Quantum incompatibility of Born probabilities

Standard quantum theory assumes that measurements are performed relative to ideal, classical reference frames that are infinitely resourceful. This paper investigates how the fundamental probabilistic structure of quantum mechanics changes when measurements are instead performed relative to non-ideal quantum reference frames (QRFs) that possess finite resources and are subject to quantum-mechanical uncertainty.

The authors introduce a formal framework for 'relational relative frequency' (RRF) operators. By treating the reference frame as a quantum system, they construct relational observables using group-theoretic twirling, which ensures that the measurements are invariant under the symmetry group of the reference frame. They compare two ways of interpreting these relational measurements: a POVM construction, which can be reduced to a classical mixture of ideal measurements, and a PVM construction, which represents a truly collective measurement of the system and the reference frame that does not require an external frame.

The study demonstrates that for non-ideal QRFs, the relative frequency operators associated with different measurements do not commute, even in the limit of an infinite number of trials. This leads to a 'Bell-type' theorem for relative frequencies, where the correlations between observers using non-ideal frames violate Bell inequalities. This suggests that the uncertainty in these frequencies is not merely a result of ignorance, but a fundamental feature of the quantum-mechanical description of reference frames. The authors propose a concrete implementation using pulsed homodyne detection in quantum optics to test these predictions.