Quantum uncertainties and Heisenberg-like uncertainty relations for a weak measurement scheme involving two arbitrary noncommuting observables

Abstract

We study theoretically a weak measurement of an observable $A$ and its impact on a subsequent strong measurement of another observable $B$. We assume that both measurements are described by the Gaussian point detectors. It is shown that a decrease of the system-detector interaction during the measurement of $A$ results in an increase of the fluctuations of $A$, while the mean value of $A$ is kept unaltered. The ratio ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}/{(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ turns out to be universal, i.e., independent of the physical system, its quantum state, and the quantity measured. Most of our attention is focused on analyzing outcomes of the subsequent strong measurement of the quantity $B$, namely, on the mean value $\overline{B}$ and the variance ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$. We derive explicit analytic corrections for $\overline{B}$ and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ arising due to the prior weak measurement of $A$. Importantly, the leading-order impact of the weak measurement of $A$ on $\overline{B}$ and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ is encoded in the terms ${\phantom{\rule{0.16em}{0ex}}}_{s}\ensuremath{\langle}\ensuremath{\psi}|[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}}^{k}]]{|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$, where $k=1,2$ and ${|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$ is the quantum state vector of our system before the measurements. Our additional contribution consists of deriving Heisenberg-like uncertainty relations for the product ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}\phantom{\rule{0.28em}{0ex}}{(\mathrm{\ensuremath{\Delta}}B)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ and analyzing the case of minimum uncertainty. Finally, our theoretical developments are illustrated numerically via studying the $s$ states of the hydrogen atom, assuming $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}$ (atomic radius) and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ (energy).