Robertson-Schrodinger 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 Schrodinger generalised the Heisenberg relation taking into account the covariance between the observables A and B. This extended inequality is known as the Robertson-Schrodinger uncertainty relation. Here, we demonstrate the less known fact that two-level quantum states, i.e., qubits, satisfy the equality of the Robertson-Schrodinger 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.

Automated summary

The uncertainty principle limits our ability to predict the outcomes of two incompatible measurements, with the Robertson-Schrödinger uncertainty relation considering the covariance between two observables. Quantum states such as qubits satisfy the equality of this uncertainty relation, and visualizing the quantities involved in these relations on the Bloch sphere can help distinguish specific properties and symmetries.