Mode displacements in multimode systems and applications to entanglement

A d-mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in Zd and the mode phase space is ZdxZd. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the d modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the d observers; and they also change the momentum with which the entanglement propagates