The usual justification for regarding POVMs as fundamental measurements is via Neumark's theorem, i.e., by showing that they can always be realized by a projective measurement in a larger Hilbert space.

That justification is sometimes problematic because for some applications is important *not* to enlarge the Hilbert space, so as to guarantee that the result you proved via POVMs is really about a Hilbert space of that dimension, and not just a shadow of a larger Hilbert space.

So, my question is, how to implement POVMs without enlarging the Hilbert space?

The only strategy I know is doing PVMs stochastically and grouping outcomes; for instance, the POVM $$\bigg\{\frac{1}{2}|0\rangle\langle0|,\frac{1}{2}|1\rangle\langle1|,\frac{1}{2}|+\rangle\langle+|,\frac{1}{2}|-\rangle\langle-|\bigg\}$$ can be implemented by measuring either $\{|0\rangle\langle0|,|1\rangle\langle1|\}$ or $\{|+\rangle\langle+|,|-\rangle\langle-|\}$ with probability $1/2$; by grouping the outcomes one can then measure the POVM $$\bigg\{\frac{1}{2}|0\rangle\langle0|+\frac{1}{2}|+\rangle\langle+|,\frac{1}{2}|1\rangle\langle1|+\frac{1}{2}|-\rangle\langle-|\bigg\}$$ or $\{I/2,I/2\}$

But this class can't be the whole set of POVMs; the Mercedes-Benz POVM (which has three outcomes proportional to $|0\rangle$ and $\frac{1}{2}|0\rangle \pm \frac{\sqrt{3}}{2}|1\rangle$) clearly can't be implemented this way. Is there a neat characterization of this class? Is there published research on it? Even better, is there another (more powerful) way of implementing POVMs without enlarging the Hilbert space?

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