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Physica Scripta

On the Two-State Vector Reformulation of Quantum Mechanics

Yakir Aharonov, Lev Vaidman A

Published Jan 1, 1998• 19 citations
Publisher

Abstract

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.

DOI
10.1238/physica.topical.076a00085

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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.
A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first preselected by the initial measurement and the second postselected 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 measurement procedure that fulfills certain requirements of ``weakness.'' We call it weak measurement. Physical meaning, underlying mathematical structure, and prospects of practical usage of weak measurements are explored.
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.
The novel physical quantity known as the weak value has attracted much attention for two reasons.First, from the perspective of applications, since the weak value can take arbitrarily large values, it is expected to be useful for detecting very small physical signals.Indeed, following an initial successful demonstration, numerous reports have supported the utility of this method, in which small signals are amplified through large weak values.Second, from the perspective of the foundations of quantum mechanics, the weak value is expected to shed light on certain quantum paradoxes.Its proponents argue that the weak value reveals the behavior of a quantum system prior to measurement.In this dissertation, we analyze the Two-State Vector Formalism (TSVF), which is the theoretical origin of the discovery of the weak value.The TSVF describes a quantum system that is conditioned by both pre-and post-selection.Therefore, the term "weak" does not necessarily imply that the formalism itself requires weak interactions.Rather, it refers to the fact that the measurement performed to detect the weak value is weak, typically characterized by a small coupling constant.While the weak value has attracted much attention, most studies have focused on the weak coupling regime, and the strong or intermediate coupling regions remain relatively unexplored.In this work, we numerically analyze systems described by the TSVF beyond the weak limit.Our findings suggest that the amount of information obtainable from the post-selected system has an upper bound.This observation motivates an information-theoretic analysis, through which we identify peculiar behavior in the information content associated with the quantum state in weak regime.
In this paper we present the two-state vector formalism of quantum mechanics. It is a time-symmetrized approach to standard quantum theory particularly helpful for the analysis of experiments performed on pre- and post-selected ensembles. Several peculiar effects which naturally arise in this approach are considered. In particular, the concept of ``weak measurements'' (standard measurements with weakening of the interaction) is discussed in depth revealing a very unusual but consistent picture. Also, a design of a gedanken experiment which implements a kind of quantum ``time machine'' is described. The issue of time-symmetry in the context of the two-state vector formalism is clarified.
In this paper we present the two-state vector formalism of quantum mechanics. It is a time-symmetrized approach to standard quantum theory particularly helpful for the analysis of experiments performed on pre- and post-selected ensembles. Several peculiar effects which naturally arise in this approach are considered. In particular, the concept of ``weak measurements'' (standard measurements with weakening of the interaction) is discussed in depth revealing a very unusual but consistent picture. Also, a design of a gedanken experiment which implements a kind of quantum ``time machine'' is described. The issue of time-symmetry in the context of the two-state vector formalism is clarified.