dorsal/arxiv
View SchemaOn an Argument of David Deutsch
| Authors | Richard D. Gill |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0307188 |
| URL | https://arxiv.org/abs/quant-ph/0307188 |
| Journal | pp. 277--292 in: M. Sch"urmann and U. Franz (eds.), Quantum Probability and Infinite Dimensional Analysis: from Foundations to Applications. QP--PQ: Quantum Probability and White Noise Analysis, Volume 18 (2005). World Scientific. |
Abstract
We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born's law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born's law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason's theorem under stronger assumptions. However, for some special cases the proof method gives positive results while using different assumptions to Gleason. This leads to the conjecture that the proof could be improved to give the same conclusion as Gleason under unitary invariance together with a much weaker functional invariance condition.
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"abstract": "We analyse an argument of Deutsch, which purports to show that the\ndeterministic part of classical quantum theory together with deterministic\naxioms of classical decision theory, together imply that a rational decision\nmaker behaves as if the probabilistic part of quantum theory (Born\u0027s law) is\ntrue. We uncover two missing assumptions in the argument, and show that the\nargument also works for an instrumentalist who is prepared to accept that the\noutcome of a quantum measurement is random in the frequentist sense: Born\u0027s law\nis a consequence of functional and unitary invariance principles belonging to\nthe deterministic part of quantum mechanics. Unfortunately, it turns out that\nafter the necessary corrections we have done no more than give an easier proof\nof Gleason\u0027s theorem under stronger assumptions. However, for some special\ncases the proof method gives positive results while using different assumptions\nto Gleason. This leads to the conjecture that the proof could be improved to\ngive the same conclusion as Gleason under unitary invariance together with a\nmuch weaker functional invariance condition.",
"arxiv_id": "quant-ph/0307188",
"authors": [
"Richard D. Gill"
],
"categories": [
"quant-ph",
"math.PR"
],
"journal_ref": "pp. 277--292 in: M. Sch\"urmann and U. Franz (eds.), Quantum\n Probability and Infinite Dimensional Analysis: from Foundations to\n Applications. QP--PQ: Quantum Probability and White Noise Analysis, Volume 18\n (2005). World Scientific.",
"title": "On an Argument of David Deutsch",
"url": "https://arxiv.org/abs/quant-ph/0307188"
},
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