multiple pair states [@bouwmeester1999]. However, because light supports only two polarization modes, the Hilbert space of each photon is limited to a dimension of two (qubits). On the other hand, entangling $d$-dimensional states denoted as qudits requires multi-mode states of light with $d\geq 2$. This can be for instance achieved using a specific discretization scheme of the transverse momentum degree of freedom. Experiments have been performed where entanglement appears in a discrete set of orbital angular momentum modes [@dada2011; @pires2009; @mair2001; @agnew2011; @fickler2012; @giovannini2012], pixel modes [@sullivan2005] and slit modes [@lima2009]. The manipulation and detection of transverse entanglement mainly rely on the ability to experimentally address the transverse momentum modes with the help of holograms or spatial light modulators. The entanglement content is theoretically quantified by the Schmidt number $K$, and is, for usual parameters of the pump laser and the SPDC crystal, on the order of 10 to 50 for transverse wave vector entanglement [@law2004; @giovannini2012].

Similar Schmidt numbers are in reach for energy-time entanglement generated by a short pump pulse [@law2000; @mikhailova2008; @brida2009] but considerably higher values of $K$ can be obtained for SPDC driven by a quasi-monochromatic pump laser. While high dimensional entanglement in the transverse momentum 