This Demonstration considers a one-qubit system. Assume that the observables in question are dichotomic (two-valued) and unbiased; such an observable

is expressed as the set of positive operators (commonly referred to as a POVM)

where

, with Bloch vector

,

, and

is a vector whose elements are the three Pauli matrices. This is the Bloch vector plotted in this Demonstration. The incompatible observables

have normalized Bloch vectors

, with associated Hermitian operators

,

, and the degree of incompatibility between

is given by

, where

denotes the operator norm. The approximating observables

and

are described in terms of the (generally subnormalized) Bloch vectors

and

, respectively. It is known that for these observables to be jointly measurable, it is necessary and sufficient that they satisfy

. If we fix

, for instance, then we define an ellipsoid

of joint measurability with respect to

:

,

and similarly we can define the ellipsoid

for observables

jointly measurable with

. It can be shown that the optimal approximators lie within the plane spanned by

and

[1], and so we restrict to this circle within the Bloch sphere, and

reduce to ellipses inscribed in this circle. As can be seen, by moving the location of

, the shape of the ellipse changes: it will be near a line along

as

nears the surface of the circle, and it approaches a circle as

nears the origin. The measure of error

used is given by the metric of Bloch vectors in the qubit case (for a generic physical interpretation in terms of differences of the associated probability distributions, see [1, 2]),

.

Geometrically, the Bloch vectors associated with observables

of constant error relative to

lie in a circle around the endpoint of

, and so, while moving around this circle we are able to keep

fixed, all the while changing the ellipse of joint measurability, allowing us to find different observables

that minimize the error

. Assuming that we fix

, by minimizing the error

under the constraint of joint measurability we find that the minimizing vector

must lie on the edge of the ellipse

; that is,

, and the circle of radius

centered at

must be tangent to

at

. This is demonstrated by choosing the first-order optimization, where the

vector with the smallest associated error is found under the constraint of joint measurability. An optimal pair of approximating observables will be such that their associated Bloch vectors will lie on the surface of their respective joint measurability ellipses, and these ellipses will be tangent to the fixed error circles at that point. This is demonstrated by choosing the optimal approximators option, which finds the closest pair of observables that minimize the pair of errors.

As originally proven in [3], the minimum error curve in the plot is given parametrically, with

:

,

.

It is shown in [4] that this boundary curve of the admissible error region reflects a tradeoff between the two approximation errors and the degrees of the unsharpness of the approximating observables

that is governed by the degree of incompatibility of the target observables

.

[1] P. Busch and T. Heinosaari, "Approximate Joint Measurements of Qubit Observables,"

*Quantum Information and Computation*,

**8**(8), 2008 pp. 797–818.

[2] P. Busch, P. Lahti and R. F. Werner, "Heisenburg Uncertainty for Qubit Measurements,"

*Physical Review A*,

**89**(1)

**,** 2014 012129.

doi:10.1103/PhysRevA.89.012129.

[4] T. Bullock and P. Busch, "Measurement Uncertainty: The Problem of Characterising Optimal Error Bounds."

arxiv.org/abs/1512.00104.