Uncertainty Relations of Variances in View of the Weak Value

• URL: http://arxiv.org/abs/2008.03094v1
• Date: Fri, 7 Aug 2020 11:45:43 GMT
• Title: Uncertainty Relations of Variances in View of the Weak Value
• Authors: Jaeha Lee, Keita Takeuchi, Kaisei Watanabe, and Izumi Tsutsui
• Abstract summary: We show that there is another inequality which underlies the Schr"odinger inequality in the same sense. The decomposition of the Schr"odinger inequality is examined more closely to analyze its structure and the minimal uncertainty states.
• Score: 1.1699566743796068
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Schr{\"o}dinger inequality is known to underlie the Kennard-Robertson inequality, which is the standard expression of quantum uncertainty for the product of variances of two observables $A$ and $B$, in the sense that the latter is derived from the former. In this paper we point out that, albeit more subtle, there is yet another inequality which underlies the Schr{\"o}dinger inequality in the same sense. The key component of this observation is the use of the weak-value operator $A_{\rm w}(B)$ introduced in our previous works (named after Aharonov's weak value), which was shown to act as the proxy operator for $A$ when $B$ is measured. The lower bound of our novel inequality supplements that of the Schr{\"o}dinger inequality by a term representing the discord between $A_{\rm w}(B)$ and $A$. In addition, the decomposition of the Schr{\"o}dinger inequality, which was also obtained in our previous works by making use the weak-value operator, is examined more closely to analyze its structure and the minimal uncertainty states. Our results are exemplified with some elementary spin 1 and 3/2 models as well as the familiar case of $A$ and $B$ being the position and momentum of a particle.

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