If one records the which-way information by entangling the photon paths with another degree of freedom there will be no destructive interference of the branch vectors at path 4 and the which-way information claims from the two single-path setups can be applied to this mixed setup also. In such a scenario the photon density matrix is not that of a pure state but mixed one. Red and blue branch vectors denote either photons with orthogonal polarizations (such as and ) or photons propagating at different times ( and ), in either case destructive interference at arm 4 cannot occur.
