$E=(21.1 \pm 0.2)$ ebits using $\Gamma(\omega_i,\omega_s)$ i.e. we observe almost no influence of the detection process on the degree of entanglement. This amount of entropy is the same as in a maximally bipartite entangled qudit state of dimension $d^2$ with $d=2^{E}\approx 2.2\times 10^6$. This demonstrates that SPDC driven by a spectrally narrow-band pump field offers a potentially very high dimensional state space to encode qudits in frequency modes. Accordingly, we calculate by numerical computation a Schmidt number $K=1/\Tr(\hat{\rho}_{i,s}^2)$ of $K\approx 1.3\times 10^6$. Figure \[fig:Gamma\] shows that the effect of the PSF leads to an effective loss in the correlation between the two photons. Consequently, the values for $E$, $d$ and $K$ are reduced to $E \approx 2.6$, as obtained by direct diagonalization of the reduced density matrix, $d \approx 6$ and $K \approx 4.9$.

![\[fig:Gamma\] Left: $\Gamma(\omega_i,\omega_s)$ for $\Delta \nu_p=5$ MHz, $L_{DC}=L_{SFG}=11.5$ mm and $G_{DC}=G_{SFG}=9$ $\mu$m. Inset: The narrow joint spectral amplitude implies a high degree of entanglement between idler and signal photon. Right: $\Gamma_{PSF}(\omega_i,\omega_s)$ taking into account the PSF with $\Delta \omega_{PSF}=9.6\times 10^{-3}$ rad/fs.](Gamma.eps){width="100.00000%"}

Discretization of the frequency space
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The state (\[eq:twophstate\]) is a continuous superposition of frequency modes. To encode quantum information in the form of qudits, 