# Re: Deutsch's quantum probability derivation

From: scerir (scerir@libero.it)
Date: Sat Jul 28 2001 - 08:58:45 MDT

> OK, I got interested in this enough to write up a formal note.

> http://hanson.gmu.edu/bornrule.pdf
> Quantum Probability From Decision Theory and Exchangeability
> by Robin Hanson, July 27, 2001.

> Deutsch's derivation of quantum probability from decision theory has been
> criticized as relying on an implausible hidden assumption. I here offer as
> a substitute an apparently more plausible exchangeability assumption.

With his classical representation theorem Bruno De Finetti
showed that a multi-trial probability assignment that is
permutation-symmetric - *exchangeable* - for an arbitrarily
large number of trials, is equivalent to a probability for
the "unknown probabilities".
I wonder whether De Finetti's theorem may have something
to do with "Quantum Probability From Decision Theory and
Exchangeability", or not.

- serafino

----------------------------------------

Unknown Quantum States: The Quantum de Finetti Representation

Carlton M. Caves, Christopher A. Fuchs, Ruediger Schack

We present an elementary proof of the quantum de Finetti representation
theorem, a quantum analogue of de Finetti's classical theorem on
exchangeable probability assignments. This contrasts with the original proof
of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies
on advanced mathematics and does not share the same potential for
generalization. The classical de Finetti theorem provides an operational
definition of the concept of an unknown probability in Bayesian probability
theory, where probabilities are taken to be degrees of belief instead of
objective states of nature. The quantum de Finetti theorem, in a closely
analogous fashion, deals with exchangeable density-operator assignments and
provides an operational definition of the concept of an "unknown quantum
state" in quantum-state tomography. This result is especially important for
information-based interpretations of quantum mechanics, where quantum
states, like probabilities, are taken to be states of knowledge rather than
states of nature. We further demonstrate that the theorem fails for real
Hilbert spaces and discuss the significance of this point.

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