Not-So-Normal Mode Decomposition

We provide a generalization of the normal mode decomposition for nonsymmetric or locality constrained situations. This allows, for instance, to locally decouple a bipartitioned collection of arbitrarily correlated oscillators up to elementary pairs into which all correlations are condensed. Similarly, it enables us to decouple the interaction parts of multimode channels into single-mode and pair interactions where the latter are shown to be a clear signature of squeezing between system and environment. In mathematical terms the result is a canonical matrix form with respect to real symplectic equivalence transformations.

Figure 1

Left: let Alice and Bob each have a collection of oscillators with arbitrary correlations between them (indicated by dashed lines). Right: there exist local canonical transformations which condense the correlations into elementary units (single modes or pairs) and eliminate all cross-correlations.

Figure 2

Left: a multimode Gaussian channel consists out of an interaction part $X$, which directly couples the input modes, and an addition of input-independent noise $Y$. Right: the interaction part is decoupled into single-mode and two-mode parts by applying canonical transformations before and after the channel. Two-mode interactions corresponding to a complex invariant $\lambda$ only occur if the global system-plus-environment evolution involves squeezing.