Robertson–Schrödinger uncertainty relation for qubits: a visual approach

The uncertainty principle sets a limit to our capacity to predict the outcomes of two incompatible measurements. The Heisenberg uncertainty relation for two arbitrary observables $A$ and $B$ is usually discussed in textbooks. However, little or no attention is paid to the fact that Schrödinger generalised the Heisenberg relation taking into account the covariance between the observables $A$ and $B$. This extended inequality is known as the Robertson–Schrödinger uncertainty relation. Here, we demonstrate the less known fact that two-level quantum states, i.e., qubits, satisfy the equality of the Robertson–Schrödinger uncertainty relation for two arbitrary observables $A$ and $B$. Taking advantage of the homomorphism between SU(2) and SO(3) groups, it is possible to map the distributions of the expectation values and variances of the observables, and the Heisenberg and covariance terms on the Bloch sphere. The graphical visualisation of the relevant quantities involved in the uncertainty relations allows us to distinguish specific properties and symmetries that are not so evident in the algebraic formalism. (paper)