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POVMs and the Two Theorems of Naimark and Sz.-Nagy

Center for Information Technology, NIH, Bethesda, MD 20892, USA
Author to whom correspondence should be addressed.
Academic Editor: Angel Garrido
Axioms 2015, 4(3), 400-411;
Received: 27 January 2015 / Revised: 5 August 2015 / Accepted: 17 August 2015 / Published: 1 September 2015
PDF [771 KB, uploaded 1 September 2015]


In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom. View Full-Text
Keywords: Naimark dilation; Sz.-Nagy resolution; quantum statistical inference Naimark dilation; Sz.-Nagy resolution; quantum statistical inference
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Malley, J.D.; Fletcher, A.R. POVMs and the Two Theorems of Naimark and Sz.-Nagy. Axioms 2015, 4, 400-411.

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