsimple model and shown to produce physically sensible histories. The theory is\ncapable of unconditional probabilistic prediction for closed quantum systems,\nand is strong enough to be falsifiable. Ideas on applying the theory to more\ncomplicated examples are discussed.",
"arxiv_id": "quant-ph/9707053",
"authors": [
"Jim McElwaine"
],
"categories": [
"quant-ph"
],
"journal_ref": "Ph.D. Thesis, Cambridge (1996)",
"title": "Approximate Consistency and Prediction Algorithms in Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/9707053"
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