How Continuous Quantum Measurements in Finite Dimensions Are Actually Discrete

We show that in finite dimensions a quantum measurement with a continuous set of outcomes can be always realized as a continuous random choice of measurements with a finite number of outcomes.

Figure 1

Left: Illustration of the POVM $E^{(\mathit{n})}$ in Eq. (3) as an example of finitely supported POVM $E^{(x)}$ in Eq. (6). In this specific example, the outcome space $\Omega$ is the unit sphere $\Omega \equiv {\mathbb{S}}^2$, the dimension of the Hilbert space is $d=2$, and only two out of the four terms in Eq. (6) are nonvanishing: $P_1=|\mathbf{k}⟩⟨\mathbf{k}|$, $P_2=|-\mathbf{k}⟩⟨-\mathbf{k}|$, $P_3=P_4=0$. The probability of finding the spin direction in a region $R\subseteq \Omega$ is zero for $R$ missing the two poles, as B in the figure, and is possibly nonzero for $R$ as A. Right: Another example of a finitely supported POVM for $d=2$, corresponding to a symmetric informationally complete POVM [20]. The POVM is made of four elements $P_i$ corresponding to ${\omega }_i$ at the vertices of a tetrahedron. The probability can be nonvanishing only if the region $R$ contains at least one of these four points, such as in A, whereas it is always zero in situations as in B.