are the eigenvectors or Schmidt modes of the reduced density operators and $\beta_j$ the corresponding eigenvalues. The Schmidt modes itself are orthogonal and form a complete basis. As can be seen in figure \[fig:schmidt\_functions\], only a few $\beta_j$ are significantly nonzero if we decompose $\Gamma_{PSF}\left(\omega_{i},\omega_{s}\right)$ with the parameters of our experimental setup. By substituting $\Gamma_{PSF}\left(\omega_{i},\omega_{s}\right)$ in (\[eq:gammatwophstate\]) and using (\[eq:gamma\_schmidt\]) one finds $$\label{eq:qudit_schmidt}
\vert\psi\rangle\rightarrow\vert\psi\rangle^{(d)}=\sum_{j=0}^{d-1}c_j\,\vert j \rangle_i \vert j\rangle_s$$ with $c_j=\sqrt{\beta_j}$. The Schmidt basis thus provides a direct way to discretize the state (\[eq:twophstate\]) into an entangled qudit state whose dimensionality is only bound by the number of nonzero $\beta_j$. For a symmetric joint spectral amplitude $\Gamma_{PSF}(\omega_i,\omega_s)$ (figure \[fig:Gamma\]) we find $f^{i}_j\left(\omega\right)=f^{s}_j\left(\omega\right)$. The Schmidt decomposition has been performed numerically by discretizing the continuous function $\Gamma_{PSF}(\omega_i,\omega_s)$ on a lattice with size $2049 \times 2049$.

\[fig:schmidt\_modes\]

![\[fig:schmidt\_functions\] Left: Exemplary, the first three Schmidt modes $f^{i,s}_j\left(\omega\right)$ of (\[eq:gamma\_schmidt\]). Right: Shown are the eigenvalues $\beta_j$ up to $j=20$.](schmidt_functions_lambda.eps){width="70.00000%"}

Experimental setup
==================

The experimental setup shown in figure \[fig:Setup\] is composed of three parts: The entangled state preparation by SPDC, the spectral manipulation with the SLM and the coincidence detection by SFG.

![\[fig:Setup\] Schematic of the experimental setup. Preparation: L0 pump beam focusing lens ($f = 