Quantum Relativity of Subsystems

One of the most basic notions in physics is the partitioning of a system into subsystems and the study of correlations among its parts. In this Letter, we explore this notion in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We demonstrate that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We further demonstrate that subalgebras which commute before imposing the symmetry constraint can translate into noncommuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame dependent.

Figure 1

The change from perspective $A$ to perspective $B$ takes the compositional form of a “quantum coordinate transformation,” ${\mathrm{\Lambda }}_{A\to B}≔{\mathcal{R}}_B^{(H)}\circ ({\mathcal{R}}_A^{(H)}{)}^{-1}$ [17, 18, 19, 20, 21, 22, 23]. This induces a transformation on the algebra observables from the perspective of $A$ to the perspective of $B$, namely, ${\mathrm{\Lambda }}_{A\to B}{\mathcal{A}}_{BC|A}{\mathrm{\Lambda }}_{A\to B}^{-1}\subseteq {\mathcal{A}}_{AC|B}$, where ${\mathcal{A}}_{ij|k}\simeq \mathcal{B}({\mathcal{H}}_{ij|k})$.