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We propose an experimental scheme to generate, in a heralded fashion,arbitrary quantum superpositions of two-mode optical states with a fixed totalphoton number $n$ based on weakly squeezed two-mode squeezed state resources(obtained via weak parametric down conversion), linear optics, and photondetection. Arbitrary $d$-level (qudit) states can be created this way where$d=n+1$. Furthermore, we experimentally demonstrate our scheme for $n=2$. Theresulting qutrit states are characterized via optical homodyne tomography. Wealso discuss possible extensions to more than two modes concluding that, ingeneral, our approach ceases to work in this case. For illustration and withregards to possible applications, we explicitly calculate a few examples suchas NOON states and logical qubit states for quantum error correction. Inparticular, our approach enables one to construct bosonic qubiterror-correction codes against amplitude damping (photon loss) with a typicalsuppression of $\sqrt{n}-1$ losses and spanned by two logical codewords thateach correspond to an $n$-photon superposition for two bosonic modes.

Heralded creation of photonic qudits from parametric down conversion using linear optics