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Classical and nonclassical randomness in quantum measurements

### Abstract

The space of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space *X* with values in , the algebra of linear operators acting on a *d*-dimensional Hilbert space, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra into . This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on *C*(*X*), of certain (but not all) unital completely positive linear maps . The extremal and C*-extremal points of are determined.

© 2011 American Institute of Physics

Received 18 October 2011
Accepted 18 November 2011
Published online 13 December 2011

Acknowledgments:
We acknowledge the support of the NSERC Discovery, PGS, and USRA programs and Nipissing University (North Bay, Canada), where this work was undertaken during an extended scientific visit of the first author. We are especially indebted to Giulio Chiribella for drawing our attention to the works (Refs. 7 and 8), and to Michael Kozdron and the referee for useful commentary on the results herein.

Article outline:

I. INTRODUCTION
II. NOTATION, TERMINOLOGY, AND ASSUMPTIONS
A. Assumption
B. States, effects, automorphisms, and POVMs
C. Completely positive linear maps of homogeneous C*-algebras
D. Convexity and C*-convexity
III. QUANTUM RANDOM VARIABLES AND INTEGRATION
A. The principal Radon-Nikodým derivative
B. Integrable functions
C. Quantum integration is a completely positive operator
D. The Γ-transform
E. Non-principal Radon-Nikodým derivatives
IV. TOPOLOGY OF
V. CLASSICAL RANDOMNESS
VI. NONCLASSICAL RANDOMNESS
A. Application: sharp measurements generate all quantum measurements through coarsening
VII. DISCUSSION
VIII. CONCLUSION

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2011-12-13

2016-02-06

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