Abner Shimony

Bell’s theorem without inequalities

It is demonstrated that the premisses of the Einstein–Podolsky–Rosen paper are inconsistent when applied to quantum systems consisting of at least three particles. The demonstration reveals that the EPR program contradicts quantum mechanics even for the cases of perfect correlations. By perfect correlations is meant arrangements by which the result of the measurement on one particle can be predicted with certainty given the outcomes of measurements on the other particles of the system. This incompatibility with quantum mechanics is stronger than the one previously revealed for two‐particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations. Both spin‐correlation and multiparticle interferometry examples are given of suitable three‐ and four‐particle arrangements, both at the gedanken and at the real experiment level.

Bell's theorem: Experimental tests and implications

Bells theorem represents a significant advance in understanding the conceptual foundations of quantum mechanics. The theorem shows that essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a single experimental arrangement. Moreover, the predictions by those theories must significantly differ from those by quantum mechanics. Experimental results evidently refute the theorems predictions for these theories and favour those of quantum mechanics. The conclusions are philosophically startling: either one must totally abandon the realistic philosophy of most working scientists, or dramatically revise out concept of space-time.

Optimal distinction between two non-orthogonal quantum states

Abstract Two procedures are developed for classifying an individual system as | p > or | q >, non-orthogonal, given an ensemble with respective proportions r and 1— r . One (generalizing Ivanovic, Dieks, and Peres) infallibly classifies some systems, leaving others unclassified. The second is statistically optimum, allowing individual errors.

Role of the Observer in Quantum Theory

In quantum theory as it is currently formulated the measurement of an observable quantity of a physical system is the occasion for a change of state of the system, except when the state prior to the measurement is an eigenstate of the observable. Two proposals for interpreting this kind of change are examined in detail, and several variant proposals are considered briefly. According to the interpretation proposed by von Neumann, and by London and Bauer the change of state is completed only when the result of the observation is registered in the observers consciousness. Although this interpretation appears to be free from inconsistencies, it is not supported by psychological evidence and it is difficult to reconcile with the intersubjective agreement of several independent observers. According to the interpretation proposed by Bohr, the change of state is a consequence of the fundamental assumption that the description of any physical phenomenon requires reference to the experimental arrangement. Bohrs proposals are valuable as practical maxims in scientific activity, but they are shown to involve the renunciation of any ontological framework in which all types of events—physical and mental, microscopic and macroscopic—can be located. It is concluded that a satisfactory account of the observation of microphysical quantities is unlikely if the present formulation of quantum theory is rigorously maintained.

Contextual Hidden Variables Theories and Bell's Inequalities

Noncontextual hidden variables theories, assigning simultaneous values to all quantum mechanical observables, are inconsistent by theorems of Gleason and others. These theorems do not exclude contextual hidden variables theories, in which a complete state assigns values to physical quantities only relative to contexts. However, any contextual theory obeying a certain factorisability conditions implies one of Bells Inequalities, thereby precluding complete agreement with quantum mechanical predictions. The present paper distinguishes two kinds of contextual theories, ‘algebraic’ and ‘environmental’, and investigates when factorisability is reasonable. Some statements by Fine about the philosophical significance of Bells Inequalities are then assessed.

Degree of Entanglementa

As Schrodinger discovered, the general principles of quantum mechanics imply that a quantum state of a many-particle system may be “entangled”, in the sense of not being a product of single-particle states. The concept of entanglement enters some deep investigations of the foundations of quantum mechanics, including the argument of Einstein-Podolsky-Rosen and the theorem of Bell. It is basic to the program of two-particle (or n-particle) interferometry. Of course, entanglement is exhibited by almost all of the interesting quantum states of atoms, molecules, nuclei, and condensed matter systems. The present report exhibits some mathematical relations among three concepts associated with entanglement. The second section proposes a general quantitative definition of the degree of entanglement E ( + ) of any n-particle quantum state and when n = 2 evaluates E ( + ) . (Note that the same symbol + is used for the quantum state and for the normalized vector in a Hilbert space that represents this state.) The third section is restricted to the case of two particles, each associated with a two-dimensional Hilbert space, called “the two-by-two case”. In this case, it is possible to calculate the two-particle fringe visibility V,*(+) and also the quantity B(+), which is the maximum deviation achievable, when the state is +, from the limit allowed by one of Bell’s inequalities. It is shown that V12(+) and B(+) are monotonic increasing functions of E ( + ) .

Jaynes's maximum entropy prescription and probability theory

Jayness prescription of maximizing the information-theoretic entropy is applied in a special situation to determine a certain set of posterior probabilities (when evidence fixing the expected value of a dynamical variable is given) and also the corresponding set of prior probabilities (when this evidence is not given). It is shown that the resulting values of these probabilities are inconsistent with the principles of probability theory. Three possible ways of avoiding this inconsistency are briefly discussed.

Insolubility of the quantum measurement problem for unsharp observables

Abstract The quantum mechanical measurement problem is the difficulty of dealing with the indefiniteness of the pointer observable at the conclusion of a measurement process governed by unitary quantum dynamics. There has been hope to solve this problem by eliminating idealizations from the characterization of measurement. We state and prove two ‘insolubility theorems’ that disappoint this hope. In both the initial state of the apparatus is taken to be mixed rather than pure, and the correlation of the object observable and the pointer observable is allowed to be imperfect. In the insolubility theorem for sharp observables , which is only a modest extension of previous results, the object observable is taken to be an arbitrary projection valued measure. In the insolubility theorem for unsharp observables , which is essentially new, the object observable is taken to be a positive operator valued measure. Both theorems show that the measurement problem is not the consequence of neglecting the ever-present imperfections of actual measurements.

Wave-packet reduction as a medium of communication

Using an apparatus in which two scalers register decays from a radioactive source, an observer located near one of the scalers attempted to convey a message to an observer located near the other one by choosing to look or to refrain from looking at his scaler. The results indicate that no message was conveyed. Doubt is thereby thrown upon the hypothesis that the reduction of the wave packet is due to the interaction of the physical apparatus with the psyche of an observer.A. Einstein(1)

The lattice of verifiable propositions of the spin‐1 system

A lattice of verifiable propositions LV for a spin‐1 system is constructed by admitting only propositions which correspond to appropriate Stern–Gerlach filters. LV is a complete, orthocomplemented, weakly modular lattice, and it satisfies the first part of the atomicity axiom of Jauch and Piron, but not the second part (the covering law) nor related axioms of Zierler and MacLaren. Doubt is therefore thrown upon the program of recovering the Hilbert space formulation of quantum mechanics from empirically justified axioms. The class of admissible states on LV is exhaustively characterized, and it is shown that there exist some nonquantal states but none that are dispersion free.