Frame perspectives for process matrices: from coordinate parametrization to spacetime representation

Abstract

We study how to implement and transform frame perspectives for quantum processes in the process-matrix formalism. We argue that, for pure processes, the causal reference frames (CRF)and time-delocalized subsystems (TDS) formalisms should be understood as coordinate parametrizations of a single perspective-neutral higher-order object. A genuine perspective arises when one endows the process with additional frame data by choosing an operational foliation into circuit fragments (events). With this distinction, existing no-go results acquire a clear scope: they rule out unitary transformations that preserve time foliation, attempting to switch perspectives while keeping the fragment boundaries -- hence the global past/future partition -- fixed. Focusing on the quantum switch, we construct explicit maps that transform perspectives unitarily at the price of reshuffling the notions of past and future. We then show that unitary transformations between perspectives can also be achieved in a different way, namely by extending the process with subsystems that define quantum reference frames and provide a shared spatiotemporal scaffold. In this extended setting, complementary CRF/TDS perspectives become unitarily related while preserving global past and future. We discuss how this frame-perspectival approach informs the broader question of empirical realizability of abstract process matrices.

Figures (3)

• Figure 1: Superposition of physically inequivalent configurations. Events $A$ and $B$ are defined as intersections of the world-lines of the target system (black arrow), Alice’s laboratory (blue line), and Bob’s laboratory (red line). The state preparation and the final measurement are represented by the yellow atom-shaped symbol and the detector symbol, respectively.
• Figure 2: Quantum coordinatization. The quantum-controlled diffeomorphism allows one to take the perspective of the geometric scaffolding—i.e. quantum coordinates—depicted as the gray grid. The transformed state on the right is expressed in those quantum coordinates, such that Alice’s worldline is localized (configuration 2), whereas Bob’s worldline is spacetime delocalized (superposition of configurations 2 and 3).
• Figure 3: Quantum coordinate transformations. Two different coordinate representations of the same process: (left) Bob's causal reference frame; (right) Alice's causal reference frame. One can change representation via a suitable quantum-controlled diffeomorphism.