A Separable, Dynamically Local Ontological Model of Quantum Mechanics

Abstract

A model of reality is called separable if the state of a composite system is equal to the union of the states of its parts, located in different regions of space. Spekkens has argued that it is trivial to reproduce the predictions of quantum mechanics using a separable ontological model, provided one allows for arbitrary violations of 'dynamical locality'. However, since dynamical locality is strictly weaker than local causality, this leaves open the question of whether an ontological model for quantum mechanics can be both separable and dynamically local. We answer this question in the affirmative, using an ontological model based on previous work by Deutsch and Hayden. Although the original formulation of the model avoids Bell's theorem by denying that measurements result in single, definite outcomes, we show that the model can alternatively be cast in the framework of ontological models, where Bell's theorem does apply. We find that the resulting model violates local causality, but satisfies both separability and dynamical locality, making it a candidate for the 'most local' ontological model of quantum mechanics. INTRODUCTION"We are all made of algebra-stuff: the elements of local reality are faithfully described not by real variables or stochastic real variables but by the elements of a certain algebra that can be represented by Hermitian matrices. [...] A failure to understand this can result in various misconceptions about quantum physics in general, and about its locality in particular. "Can we explain quantum phenomena in terms of an underlying model of reality? Traditionally, such explanations took the form of hidden variable models. Bell's theorem places a severe constraint on these models, demonstrating that (given certain reasonable assumptions) no hidden variable model can reproduce the predictions of quantum mechanics and at the same time satisfy local causality [2,3]. Since Bell's work, hidden variable models have been generalised and extended to the framework of ontological models, introduced by Harrigan & Spekkens [4,5]. While the framework is not general enough to capture all possible interpretations of quantum mechanics, it is sufficiently general to permit a clear exposition of the various definitions of locality that are central to understanding the implications of Bell's theorem.We will focus on ontological models set in relativistic space-time, in which one can ascribe a real, physical state, called the ontic state, to the physical systems located in any spatial region at one time (more generally, on a space-like hyper-surface). Given a foliation of spacetime into a family of time-slices (or hyper-surfaces), the ontic state evolves with respect to the global time parameter according to some dynamical laws. Measurements performed in a region of space-time produce outcomes with probabilities that depend only on the ontic state in the region where the measurement is performed. We do not make any assumptions about the kinematics and evolution of the ontic states, other than that they reproduc...

Quantum non-locality co-exists with locality

“…where the maximization and minimization go over projectors R [25]. Indeed, if R ≤ P , and R ≤ Q in (4), then due to (1,3), S R is a subspace of both S P and S Q . The maximal such subspace is S P ∧Q .…”

“…Physics holds signal locality thereby the state of an isolated system (particle) cannot be changed by manipulating other particles that interacted with it in the past. Attempts to make this notion stronger created a bunch of quantum concepts known as non-locality [1][2][3]. Two main aspects of non-locality were identified: (i) application of the projection postulate to an entangled subsystem that changes (steers) the state of another subsystem arbitrary far away [1].…”

“…For [P, Q] ≡ P Q−QP = 0 we have from (3)(4)(5) the ordinary features of classical characteristic functions…”

“…Instead they focus on explaining quantum correlations via hidden variables, i.e., they assume structures that are additional with respect to quantum probability. Bell's inequalities can be derived assuming that the hidden variables -in addition to signal locality-hold the outcome independence feature (frequently also called local causality) [2,3,9]. Alternatively, Bell's inequality can be derived without hidden variables, but assuming that frequencies obtained via different non-commuting measurements can be embedded into a single Kolmogorovian space [10].…”