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†|Omega, with f† a homogeneous polynomial in the mode creation operators, we propose two sets of LU invariants: spectral invariants, which are the eigenvalues of the operator ff†, 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 fk(f†)k. We provide a scheme for experimental measurement of the later, as related to the postselection probability of creating state f†k|Omegafrom k copies of f†|Omega. © 2014 American Physical Society.We would like to thank David Bar-Moshe, Anton Petrunin, Karol Zyczkowski, Antonio Acín, and Michał Studzinski for fruitful discussions. We acknowledge the support of the Spanish MINCIN/MINECO project TOQATA (Grant No. FIS2008-00784), EU Integrated Projects AQUTE and SIQS, CHISTERA project DIQUIP, ERC AdG OSYRIS, the John Templeton Foundation, a Polish National Science Centre grant under Contract No. DEC-2011/01/M/ST2/00379 and Grant No. FIS2012-33642 from the Spanish government