Operationally induced preferred basis in unitary quantum mechanics
Vitaly Pronskikh
TL;DR
This work tackles the persistent preferred-basis and definite-outcome aspects of quantum measurement within standard unitary quantum mechanics by positing a structural-operational split: the noncommutative quantum sector and a commutative Boolean readout sector connected by a non-unitary instrument/POVM. Born probabilities are argued to be uniquely determined by Gleason-type theorems (with Busch's extension for qubits) as the interface measure mapping quantum amplitudes to classical frequencies, provided additivity and basis-independence hold. The detector-induced POVM, formalized through a detector coupling and coarse-grained readout, induces the relevant basis for recorded outcomes, as illustrated by a minimal qubit–pointer model in which $E_{\ a}$ takes the form $E_{\pm}=\frac{1}{2}(\mathbb{I}\pm \eta\,\sigma_z)$ with $\eta$ set by detector resolution. Decoherence stabilizes records but does not by itself select outcomes; a non-composability lemma shows that joint assignments of nested outcomes require a joint instrument, relocating randomness to the operational transition rules. The framework remains within standard quantum theory while offering concrete experimental touchpoints (POVM tomography, joint-measurability tests) to characterize measurement devices and multi-observer scenarios, and it clarifies where randomness and classicality enter the theory.
Abstract
The preferred-basis problem and the definite-outcome aspect of the measurement problem persist even if the detector is modeled unitarily, because experimental data are necessarily represented in a Boolean event algebra of mutually exclusive records whereas the theoretical description is naturally formulated in a noncommutative operator algebra with continuous unitary symmetry. This change of mathematical type constitutes the core of the 'cut': a structurally necessary interface from group-based kinematics to set-based counting. In the presented view the basis relevant for recorded outcomes is not determined by the system Hamiltonian alone; it is induced by the measurement mapping, i.e., by the detector channel together with the coarse-grained readout that defines an instrument. The probabilistic mapping is anchored in symmetry and measure theory: by Gleason-type uniqueness (Gleason for projections in $d>2$ and Busch's extension for Positive Operator-Valued Measures (POVMs) including $d=2$), the trace rule is the unique probability measure consistent with additivity over exclusive events and basis-independence of the unitary sector. A compact qubit--pointer model yields an induced unsharp POVM $E_\pm=\tfrac12(\id\pm η\,σ_z)$ with $η$ fixed by pointer resolution, displaying explicitly how the detector induces the relevant basis. Finally, nested-observer paradoxes are tightened into a non-composability lemma: joint assignment of outcome propositions is obstructed unless a joint instrument exists. This relocates the origin of randomness to the stochasticity of the transition rules.