Covariant measurements and uncertainty relations

Paolo Gibilisco, Tommaso Isola

Julio C. Gutiérrez-Vega

Abstract 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.

Günter Ludyk

Reinhard F. Werner

We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (\Delta P)(\Delta Q)\geq C\hbar, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision \Delta Q of the position measurement, and the perturbation \Delta P of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.

Reinhard F. Werner

We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (ΔP)(ΔQ) ≥ Ch, where the 'uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision ΔQ of the position measurement, and the perturbation ΔP of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.