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arXiv:quant-ph/0607208 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Non-statistical weak measurements

Jeff Tollaksen

Published Apr 27, 2007• 4 citations
arXivHTMLPDF

Abstract

Weak values are the outcomes of weak measurements and were discovered by studying the time-symmetric aspects of quantum mechanics. This resulted in new approaches to quantum cryptography and new resources for quantum information. In this article, new non-statistical aspects of weak measurements are introduced. Contary to past results, this outcome is not rare, suggesting that weak values are a property of every individual pre-and-post-selected system.

DOI
10.1117/12.719329

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Weak values are the outcomes of weak measurements and were discovered by studying the time-symmetric aspects of quantum mechanics. This resulted in new approaches to quantum cryptography and new resources for quantum information. In this article, new non-statistical aspects of weak measurements are introduced. Contary to past results, this outcome is not rare, suggesting that weak values are a property of every individual pre-and-post-selected system.
The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.
A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first pre-selected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measuring procedure that fulfills certain requirements of “weakness”. We call it weak measurement . Physical meaning and underlying mathematical structure of weak measurements are explored.
Recent controversy regarding the meaning and usefulness of weak values is reviewed. It is argued that in spite of recent statistical arguments by Ferrie and Combes, experiments with anomalous weak values provide useful amplification techniques for precision measurements of small effects in many realistic situations. The statistical nature of weak values is questioned. Although measuring weak values requires an ensemble, it is argued that the weak value, similarly to an eigenvalue, is a property of a single pre- and post-selected quantum system.This article is part of the themed issue 'Second quantum revolution: foundational questions'.