Abstract
The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.
Highlights
Erwin Schrödinger shared with Einstein an enormous puzzlement about the implications of the laws that were being uncovered in the investigation of atomic processes
We will address a problem that underlines some of the previous discussions and that consists of determining whether or not, starting with a composite system made up of a classical and a quantum part, and which is in a separable state, it is possible to build an entangled state by means of a unitary evolution of the system
The notion of classical systems is discussed, and for countable groupoids, it is found that a classical system in the sense of the quantum logic of Birkhoff and von Neumann is necessarily associated with a totally disconnected groupoid with Abelian isotropy subgroups, and vice versa
Summary
Erwin Schrödinger shared with Einstein an enormous puzzlement about the implications of the laws that were being uncovered in the investigation of atomic processes. Gedankenexperiment [1], Einstein, Podolski, and Rosen showed the conflicting relation between “Elements of Physical Reality” and the notions of separability and independence in quantum mechanics (see the recent analysis of such a situation in the recent paper [2]). Schrödinger showed his bewilderment in a series of reflections summarized by his famous experiment involving a macroscopic body (a cat) and a quantum system [3], where he argued about the conflict between “common sense” and what we refer to as an entangled state between a cat and some radioactive material. The formulation of the problem in the groupoid formalism will allow us to present an example that escapes Raggio’s theorem in the sense that, even if both algebras of the systems are non-Abelian, there are separable states that are sent to separable states by all unitary dynamics of the composite system
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