Pedagogical Review of Quantum Measurement Theory with an Emphasis on Weak Measurements

The quantum theory of measurement has been with us since quantum mechanics (QM) was invented. It has recently been invigorated, partly due to the increasing interest in quantum information science. In this, partly pedagogical review, I attempt to give a self-contained overview of the basis of (non-relativistic) QM measurement theory expressed in density matrix formalism. I will not dwell on the applications in quantum information theory; it is well covered by several books in that field. The focus is instead on applications to the theory of weak measurement, as developed by Aharonov and Vaidman and their collaborators. Their development of weak measurement combined with what they call ‘post-selection’ – judiciously choosing not only the initial state of a system (‘pre-selection’) but also its final state – has received much attention recently. Not the least has it opened up new, fruitful experimental vistas, like novel approaches to amplification. But the approach has also attached to it some air of mystery. I will attempt to ‘de-mystify’ it by showing that (almost) all results can be derived in a straight-forward way from conventional QM. Among other things, I develop the formalism not only to first order but also to second order in the weak interaction responsible for the measurement. This also allows me to derive, more or less as a by-product, the master equation for the density matrix of an open system in interaction with an environment. One particular application I shall treat of the weak measurement is the so called Leggett-Garg inequalities, a k a ‘Bell inequalities in time’. I also give an outline, even if rough, of some of the ingenious experiments that the work by Aharonov, Vaidman and collaborators has inspired. If anything is magic in the weak measurement + post-selection approach, it is the interpretation of the so called weak value of an observable. Is it a bona fide property of the system considered? I have no answer to this question; I shall only exhibit the pros and cons of the proposed interpretation