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.
The general idea is that two incompatible observables (such as two different spin-1/2 components) may be approximated by two compatible unsharp observables , respectively. For qubits, however, the general quantum mechanical problem may be represented using the Bloch sphere and thus simplifies to geometric relations: the observables are represented by vectors and the approximators are represented by vectors of length less than unity. For a given pair , the trade-off relation then reads
which is subject to the compatibility of the approximators and that is ensured by the inequality
You can explore choices of constellations that lead to optimal approximations in the sense of equality in (1) subject to (2), illustrating the proof in . There are three important steps in the proof illustrated here:
1. The compatibility of the approximators and : the two ellipses (blue and red) are specified by the relation , which follows from (2).
2. The symmetrization of the approximators: for any given nonsymmetric pair of approximators, it is always possible to obtain a pair of symmetric approximators, resulting in an approximation that is at least as good as the nonsymmetric pair. Enabling the appropriate setting shows a symmetric pair of vectors (green), which is derived from the pair . The green slider that is shown on the scale on the right-hand side indicates the (improved) uncertainty associated with the symmetrized pair, the lowest point being determined by the lower bound in (1). The symmetrized approximators are derived as follows, denoting the left-hand side approximator by and the other by :
The following calculation shows that the symmetrized pair results in an approximation that is at least as good as for the pair :
The final expression is obtained using the triangle inequality and the symmetry is and .
3. The compatibility of the symmetrized approximators: it is important to note that the compatibility of the approximators and immediately implies compatibility of the symmetrized approximators. A straightforward calculation (using the definitions of and , the triangle inequality, and a similar symmetry consideration to the one in 2) confirms this:
The final expression is bounded by 2 (see (2)), that is, the compatibility of and , showing that the compatibility of the symmetrized approximators, represented by vectors and , follows.
Finally, the additional settings enable more information to be displayed, most notably the locations for the approximators that give minimal uncertainty.
The -slider lets you explore different sets of by specifying the angle between and . No generality is lost in assuming such a symmetric setup; this merely corresponds to a convenient choice of coordinate system.
All locations of minimal uncertainty (for different , changed via -slider) are located on what is referred to as the "triangle of best approximations" corresponding to .
 P. Busch, P. Lahti, and R. F. Werner, "Heisenberg Uncertainty for Qubit Measurements." arxiv.org/abs/1311.0837.
 P. Busch and T. Heinosaari, "Approximate Joint Measurements of Qubit Observables." arxiv.org/abs/0706.1415.