The meaning of "anomalous weak values" in quantum and classical theories

TL;DR

This paper explains the phenomenon of anomalous weak values in quantum mechanics, showing they can take any real number depending on pre- and post-selected states, a behavior not found in classical statistics.

Contribution

It demonstrates that anomalous weak values arise naturally from quantum uncertainty and cannot be replicated in classical probability, clarifying their quantum-specific nature.

Findings

Weak measurements can yield arbitrary real values.

Anomalous weak values involve negative probability weights.

This phenomenon is unique to quantum systems, not classical ones.

Abstract

The readings of a highly inaccurate "weak" quantum meter, employed to determine the value of a dichotomous variable $S$ without destroying the interference between the alternatives,may take arbitrary values. We show that the expected values of its readings may take any real value, depending on the the choice of the states in which the system is pre- and post-selected. Some of these values must fall outside the range of eigenvalues of $A$, in which case they may be expressed as "anomalous" averages obtained with negative probability weights, constructed from available probability amplitudes. This behaviour is a natural consequence of the Uncertainty Principle. The phenomenon of "anomalous weak values" has no non-trivial analogue in classical statistics.