Colloquium: Understanding quantum weak values: Basics and applications

Abstract

Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here, a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation of how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration is used to discuss and clarify each of these applications.

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.…”

Heisenberg scaling with weak measurement: a quantum state discrimination point of view

“…Reference [10] points out that there are three possible sources of information in the measurement: the probability of the post-selection projection p + , and the information in the two meter states, |φ ± (in principle, the correlations between these outcomes also have information in them). The Fisher information contained in these channels is then calculated, and curiously, while the meter states have Fisher information that scales with N = |α| 2 (yielding the standard quantum limit), the probability of the post-selection has a Fisher information that scales as N 2 = |α| 4 , giving Heisenberg scaling in the photon number for the precision of estimating g. The main purpose of this paper is to give physical insight into why Heisenberg scaling for the parameter g can be obtained at all, and further, why it comes mainly from the probability of projecting on the system state, as opposed to mining the meter states for information, as is usually done in weak value amplification experiments [3]. Zhang, Datta, and Walmsley write that "How this conditioning step using a classical measurement apparatus achieves a precision beyond the standard quantum limit is therefore an interesting open question."…”