Weak value

Weak Measurements are a type of Quantum measurement , where the measured system is very weakly coupled to the measuring device. After the measurement the measuring device is shifted by what is called the "weak value". The system is not disturbed by the measurement. Although this may seem to contradict some basic aspects of quantum theory, the formalism lies within the boundaries of the theory and does not contradict any fundamental concept.

The idea of weak measurements and weak values, first developed by Yakir Aharonov, David Albert and Lev Vaidman is especially useful for gaining information about pre and post selected systems described by the two state vector formalism. Since a "strong" perturbative measurement can both upset the outcome of the post selection and tamper with all subsequent measurement, weak nonperturbative measurements may be used to learn about such systems during their evolution.

If $|\phi _{1}\rangle$ and $|\phi _{2}\rangle$ are the pre- and post-selected quantum mechanical states, the weak value of the observable ${\hat {\mathbf {A} }}$ is defined as

$A_{w}={\frac {\langle \phi _{1}|{\hat {\mathbf {A} }}|\phi _{2}\rangle }{\langle \phi _{1}|\phi _{2}\rangle }}.$

The weak value of the observable becomes large when the post-selected state, $|\phi _{2}\rangle$, approaches being orthogonal to the pre-selected state, $|\phi _{1}\rangle$. In this way, by properly choosing the two states, the weak value of the operator can be made arbitrarily large, and otherwise small effects can be amplified.