Fig. 2: Natural subsystem decompositions.

The full invariant system can be decomposed in a way that is natural to \({\mathsf{A}}\) (vertical, orange “threads'') and in a way that is natural to \({\mathsf{B}}\) (horizontal, green threads). A QRF is a preferred factorisation of the invariant system, and a QRF transformation is a change from one preferred factorisation to another. In this illustration, when 2 different subsystems overlap it means that their corresponding operators don’t commute in general. In this way, when \({\mathsf{A}}\) refers to “the system,” she is actually referring to the subsystem \({\mathsf{A}}| {\mathsf{B}}\), which overlaps with \({\mathsf{S}}| {\mathsf{B}}\) and \({\mathsf{A}}| {\mathsf{B}}\) from the point of view of \({\mathsf{B}}\). Note that the inclusion of the subsystems \(\overline{{\mathsf{SB}}| {\mathsf{A}}}\) and \(\overline{{\mathsf{SA}}| {\mathsf{B}}}\) is essential to find a unitary relation between \({\mathsf{A}}\)'s and \({\mathsf{B}}\)'s tensor product factorisations.