We investigate the transformation rule of a single particle wave-function under a change of reference frame. A postulate is raised, and some fundamental aspects regarding the reference-frame independence of quantum probabilities are explored.
The textbook projection postulate of quantum mechanics stands in tension with the symmetries of relativistic spacetime: it is stated as an update applied at a single instant to the entire composite state, which presupposes a preferred simultaneity surface, and its description of the distant partner of an entangled pair invites the reading that measurement physically changes that partner. We argue that this tension arises from one removable ingredient of the axiom, and we make precise what that ingredient is and is not. Separating the postulate into three logically independent claims—definite-outcome realization, Born-rule probability, and a globally-timed projection—we show that the third can be replaced by a locally triggered rule: branch selection upon completed local decoherence, in the einselected pointer basis (Postulate M). The measurement operator itself is not modified: the update at a site A is of the local form PiA⊗IB under both the standard postulate and Postulate M, and the two rules yield identical post-measurement states and probabilities, the standard rule being recovered as the vanishing-decoherence-time limit. Postulate M is therefore a reinterpretation of standard quantum mechanics rather than a modification of it; what it changes is the trigger of the update, from a global instant to a local decoherence event, so that the axiom can be stated without reference to any foliation. Parameter independence holds exactly, as it does in the standard theory; the final branch structure and joint outcome statistics are foliation-independent; and the exchange symmetry of the entangled pair is respected. We illustrate the framework in the Einstein–Podolsky–Rosen (EPR) scenario with one and with both particles coupled to local spin environments. The Born-rule probabilities in the pointer basis are exactly conserved throughout decoherence, while the maximal CHSH (Clauser–Horne–Shimony–Holt) value, obtained from the Horodecki criterion, is Smax=21+∣z(t)∣2 in terms of the decoherence factor z(t), interpolating between the Tsirelson bound and the classical bound; this corrects the constant-22 claim of Ref. [10]. The single symmetry that quantum mechanics does break, outcome independence, is located entirely in the nonseparability of the global state, never in the measurement operator: it transmits no signal and selects no frame. The single-outcome question—which branch is realized—is left open.
In the literature on quantum reference frames, the internal (relative) properties of a system are defined as those which are preserved under an arbitrary change of reference frame. For a system of quantum spins, these are all properties preserved by proper spatial rotations of the laboratory. However, this approach does not account for the hypothetical possibility of the laboratory becoming entangled to the system, as described by a second laboratory (the `Wigner's friend' scenario), in which case the relationship between the two laboratories is not a rotation, but is fundamentally quantum. To overcome this limitation, we re-define the reference frame transformations to be those that preserve the fidelities between subsystems. This enables us to derive U(2) as the correct symmetry group for transformations of a system of N spin-half particles. Next, we propose that systems having the same internal properties should be regarded as physically equivalent in the absence of an external frame. Remarkably, this implies that a single spin in a superposition relative to a spin magnet is equivalent to a macroscopic superposition of the magnet relative to the spin. We discuss the implications of this result for the Wigner's friend paradox.
We investigate the physics of quantum reference frames. Specifically, we study several simple scenarios involving a small number of quantum particles, whereby we promote one of these particles to the role of a quantum observer and ask what is the description of the rest of the system, as seen by this observer? We highlight the interesting aspects of such questions by presenting a number of apparent paradoxes. By unravelling these paradoxes, we obtain a better understanding of the physics of quantum reference frames.