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arXiv:1212.2379 · International Journal of Quantum Information

THE PHYSICS OF QUANTUM INFORMATION: COMPLEMENTARITY, UNCERTAINTY, AND ENTANGLEMENT

Joseph M. Renes

Published Dec 1, 2013• 14 citations
arXivHTMLPDF

Abstract

Complementarity is one of the central mysteries of quantum mechanics, dramatically illustrated by the wave-particle duality in Young's double-slit experiment, and famously regarded by Feynman as "impossible, absolutely impossible to describe classically, [and] which has in it the heart of quantum mechanics" (emphasis original).1 The overarching goal of this thesis is to demonstrate that complementarity is also at the heart of quantum information theory, that it allows us to make (some) sense of just what information "quantum information" refers to, and that it is useful in understanding and constructing quantum information processing protocols.

DOI
10.1142/s0219749913300027

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The double-slit experiment

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Complementarity is one of the central mysteries of quantum mechanics, dramatically illustrated by the wave-particle duality in Young's double-slit experiment, and famously regarded by Feynman as "impossible, absolutely impossible to describe classically, [and] which has in it the heart of quantum mechanics" (emphasis original).1 The overarching goal of this thesis is to demonstrate that complementarity is also at the heart of quantum information theory, that it allows us to make (some) sense of just what information "quantum information" refers to, and that it is useful in understanding and constructing quantum information processing protocols.
Complementarity, e.g., the wave–particle duality of nature, lies at the heart of quantum mechanics. It distinguishes the world of quantum phenomena from the reality of classical physics. In the 1920s, quantum theory as we know it today was still new, and examples used to illustrate complementarity emphasized the position (particle-like) and momentum (wave-like) attributes of a quantum mechanical object, be it a photon or a massive particle. This is the historical reason why complementarity is often superficially identified with the so-called wave-particle duality of matter . Complementarity, however, is a more general concept. We say that two observables are complementary if precise knowledge of one of them implies that all possible outcomes of measuring the other one are equally probable. We may illustrate this by two extreme examples. The first example consists of the position and momentum (along one direction) of a particle: if, say, the position is predetermined then the result of a momentum measurement cannot be predicted, all momentum values are equally probable (in a large range). The second extreme involves two orthogonal spin components of a spin- 1/2 particle: if, say, the vertical spin component has a definite value ( up or down ) then upon measuring a horizontal component both values ( left or right , for instance) are found, each with a probability of 50%. Thus, in the microcosmos complete knowledge in the sense of classical physics is not available. The classic example of the merger of wave and particle behavior is provided by Young's double-slit experiment.
Complementarity lies at the heart of conceptual foundation of orthodox quantum mechanics. The wave-particle duality makes it impossible to tell which slit each particle passes through and still observe an interference pattern in a Young's double-slit experiment. In this paper, this duality is appraised under the standard formulation of quantum mechanics for atom interferometers. It is found that the internal freedoms like electronic states of an atom can be used to detect the which-path information of each atom while uphold the interference pattern of atoms when the center-of-mass motion of atoms is detected.
A critical re-examination of the double-slit experiment and its variants is presented to clarify the nature of what Feynman called the “central mystery” and the “only mystery”of quantum mechanics, leading to an interpretation of complementarity in which and description rather than a wave or particle' description is valid for the same experimental set up, with the wave culminating in the particle sequentially in time. This interpretation is different from Bohr's but is consistent with the von Neumann formulation as well as some more recent interpretations of quantum mechanics.
A critical re-examination of the double-slit experiment and its variants is presented to clarify the nature of what Feynmann called the ``central mystery'' and the ``only mystery'' of quantum mechanics, leading to an interpretation of complementarity in which a `wave {\em and} particle' description rather than a `wave {\em or} particle' description is valid for the {\em same} experimental set up, with the wave culminating in the particle sequentially in time. This interpretation is different from Bohr's but is consistent with the von Neumann formulation as well as some more recent interpretations of quantum mechanics.
One of the most remarkable features of quantum physics is that attributes of quantum objects, such as the wavelike and particlelike behaviors of single photons, can be complementary in the sense that they are equally real but cannot be observed simultaneously. Quantum measurements, serving as windows providing views into the abstract edifice of quantum theory, are basic tools for manifesting the intrinsic behaviors of quantum objects. However, a quantitative formulation of complementarity that highlights its manifestations in general measurement scenarios remains elusive. Here we develop a general framework for demonstrating quantum complementarity in the form of information exclusion relations (IERs), which incorporates the wave-particle duality relations as particular examples. Moreover, we explore the applications of our theory in entanglement detection and elucidate that our IERs lead to an extended form of entropic uncertainty relations, providing insights into the connection between quantum complementarity and the preparation uncertainty.
What is the most beautiful experiment in physics? More than 200 readers replied when Robert Crease asked this question in May, and their top five choices include classic experiments by Galileo, Millikan, Newton and Thomas Young (see pp19–20). But uniquely among the top 10, the most beautiful experiment in physics – the double-slit interference experiment with electrons – does not have the name of an individual associated with it. Most discussions of double-slit experiments with particles refer to Feynman's quote in his 1963 lectures: "[It] is absolutely impossible to explain in any classical way and has in it the heart of quantum mechanics. In reality, it contains the only mystery." But who actually carried out the experiment? Standard reference books offer no answer to this question but a search through the literature does reveal several unsung experimental heroes.