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The variance of an observable in a pre-selected quantum system, which isalways real and non-negative, appears as an increase in the probe wave packetwidth in indirect measurements. Extending this framework to pre- andpost-selected systems, we formulate a complex-valued counterpart of thevariance called "weak variance." In our formulation, the real and imaginaryparts of the weak variance appear as changes in the probe wave packet width inthe vertical-horizontal and diagonal-antidiagonal directions, respectively, onthe quadrature phase plane. Using an optical system, we experimentallydemonstrate these changes in the probe wave packet width caused by the realnegative and purely imaginary weak variances. Furthermore, we show that theweak variance can be expressed as the variance of the weak-valued probabilitydistribution in pre- and post-selected systems. These operational andstatistical interpretations support the rationality of formulating the weakvariance as a complex counterpart of the variance in pre- and post-selectedsystems.

Complex counterpart of variance in quantum measurements for pre- and post-selected systems