Unknown quantum states: The quantum de Finetti representation
Abstract
We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog 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 densityoperator assignments and provides an operational definition of the concept of an "unknown quantum state" in quantumstate tomography. This result is especially important for informationbased 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 September 2002
 DOI:
 10.1063/1.1494475
 arXiv:
 arXiv:quantph/0104088
 Bibcode:
 2002JMP....43.4537C
 Keywords:

 03.65.Ta;
 03.67.a;
 03.65.Ge;
 02.50.Cw;
 Foundations of quantum mechanics;
 measurement theory;
 Quantum information;
 Solutions of wave equations: bound states;
 Probability theory;
 Quantum Physics
 EPrint:
 30 pages, 2 figures