Products of weak values: uncertainty relations, complementarity and incompatibility

TL;DR

This paper explores how products of weak values can derive quantum uncertainty, complementarity, and incompatibility relations, revealing new bounds and interpretations for quantum measurements and probabilities.

Contribution

It introduces a product representation formula linking weak values to uncertainty relations and provides new bounds on incompatible observables in weak measurement contexts.

Findings

Derives Heisenberg uncertainty from classical complex variables.

Establishes upper bounds on products of weak values for incompatible observables.

Quantifies anomalous weak probabilities through tradeoff relations.

Abstract

The products of weak values of quantum observables are shown to be of value in deriving quantum uncertainty and complementarity relations, for both weak and strong measurement statistics. First, a 'product representation formula' allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. We show this formula also leads to strong uncertainty relations for unitary operators, and underlies an interpretation of weak values as optimal (complex) estimates of quantum observables. Furthermore, we show that two incompatible observables that are weakly and strongly measured in a weak measurement context obey a complementarity relation under the interchange of these observables, in the form of an upper bound on the product of the corresponding weak values. Moreover, general tradeoff relations between weak purity, quantum purity and quantum incompatibility, and also between weak and strong joint probability distributions, are obtained based on products of real and imaginary components of weak values, where these relations quantify the degree to which weak probabilities can take anomalous values in a given context.