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Physics-Uspekhi

Basic quantum mechanical concepts from the operational viewpoint

Давид Николаевич Клышко

Published Sep 30, 1998• 48 citations
Publisher

Abstract

The physical meaning of the basic quantum mechanical concepts (such as the wave function, reduction, state preparation and measurement, the projection postulate, and the uncertainty principle) is clarified using realistic experimental procedures and employing classical analogies whenever possible. Photon polarization measurement and particle coordinate and momentum measurement are considered as examples, as also are Einstein–Podolsky–Rosen correlations, Aharonov–Bohm effects, quantum teleportation, etc. Various nonclassicality criteria of quantum models, including photon antibunching and the violation of the Bell inequality, are discussed.

DOI
10.1070/pu1998v041n09abeh000441

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The physical meaning of the basic quantum mechanical concepts (such as the wave function, reduction, state preparation and measurement, the projection postulate, and the uncertainty principle) is clarified using realistic experimental procedures and employing classical analogies whenever possible. Photon polarization measurement and particle coordinate and momentum measurement are considered as examples, as also are Einstein–Podolsky–Rosen correlations, Aharonov–Bohm effects, quantum teleportation, etc. Various nonclassicality criteria of quantum models, including photon antibunching and the violation of the Bell inequality, are discussed.
In this chapter the basic concepts of quantum theory are formulated via a self-consistent set of six postulates. These six postulates deal with the following: operator images of physical observables; properties of state vectors and wave functions; the connection with experimental measurements; coordinate-space forms of those operators having classical analogs; the primary dynamical equation of quantum mechanics; and the completeness of the eigenvectors of observables. The postulates are the framework on which the edifice of quantum mechanics is constructed. A separate section is devoted to the statement of each postulate and to a few examples chosen to illustrate or apply it. A comprehensive set of applications is presented in Chapters 6 and 7, where various 1-D systems are used as the vehicles for illustrating the postulates. The basic theory of this chapter is extended and developed in Chapters 8 and 9. Since probability is a key interpretational concept, a brief introduction to this topic is given in an appendix. We emphasize that our formulation of quantum theory (in this chapter and throughout this book) is restricted to the case of pure states. This is not a major restriction. It means that, when we consider ensembles of individual quantum systems (e.g., single particles, atoms, nuclei, etc.), all of the systems in the ensemble are assumed to be in the same quantum state, not necessarily an eigenstate of a quantal operator. Ensembles for which this assumption does not hold are described by a density operator or density matrix rather than by a pure state; treatments of such cases can be found in references cited later.
This textbook provides a physical understanding of what photons are and of their properties and applications. Special emphasis is made in the text to entangled photon pairs which exhibit quantum mechanical correlations over manifestly macroscopic distances. Such photon pairs make possible such exciting techniques as teleportation and quantum cryptography, as well as the physical realisation of Einstein-Podolsky-Rosen type experiments. In addition, nonclassical properties of light, such as photon antibunching and squeezing, as well as quantum phase measurement and optical tomography are discussed. The author describes relevant experiments and elucidates the physical ideas behind them. This book will be of interest to undergraduates and graduate students studying optics, and to any physicist with an interest in the mysteries of the photon and exciting modern work in quantum cryptography and teleportation.
It is shown that when properly analyzed using principles consistent with the use of a Hilbert space to describe microscopic properties, quantum mechanics is a local theory: one system cannot influence another system with which it does not interact. Claims to the contrary based on quantum violations of Bell inequalities are argued to be incorrect. A specific example traces a violation of the CHSH Bell inequality in the case of a spin-3/2 particle to the noncommutation of certain quantum operators in a situation where (non)locality is not an issue. A consistent histories analysis of what quantum measurements measure, in terms of quantum properties, is used to identify the basic problem with derivations of Bell inequalities: the use of classical concepts (hidden variables) rather than a probabilistic structure appropriate to the quantum domain. A difficulty with the original Einstein-Podolsky-Rosen (EPR) argument for the incompleteness of quantum mechanics is the use of a counterfactual argument which is not valid if one assumes that Hilbert-space quantum mechanics is complete; locality is not an issue. The quantum correlations that violate Bell inequalities can be understood using local quantum common causes. Wave-function collapse and Schr\"odinger steering are calculational procedures, not physical processes. A general Principle of Einstein Locality rules out nonlocal influences between noninteracting quantum systems. Some suggestions are made for changes in terminology that could clarify discussions of quantum foundations and be less confusing to students.
In classical physics, the ‘state’ of a particle, at any particular time, is taken to involve a definite position in space and a definite motion; and the state of a system of particles is taken to involve a definite position in space and a definite motion for each of the particles. The situation in relation to quantum mechanics is completely different. As in classical physics, calculations can be made concerning the position and motion of a particle. However, this representation is not by means of symbols for scalar or vector quantities: it is by a more complex mathematical object. This quantum state does not indicate or represent position or motion in any direct way: what it does in substance is to enable calculation of the probability of the particle in question being found to have a position or momentum if an appropriate measurement is made.
The standard presentation of the principles of quantum mechanics is critically reviewed both from the experimental/operational point and with respect to the request of mathematical consistency and logical economy. A simpler and more physically motivated formulation is discussed. The existence of non commuting observables, which characterizes quantum mechanics with respect to classical mechanics, is related to operationally testable complementarity relations, rather than to uncertainty relations. The drawbacks of Dirac argument for canonical quantization are avoided by a more geometrical approach.
An operational concept of locality whose quantum violation is indicated independently of any other assumption(s) seems to be lacking in the quantum foundations literature so far. Bell's theorem only shows that quantum correlations violate the conjunction of the ontological assumptions of localism and determinism. Taking a cue from computational complexity theory, here we define such a concept of locality in terms of a class of operational decision problems, and propose that Einstein-Podolsky-Rosen (EPR) steering in its basic asymmetric formulation gives a specific realization of this class. Various nonclassical (convex) operational theories, including quantum mechanics, are shown to be nonlocal according to this operational criterion. We discuss several ramifications arising from this result. It indicates a basic difference between quantum no-signaling and relativistic no-signaling, and suggests an information theoretic derivation of the former from other basic principles in the convex framework. It elucidates the connection between two different bounds on nonlocality, due to Oppenheim and Wehner (2010) and Banik et al. (2015), thereby highlighting the interplay and contrast between uncertainty and measurement incompatibility. Our result provides an argument supporting the realist interpretation of the quantum state, and helps clarify why Spekkens' toy model (2007) features steering but not Bell nonlocality.