We investigate the invariance structure of observable predictions in the Intrinsic Observability Framework (IOF), building on the Access-Induced Reduction Principle (ARP) introduced in Ref. [1]. While IOF allows for non-unique decompositions of a quantum system into observable and hidden degrees of freedom, physical predictions must remain invariant under admissible transformations of such decompositions. We formalize admissible transformations as those that preserve observable probability distributions derived from reduced states. Furthermore, we clarify the distinction between invariant statistical structures and non-invariant configurations of individual measurement outcomes. This establishes a consistent framework in which structural freedom coexists with empirical determinacy.
As physics searches for invariants in observations, this paper looks for invariants of probabilistic observation without assuming physical structure. Structure emerges from the basic assumption of science that new information shall lead to more accurate knowledge of the invariants. This leads to statistically unique random variables for expressing observed information: Complex probability amplitudes. Predictions are also just random variables computed from observed data, and must become more accurate with more observations as input. This singles out the quantum mechanical superposition principle. The external conditions of a probabilistic experiment can themselves be monitored at the most detailed level, resulting in observation of coicidence probabilities. The invariants of any multi-coincidence experiment are the same as those of a one-event experiment with the same number of possible outcomes. An observable probability turns out to be controllable by two independent experimental conditions, naturally parametrized as a direction in a 3-dimensional space. In summary, the probabilistic paradigm itself defines a unique method of forming concepts and making predictions. The method appears irrefutable within probability, because, whenever a prediction turns out wrong the existence of an as yet unmonitored condition is postulated, and a formal way to incoroporate it is shown. The Hilbert space formalism of quantum theory seems to be isomorphic to this method.
We explore a geometric approach to generating local
SU(2) and
SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or ``gauge'' invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of
d-level quantum systems. We give an operational interpretation of these invariants in terms of observables.