Introduction to Weak Measurements and Weak Values

Abstract

W e present a short review of the theory of weak measurement. This should serve as a map for the theory and an easy way to get familiar with the main results, problems and paradoxes raised by the theory. Quanta 2013; 2: 7-17.

Quantum Weak Values and Logic: An Uneasy Couple

“…As Aharonov, Albert and Vaidman [5] originally showed -and as has later been exploited by many authors in many different ways [6][7][8][9][10][11][12] -reducing the strength of the intermediate measurement interaction between the system and the measuring device (here called the "meter') has astonishing and fruitful implications. In particular, in the weak measurement limit, the change of the mean value Q of the meter's pointer variable Q and of the mean value P of the meter's conjugate momentum are directly proportional to, respectively, the real and the imaginary part of the weak value…”

“…[2,3] for reviews and further references. 1 It is also the view taken in the approach by Aharonov and his many collaborators, starting with paper [4] (in the sequel referred to as ABL) and continuing in the development of the concepts of weak measurements and weak values [5]; for some reviews see [6][7][8][9][10][11][12]. In this paper, I shall mostly follow this line of thought of Aharonov and collaborators.…”

“…A basic one is that a weak value does indeed involve an incoming, preselected, state |in as well as a final, postselected state | f for the system under investigation [5][6][7][8][9][10][11][12]. The interest is then focused on what may be said regarding the system at an intermediate time, i.e.…”

“…The main new aspect in the weak value approach compared to the original ABL approach [4] (which involves invoking ordinary projective/strong measurements) is that one now invokes weak intermediate measurements. As usually presented [5][6][7][8][9][10][11][12], a weak measurement crucially involves a von Neumann type interaction with a measuring device, a meter. It is in the limit of the strength of this measuring interaction tending to zero that a weak value emerges.…”

What Is a Quantum-Mechanical “Weak Value” the Value of?

“…The basic idea of weak measurement + post-selection has since attracted much interest; for recent reviews with further references see, e.g., [2][3][4]. Not the least has the field opened up new tools for experimentalists to investigate aspects of phenomena that were thought impossible earlier.…”