Heisenberg-Type Uncertainty Relation for Qubits

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Consider a pair of sharp qubit observables . These observables do not commute and cannot be measured jointly and precisely; they are incompatible. However, allowing for imprecision in the measurement makes it possible to measure approximations to the noncommuting pair . The approximators would ideally be chosen so that they give the best possible approximation to without being incompatible themselves. In this Demonstration, you can explore this choice of : representing by vectors on a vertical slice of the Bloch sphere, the described physical structure translates into geometric constraints. The vectors and (both represented by black arrows) are to be approximated by the vectors (blue arrow) and (red arrow), while must be contained in the blue ellipse and must be contained in the red ellipse. You can drag the vectors . The scale on the right-hand side indicates how well the current choice of is approximating . No choice of approximators can do better than the minimum of this scale.

Contributed by: Johannes Biniok and Paul Busch (August 2014)
Open content licensed under CC BY-NC-SA




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