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Within photonic quantum computing, one of the ways to represent information is the dual-rail representation of single-photon states ($c_0|01\rangle \ + \ c_1|10\rangle$). Is it possible to utilize multi-photon states (for example, two optical cavities with total energy $2\hbar\omega$, using a state like $c_0|02\rangle \ + \ c_1|20\rangle \ + \ c_2|11\rangle$, where $|02\rangle$ represents two photons in one cavity, $|20\rangle$ represents two photons in the other cavity, and $|11\rangle$ represents one photon in each cavity?

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Yes. The kets themselves can have arbitrary labels, and it's just for you to establish the connection between them and the physical scenario. There's no reason why you can't have the physical scenario you've specified and, indeed, people frequently do.

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This is very much possible. And is a very general technique of how product systems in composite states are coupled. Here of the form $|n_1\rangle |n_2\rangle$. This kind of general ket is a solution of the Hamiltonian interaction/coupling terms like $V\sim (a_1^\dagger a_2 +h.c)$ which describe the exchange of one quanta (between the two optical cavities here). $a_i (a_i^\dagger)$ are the annihilation and creator operators for $i^{th}$ cavity in second quantisation. It follows similarly for more systems coupled in the form of product states. A very general and frequently used is a spin (with an excited and ground state as $|e\rangle,|g\rangle$) in an optical cavity with solutions like $|\psi\rangle=c_1|e,n\rangle+c_2|g,n+1\rangle$ where $n$ is the number of photons in the cavity, and so on for more complicated systems.

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