Weak Values and Quantum Properties

Enrico Rebufello, Fabrizio Piacentini, Alessio Avella +7 more

Despite being very influential on both foundations and applications of quantum mechanics, weak values are still somewhat controversial. Although there are some indications that weak values are physical properties of a single quantum system, the common way weak values are presented is statistical: it is commonly believed that for measuring weak values one has to perform many weak measurements over a large ensemble of pre- and postselected particles. Other debates surround the anomalous nature of weak value and even their quantumness. To address these issues, we present some preliminary data showing that anomalous weak values can be measured using just a single detection, i.e. with no statistics. In our experiment, a single click of a detector indicates the weak value as a single photon property, which moreover lies well beyond the range of eigenvelues of the measured operator. Importantly, the uncertainty with which the weak values is measured is smaller than the difference between the weak value and the closet eigenvalue. This is the first experimental realization of robust weak measurements.

D. Sokolovski, Elena V. Akhmatskaya

Lev Vaidman A

Lev Vaidman A, Alon Ben-Israel, Jan Dziewior +6 more

It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.

Xavier Oriols

The time derivative of a physical property often gives rise to another meaningful property. Since weak values provide empirical insights that cannot be derived from expectation values, this paper explores what physical properties can be obtained from the time derivative of weak values. It demonstrates that, in general, the time derivative of a weak value that is invariant under an electromagnetic gauge transformation is neither a weak value nor a gauge-invariant quantity. Left- and right-hand time derivatives of a weak value are defined, and two necessary and sufficient conditions are presented to ensure that they are also gauge-invariant weak values. Finite-difference approximations of the left- and right-hand time derivatives of weak values are also presented, yielding results that match the theoretical expressions when these are gauge invariant. With these definitions, a local Ehrenfest-like theorem can be derived, giving a natural interpretation for the time derivative of weak values. Notably, a single measured weak value of the system's position provides information about two additional unmeasured weak values: the system's local velocity and acceleration, through the first- and second-order time derivatives of the initial weak value, respectively. These findings also offer guidelines for experimentalists to translate the weak value theory into practical laboratory setups, paving the way for innovative quantum technologies. An example illustrates how the electromagnetic field can be determined at specific positions and times from the first- and second-order time derivatives of a weak value of position.

B. E. Y. Svensson