Hidden variables and locality

Bell's problem of the possibility of a local hidden variable theory of quantum phenomena is considered in the context of the general problem of representing the statistical states of a quantum mechanical system by measures on a classical probability space, and Bell's result is presented as a generalization of Maczynski's theorem for maximal magnitudes. The proof of this generalization is shown to depend on the impossibility of recovering the quantum statistics for sequential probabilities in a classical representation without introducing a randomization process for the hidden variables. Hidden variable theories that exclude such a randomization process are termed strict, and it is shown that the class of local hidden variable theories is included in the class of strict theories. A counterargument by Freedman and Wigner is evaluated with reference to Clauser's extension of a hidden variable model proposed by Bell.     

Hidden variable models for quantum theory cannot have any local part

It was shown by Bell that no local hidden variable model is compatible with quantum mechanics. If, instead, one permits the hidden variables to be entirely nonlocal, then any quantum mechanical predictions can be recovered. In this Letter, we consider general hidden variable models which can have both local and nonlocal parts. We show the existence of (experimentally verifiable) quantum correlations that are incompatible with any hidden variable model having a nontrivial local part, such as the model proposed by Leggett.     

Approximate Hidden Variables

The usual definition of (non-contextual) hidden variables is found to be too restrictive, in the sense that, according to it, even some classical systems do not admit hidden variables. A more general concept is introduced and the term approximate hidden variables is used for it. This new concept avoids the aforementioned problems, since all classical systems admit approximate hidden variables. Standard quantum systems do not admit approximate hidden variables, unless the corresponding Hilbert space is 2-dimensional. However, an appropriate non-standard quantum system, which arises by focussing on momentum and position and neglecting the remaining observables, admits approximate hidden variables. Systems associated with JBW-algebras (resp. von Neumann algebras) and satisfying some mild conditions admit approximate hidden variables iff they are classical, that is, iff the corresponding JBW-algebra (resp. von Neumann algebra) is associative (resp. commutative).     

Nonlocality and conservation laws in hidden variable theories

It is shown that any hidden variable model that reproduces quantum mechanics for a single particle must either be nonlocal or violate conservation of momentum. This is established by deriving an inequality which must hold in any local, momentum-conserving hidden variable model for a modified form of the double-slit experiment. It is then shown that any hidden variable model that reproduces quantum mechanics must violate the inequality. The inconsistency between the classical and quantum views of the world is therefore demonstrated in a new way.     

Hidden Variables with Nonlocal Time

To relax the apparent tension between nonlocal hidden variables and relativity, we propose that the observable proper time is not the same quantity as the usual proper-time parameter appearing in local relativistic equations. Instead, the two proper times are related by a nonlocal rescaling parameter proportional to |ψ|², so that they coincide in the classical limit. In this way particle trajectories may obey local relativistic equations of motion in a manner consistent with the appearance of nonlocal quantum correlations. To illustrate the main idea, we first present two simple toy models of local particle trajectories with nonlocal time, which reproduce some nonlocal quantum phenomena. After that, we present a realistic theory with a capacity to reproduce all predictions of quantum theory.     

Interaction between classical and quantum systems and the measurement of quantum observables

Quantum mechanics presumes classical measuring instruments with which they interact. The problem of measurement interaction between classical and quantum systems is posed and solved. The restriction to compatible measurements comes about naturally as the condition for the integrity of the classical system. A technical device is the perspective on classical mechanics as quantum mechanics with essentially hidden dynamical variables. Work supported in part by U.S. Atomic Energy Commission, ERDA.     

Nonlocality criteria for quantum teleportation

An upper bound for the fidelity of quantum teleportation explainable by local hidden variables is derived. This bound is larger than the fidelity corresponding to product states, i.e. to local quantum states. This is relevant for the study of mixed states. In particular, the fidelity of Werner's mixed state, known to be larger than the fidelity of product states, is found to be smaller than the fidelity explainable by local hidden variables. Hence the fidelity of Werner's mixed state does not exhibit nonclassical aspects.     

INDISTINGUISHABLE PARTICLES AND HIDDEN VARIABLES: PART II

In a previous work it was shown that it is possible to deal with collections of indistinguishable elementary particles in a set-theoretical framework, by using hidden variables. We propose in the present paper a set-theoretical axiomatics for collections of indiscernibles with no explicit mention to hidden variables. We also show, in this context, the fundamental role of the (micro) state in the process of individuation of classical and quantum particles. Finally, we discuss the importance of the axiom of choice in Zermelo-Fraenkel set theory in the context of quantum distributions of bosons and fermions.     

Bayes' rule and hidden variables

We show that a quantum system admits hidden variables if and only if there is a rich set of states which satisfy a Bayesian rule. The result is proved using a relationship between Bayesian type states and dispersion-free states. Various examples are presented. In particular, it is shown that for classical systems every state is Bayesian and for traditional Hilbert space quantum systems no state is Bayesian.     

Dynamics of a qubit as a classical stochastic process with time-correlated noise: minimal measurement invasiveness

So far it has been shown that the quantum dynamics cannot be described as a classical Markov process unless the number of classical states is uncountably infinite. In this Letter, we present a stochastic model with time-correlated noise that exactly reproduces any unitary evolution of a qubit and requires just four classical states. The invasive updating of only 1 bit during a measurement accounts for the quantum violation of the Leggett-Garg inequalities. Unlike in a pilot-wave theory, the stochastic forces governing the jumps among the four states do not depend on the quantum state but only on the unitary evolution. This model is used to derive a local hidden variable model, augmented by 1 bit of classical communication, for simulating entangled Bell states.     

Hiding classical data in multipartite quantum states

We present a general technique for hiding a classical bit in multipartite quantum states. The hidden bit, encoded in the choice of one of two possible density operators, cannot be recovered by local operations and classical communication without quantum communication. The scheme remains secure if quantum communication is allowed between certain partners, and can be designed for any choice of quantum communication patterns to be secure, but to allow near perfect recovery for all other patterns. No entanglement is needed since the hiding states can be chosen to be separable. A single ebit of prior entanglement is not sufficient to break the scheme.     

A competitive game whose maximal Nash-equilibrium payoff requires quantum resources for its achievement

While it is known that shared quantum entanglement can offer improved solutions to a number of purely cooperative tasks for groups of remote agents, controversy remains regarding the legitimacy of quantum games in a competitive setting. We construct a competitive game between four players based on the minority game where the maximal Nash-equilibrium payoff when played with the appropriate quantum resource is greater than that obtainable by classical means, assuming a local hidden variable model.     

Simulating POVMs on EPR pairs with 5.7 bits of expected communication

We present a classical protocol for simulating correlations obtained by bipartite POVMs on an EPR pair. The protocol uses shared random variables (also known as local hidden variables) augmented by 5.7 bits of expected communication.     

Hidden duality and associated instabilities of Tomonaga–Luttinger liquid on lattice

Hidden duality and its associated instabilities of the spinless Luttinger liquid on lattice are reported. The local quantum fluctuations due to the general multi-particle umklapp and other processes and the long-distance chiral modes compete and as a result produce a hierarchy of exotic charge density instabilities. Explicit bosonic quantum operators for the local density fluctuations are constructed and are used to make identification of the Luttinger liquid with the classical 2D Coulomb gas with -term and with the rich hidden duality.     

Statistical definition of observable and the structure of statistical models

A purely statistical characterization of measurements of observables (described by spectral measures in conventional formalism of quantum mechanics) is given in the framework of the general statistical (convex) approach. The relation to physical premises underlying the conventional notion of observable is discussed. Structural aspects of general statistical models such as central decomposition and characterization of classical models are considered. It is shown by explicit construction that an arbitrary statistical model admits a formal introduction of “hidden variables” preserving the structural properties of a single statistical model. The relation of this result to other theorems on hidden variables is discussed.     

Contextualist viewpoint to Greenberger–Horne–Zeilinger paradox

We present probabilistic analysis of the Greenberger–Horne–Zeilinger (GHZ) scheme in the contextualist framework, namely under the assumption that distributions of hidden variables depend on settings of measurement devices. On one hand, we found classes of probability distributions of hidden variables for that the GHZ scheme does not imply a contradiction between the local realism and quantum formalism. On the other hand, we found classes of probability distributions of hidden variables for that the GHZ scheme still induce such a contradiction (despite variations of distributions). It is also demonstrated that (well known in probability theory) singularity/absolute continuity dichotomy for probability distributions is closely related to the GHZ paradox. Our conjecture is that this GHZ coupling between singularity/absolute continuity dichotomy and incompatible/compatible measurements might be a general feature of quantum theory.     

Causal action-at-a-distance interpretation of the aspect-Rapisarda experiments

Recent results in EPR-type experiments on singlet photon pairs which establish (a) the non-existence of local hidden variables, (b) the existence of quantum superluminal correlations between the action of independent parts of a measuring device separated by space-like intervals, are interpreted causally within the frame of the stochastic interpretation of quantum mechanics.     

The End of a Classical Ontology for Quantum Mechanics?

In this paper, I argue that the Shrapnel–Costa no-go theorem undermines the last remaining viability of the view that the fundamental ontology of quantum mechanics is essentially classical: that is, the view that physical reality is underpinned by objectively real, counterfactually definite, uniquely spatiotemporally defined, local, dynamical entities with determinate valued properties, and where typically ‘quantum’ behaviour emerges as a function of our own in-principle ignorance of such entities. Call this view Einstein–Bell realism. One can show that the causally symmetric local hidden variable approach to interpreting quantum theory is the most natural interpretation that follows from Einstein–Bell realism, where causal symmetry plays a significant role in circumventing the nonclassical consequences of the traditional no-go theorems. However, Shrapnel and Costa argue that exotic causal structures, such as causal symmetry, are incapable of explaining quantum behaviour as arising as a result of noncontextual ontological properties of the world. This is particularly worrying for Einstein–Bell realism and classical ontology. In the first instance, the obvious consequence of the theorem is a straightforward rejection of Einstein–Bell realism. However, more than this, I argue that, even where there looks to be a possibility of accounting for contextual ontic variables within a causally symmetric framework, the cost of such an account undermines a key advantage of causal symmetry: that accepting causal symmetry is more economical than rejecting a classical ontology. Either way, it looks like we should give up on classical ontology.