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

The variance of an observable for a preselected quantum system, which is always real and non-negative, appears in the increase of the probe wave packet width in indirect measurement. We extend this framework to pre- and postselected systems, and formulate a complex-valued counterpart of the variance called "weak variance." In our formulation, the real and imaginary parts of the weak variance appear in the changes in the probe wave packet width in vertical-horizontal and diagonal-antidiagonal directions on the quadrature phase plane, respectively. Using an optical system, we experimentally demonstrate these changes in a probe wave packet width caused by the real negative and pure imaginary weak variances. Furthermore, we describe that the weak variance can be expressed as the variance of the weak-valued probability distribution for pre- and post-selected systems. These operational and statistical interpretations support that our formulation of the weak variance is reasonable as a complex counterpart of the variance for pre- and post-selected systems.

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