Quantum orientation, Noether structure, composition of systems and operations
arXiv:2607.04899
The paper introduces a "Noether structure"—the dual role of Hermitian operators as observables and symmetry generators—to complement the statistical framework of quantum theory, showing how it underlies system composition and the definition of completely positive (orientation‑preserving) operations in finite‑dimensional quantum systems.
Abstract
In this paper we argue that, in addition to the statistical structure of quantum theory, another structure, referred to here as the ``Noether structure," is necessary to describe the composition of systems and to define completely positive operations. A Noether structure reflects the dual role of Hermitian operators as observables on the one hand and as generators of symmetry transformations on the other. This idea has been expressed in a similar form in the works of Alfsen and Shultz, who investigated the conditions under which the Jordan product can be extended to an associative product of operator algebras. Our investigations into the Noether structure and the composition of systems are limited to the finite-dimensional case and establish a connection to completely positive operations. In the case of pure operations, the latter can be characterized as orientation-preserving maps.
Contains some corrections, extensions and further references