Dimensionality of the spatiotemporal entanglement of parametric down-conversion photon pairs

In this work, the Schmidt number of the two-photon state generated by parametric down-conversion (PDC) is
evaluated in the framework of a fully spatiotemporal model for PDC. A comparison with the results obtained in
either purely spatial or purely temporal models shows that the degree of entanglement of the PDC state cannot be
trivially reduced to the product of the Schmidt numbers obtained in models with lower dimensionality, unless the
detected bandwidth is very narrow. This result is a consequence of the nonfactorability of the state in the spatial
and temporal degrees of freedoms of twin photons. In the limit of a broad pump beam, we provide a geometrical
interpretation of the Schmidt number as the ratio between the volume of the phase-matching region and of a
correlation volume.