The Weirdness Theorem and the Origin of Quantum Paradoxes

Abstract

We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics.

Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality

“…Whenever L is a tractable QEO, we also call the corresponding GPT tractable. In this case, we can interpret the GPT as an algorithmic (bounded) rationality theory [32]. These tractable theories model the idea that rationality (expressed by the property (A) in CPT) is limited by the available computationally resources for decision making.…”

“…The functional L : F → R is a real-valued operator defined on the space of real-valued functions F. Therefore, as in GPTs, QEOs are defined on a real-vector space. This means that, despite the aforementioned difference, the two approaches are formally equivalent when expressed using order unit spaces and using duality [32,Th.3].…”

“…In this work, we present a different but equivalent way to formulate QT as a GPT, that is in terms of quasi-expectation operators (QEOs). This approach is dual to the algorithmic (bounded) rationality theory we introduced in [32]. This formulation departs from standard GPT in three ways.…”