On the Hilbert space $\mathbb{R}^2$, what is a concrete example of a positive operator-valued measure that is not a projection-valued measure?

Question

I am looking for a purely mathematical example. I tried looking for a set of symmetric matrices $\{F_1,F_2\}$ such that $F_1+F_2=I$ but I cannot seem to find an example.

Why not take F_1 to be the diagonal matrix with .5 and .4 on the diagonal and F_2 the diagonal matrix with .5 and .6? — Martin, Jan 13 '17 at 14:14
There's a POVM that distinguishes |0> from |0>+|1> without making "silent" errors (it has a "distinguishing failed, input lost" result instead). Pretty sure the corresponding PVM requires ancilla and prep operations. — Craig Gidney, Jan 13 '17 at 16:00
To expand Martin's comment, you need also to specify the borel sets that map to either one of the two matrices. The easiest example would be, as a measure on $\mathbb {R} $, the measure $\mu $ such that $\mu ([0])=F_1$ and $\mu ([1]) =F_2$ and $\mu (B) =0$ for any other borel set $B $ disjoint from those two. — yuggib, Jan 14 '17 at 07:40
Might [math.se] (maybe even [mathoverflow.se]?) be better suited for this question? — Kyle Kanos, Jan 14 '17 at 18:29
Acronyms: "Positive Operator Valued Measure" and "Projection Valued Measure". — Ben Grossmann, Jan 17 '17 at 04:14

score 2 · Accepted Answer · answered Jan 17 '17 at 04:10

To do an example as you want, you take $F_1$ to be any positive matrix with $F_1\leq I$ (equivalently, $F_1$ is selfadjoint, and its eigenvalues are in $[0,1]$), and then take $F_2=I-F_1$. As a simplest example you could take $$ F_1=\begin{bmatrix}1/2&0\\0&1/3\end{bmatrix},\ \ \ F_2=\begin{bmatrix}1/2&0\\0&2/3\end{bmatrix}. $$

On the Hilbert space $\mathbb{R}^2$, what is a concrete example of a positive operator-valued measure that is not a projection-valued measure?

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