Multiphoton states related via linear optics

We investigate which pure states of n photons in d modes can be transformed into each other via passive linear optics, without postselection. In other words, we study the local unitary (LU) equivalence classes of symmetric many-qudit states. Writing our state as f(dagger)vertical bar Omega >, with f(dagger) a homogeneous polynomial in the mode creation operators, we propose two sets of LU invariants: spectral invariants, which are the eigenvalues of the operator f f(dagger), and moments, each given by the norm of the symmetric component of a tensor power of the initial state, which can be computed as vacuum expectation values of f(k)(f(dagger))(k). We provide a scheme for experimental measurement of the later, as related to the postselection probability of creating state f(dagger k)vertical bar Omega > from k copies of f(dagger)vertical bar Omega >.