What is the most general (ideal) quantum coordinate transformation?

Our work builds on the foundational question of how reference frames should be treated when the frame itself is a quantum system. In particular, we consider the perspective-neutral approach for a system described by a group G. Under certain assumptions, it has been shown that the known quantum reference frame transformations between single particle-subsystems are elements of a much larger symmetry group. This group contains additional quantum coordinate transformations, such as those mapping to the center-of-mass of a given configuration.

We extend the formalism to encompass more general representations, rather than only product representations. In particular, we show that quantum coordinate transformations can be understood as symmetry transformations conditioned on invariant projective measurements. Consequently, we can examine systems without a fixed subsystem tensor-product-structure, such as bosons. We define quantum coordinate transformations for bosonic systems and illustrate how the choice of a frame corresponds to the choice of distinguishing labels for bosonic states. Conceptually, this sheds light on the entanglement of indistinguishable particles: we analyze how the entanglement structure corresponds to a tensor factorization for a given frame.