Quantum theory and determinism

Gino Tarozzi

Niall Shanks

Victoria J Wright

Abstract Quantum theory is indeterministic, but not completely so. When a system is in a pure state there are properties it possesses with certainty, known as actual properties. The actual properties of a quantum system (in a pure state) fully determine the probability of finding the system to have any other property. We will call this principle, wherein the deterministic elements of a theory completely characterise the probabilistic elements, intermediate determinism. In dimensions of at least three, intermediate determinism in quantum theory is guaranteed by the structure of its lattice of properties. This observation follows from Gleason’s theorem, which is why it fails to hold in dimension two. In this work we extend the idea of intermediate determinism from properties to measurements. Under this extension intermediate determinism follows from the structure of quantum effects for separable Hilbert spaces of any dimension, including dimension two. Then, we find necessary and sufficient conditions for a general probabilistic theory to obey intermediate determinism. We show that, although related, both the no-restriction hypothesis and a Gleason-type theorem are neither necessary nor sufficient for intermediate determinism.

Giacomo Mauro D’Ariano, Franco Manessi, Paolo Perinotti

Causality has often been confused with the notion of determinism. It is mandatory to separate the two notions in view of the debate about quantum foundations. Quantum theory provides an example of causal non-deterministic theory. Here we introduce a toy operational theory that is deterministic and non-causal, thus proving that the two notions of causality and determinism are totally independent.

Grete Hermann

Eftichios Bitsakis

Four forms of physical determination are presently known in physics: classical or mechanical determination, related to classical mechanics; dynamical determination, related to the relativistic theories of electromagnetism and gravitation; classical statistical determination, related to classical statistical theories; and quantum statistical determination, related to modern microphysics. The concept of interaction is a fundamental on-tological category, a moment of which is the causal relation; interaction is also local, taking place at finite velocities. Following the evolution of physical theories, it is possible to attest to the historicity of the concept of interaction and, accordingly, of the categories of interaction and determination. According to the Copenhagen School, causality and locality are incompatible with quantum mechanics. On the contrary, according to the point of view developed here, realism, causality and locality are simultaneously compatible in the domain of microphysics.