What Are the Minimal Conditions Required to Define a Symmetric Informationally Complete Generalized Measurement?

Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.

Figure 1

The probabilities which correspond to a quantum state. The green triangle is the surface with coordinates $p_1$, $p_2$, $p_3$ where $p_1+p_2+p_3=1$. The blue ellipsoid is the probability distribution associated with a SIC POVM (i.e., $b=1/12$), while the yellow ellipsoid is the distribution associated with the semi-SIC POVM where $b=2/25$. Note that the areas of both distributions are located entirely above the green triangle.