The meaning of “anomalous weak values” in quantum and classical theories

• Average reading of a weak meter can take any value, depending on the transition.
• No information about intrinsic properties of the measured system, e.g., the size of a spin.
• This is a direct consequence of the Uncertainty Principle, which forbids distinguishing between interfering alternatives.
• Some of the average have to lie outside the spectrum of the measured operator, i.e., be “anomalous”.
• There can be no anomalous averages in a purely classical theory.

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 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 S, 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.