Real-Valued Observables and Quantum Uncertainty

TL;DR

This paper generalizes the quantum uncertainty principle to mixed states with covariance terms, explores real-valued observables, and discusses their sharpness, conjugates, and coarse graining, with illustrative examples.

Contribution

It introduces a generalized uncertainty principle applicable to mixed states, including real-valued observables and their properties, expanding the theoretical framework.

Findings

Uncertainty inequality characterized for faithful states.

Conditions for equality in the uncertainty relation identified.

Analysis of real-valued observables and their conjugates included.

Abstract

We first present a generalization of the Robertson-Heisenberg uncertainty principle. This generalization applies to mixed states and contains a covariance term. For faithful states, we characterize when the uncertainty inequality is an equality. We next present an uncertainty principle version for real-valued observables. Sharp versions and conjugates of real-valued observables are considered. The theory is illustrated with examples of dichotomic observables. We close with a discussion of real-valued coarse graining.