Eur. Phys. J. D (2014) 68: 338
https://doi.org/10.1140/epjd/e2014-50368-y

Regular Article

Wigner representation for polarization-momentum hyperentanglement generated in parametric down-conversion, and its application to complete Bell-state measurement

¹ Departamento de Física Aplicada III, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, 41092 Sevilla, Spain
² Centro Asociado de la Universidad Nacional de Educación a Distancia de Las Palmas de Gran Canaria, 35004 Las Palmas de Gran Canaria, Spain
³ Grupo de Ingeniería Térmica e Instrumentación, Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain

^a e-mail: acasado@us.es

Received: 9 May 2014
Received in final form: 22 July 2014
Published online: 6 November 2014

Abstract

We apply the Wigner function formalism to the study of two-photon polarization-momentum hyperentanglement generated in parametric down-conversion. It is shown that the consideration of a higher number of degrees of freedom is directly related to the extraction of additional uncorrelated sets of zeropoint modes at the source. We present a general expression for the description of the quantum correlations corresponding to the sixteen Bell base states, in terms of four beams whose amplitudes are correlated through the stochastic properties of the zeropoint field. A detailed analysis of the two experiments on complete Bell-state measurement included in [Walborn et al., Phys. Rev. A 68, 042313 (2003)] is made, emphasizing the role of the zeropoint field. Finally, we investigate the relationship between the zeropoint inputs at the source and the analysers, and the limits on optimal Bell-state measurement.