arXiv:2108.04086v2 [quant-ph] 23 Sep 2021 REAL POVMS ON THE PLANE: INTEGRAL QUANTIZATION, NAIMARK THEOREM AND LINEAR POLARISATION OF THE LIGHT ROBERTO BENEDUCI A , EMMANUEL FRION B , JEAN-PIERRE GAZEAU C AND AMEDEO PERRI A A Dipartimento di Fisica, Università della Calabria and Istituto Nationale di Fisica Nucleare, Gruppo c. Cosenza, 87036 Arcavacata di Rende (Cs), Italy B Helsinki Institute of Physics, P . O. Box 64, FIN-00014 University of Helsinki, Finland C Université de Paris, CNRS, Astroparticule et Cosmologie 75013 Paris, France ABSTRACT. We investigate two aspects of the elementary example of real POVMs on the Euclidean plane, namely their status as quantum observables and their role as quantizers in the integral quantization procedure. The compatibility of POVMs in the ensuing quantum formalism is discussed, and a Naimark dilation is found for the quantum operators. A physical situation is discussed, where we describe the linear polarization of the light with the use of Stokes parameters. In particular, the case of sequential measurements in a real bidimensional Hilbert space is addressed. An interpretation of the Stokes parameters in the framework of unsharp or fuzzy observables is given. Finally, a necessary condition for the compatibility of two dichotomic fuzzy observables which provides a condition for the approximate joint measurement of two incompatible sharp observables is found. E-mail address: roberto.beneduci@unical.it, emmanuel.frion@helsinki.fi, gazeau@apc.in2p3.fr . Date: September 24, 2021. 1