Motivation
In a d-dimensional Hilbert space when averaging over all quantum pure states the natural group is SU(d), the special unitary group of dimension d. The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries.
A particularly widely-used example of this is the spin system. For this system the relevant group is SU(2) which is the group of all 2x2 unitary operators. Since every 2x2 unitary operator is a rotation of the Bloch sphere, the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is the rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.
Another recent application is the fact that a symmetric informationally complete POVM is also a spherical 2-design. Also, since a 2-design must have more than elements, a SIC-POVM is a minimal 2-design.
Read more about this topic: Quantum T-design
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