Uncertainty Relations of Variances in View of the Weak Value

Abstract

The Schrödinger 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 Schrödinger inequality in the same sense.The key component of this observation is the use of the weak-value operator Aw(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 Schrödinger inequality by a term representing the discord between Aw(B) and A. In addition, the decomposition of the Schrödinger 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.

Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable

“…On the other hand, Lee et al in Ref. [53] have derived separately the two lower bounds by using the weak value. However, their derivation fully employs the machinery of linear operators in Hilbert space so that its correspondence with classical uncertainty relation is not clear.…”

“…See also Refs. [45,53,59]. Note however that, there, the authors consider the estimation of the observable  based on the measurement of another observable B, whereas, here, we make an estimate of the c-valued physical quantity Ã(b n , ξ|ψ) based on information on another c-valued physical quantity B(b n , ξ|ψ) = b n .…”