  Optical multi-mode systems provide large scale Hilbert spaces that can be
accessed and controlled using single photon sources, linear optics and photon
detection. Here, we consider the bipartite entanglement generated by coherently
distributing $M$ photons in $M$ modes to two separate locations, where linear
optics and photon detection is used to verify the non-classical correlations
between the two $M$-mode systems. We show that the entangled state is symmetric
under mode shift operations performed in the two systems and use this symmetry
to derive correlations between photon number distributions detected after a
discrete Fourier transform (DFT) of the modes. The experimentally observable
correlations can be explained by a simple and intuitive rule that relates the
sum of the output mode indices to the eigenvalue of the input state under the
mode shift operation. Since the photon number operators after the DFT do not
commute with the initial photon number operators, entanglement is necessary to
achieve strong correlations in both the initial mode photon numbers and the
photon numbers observed after the DFT. We can therefore derive entanglement
witnesses based on the experimentally observable correlations in both photon
number distributions, providing a practical criterion for the evaluation of
large scale entanglement in optical multi-mode systems. Our method thus
demonstrates how non-classical signatures in large scale optical quantum
circuits can be accessed experimentally by choosing an appropriate combination
of modes in which to detect the photon number distributions that characterize
the quantum coherences of the state.
