Uncertainty Relations of Variances in View of the Weak Value

TL;DR

This paper introduces a new uncertainty relation involving the weak-value operator, revealing additional bounds beyond the Schr{"o}dinger inequality and analyzing minimal uncertainty states with practical examples.

Contribution

It uncovers a novel uncertainty inequality based on weak values, extending the Schr{"o}dinger relation and providing insights into minimal uncertainty states.

Findings

New inequality supplements Schr{"o}dinger bound with weak-value discord term

Decomposition of Schr{"o}dinger inequality analyzed for structure and minimal states

Examples include spin models and position-momentum uncertainty

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.