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

, (1)

which is subject to the compatibility of the approximators

and

that is ensured by the inequality

. (2)

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 [1]. 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

.

Use these links to search for Demonstrations related to these topics:

Qubit and

Bloch Sphere.[1] P. Busch, P. Lahti, and R. F. Werner, "Heisenberg Uncertainty for Qubit Measurements."

arxiv.org/abs/1311.0837.