Next Article in Journal
Some Aspects of Extended Kinetic Equation
Previous Article in Journal
Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and Its Applications in Various Disciplines
Open AccessArticle

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; https://doi.org/10.3390/axioms4030400
Received: 27 January 2015 / Revised: 5 August 2015 / Accepted: 17 August 2015 / Published: 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
MDPI and ACS Style

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

Show more citation formats Show less citations formats

Article Access Map

1
Only visits after 24 November 2015 are recorded.
Back to TopTop