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

Kazuhisa Ogawa,Akihisa Tomita,Natsuki Abe,Hirokazu Kobayashi

doi:10.1103/physrevresearch.3.033077

Kazuhisa Ogawa, Akihisa Tomita + Show 2 more

Open Access

https://doi.org/10.1103/physrevresearch.3.033077

Copy DOI

Abstract

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

Highlights

The outcomes of quantum measurements show probabilistic behavior
The probe wave packet remains a pure state while its shape changes; the weak variance does not represent the statistical nature of the probabilistic mixture of probe wave packets and is consistent with the claim of Vaidman et al Considering higher orders of θ, the probe wave packet undergoes a non-Gaussian transformation specified by the higher order weak moments An w, as described at the end of this paper
We introduced the weak variance σw2 (A) as a complex counterpart of the variance in pre- and postselected systems

Summary

INTRODUCTION

The outcomes of quantum measurements show probabilistic behavior. This characteristic, which is not observed in classical systems, has been the root of many fundamental arguments in quantum theory [1]. For pre- and postselected systems, any counterparts of the variance cannot be observed in the typical framework of the weak measurement [3], in which the probe wave packet width does not change because the second- and higher order terms of the coupling strength are ignored. We focus on the recent studies reporting that when considering the second- and higher order terms of the coupling strength, the probe wave packet width can increase and decrease under appropriate pre- and postselection conditions [22,23] If these reported phenomena are interpreted to result from a counterpart of the variance in pre- and postselected systems, it may be possible to formulate an effective variancelike quantity that can be negative. We formulate a counterpart of the higher order moment and investigate its operational and statistical meanings and applications

WEAK VARIANCE APPEARING IN INDIRECT MEASUREMENT FOR PRE- AND POSTSELECTED

EXPERIMENTAL DEMONSTRATION OF WEAK VARIANCES

WEAK VARIANCE AS A STATISTIC OF THE WEAK-VALUED PROBABILITY DISTRIBUTION

CONCLUSION