# In Neumark's extension, can the PVMs be rank one projectors on the total enlarged space or does it have to be projectors on the ancillary space alone?

Starting with the definitions used.

A PVM is a set $$\mathcal{P} = \{P_i: P_i^2 = P_i, P_iP_j = \delta_{ij}P_j, \sum{P_i} = \mathbf{I}\}_{i,j=1}^n$$, where $$n\leq d$$ on a Hilbert space $$\mathcal{H}^d$$ of dimension $$d$$

A POVM is a set $$\mathcal{M} = \{A_i : A_i \geq 0, \sum{A_i }= \mathbf{I}\}_{i=1}^m$$ on a Hilbert space $$\mathcal{H}^d$$ of dimension $$d$$.

"Neumark's theorem states that any rank one POVM can be realised as a PVM on a higher-dimensional space".

$$\textbf{Confusion 1:}$$ I have seen the enlargement of the Hilbert space being done in two different ways.

$$\textbf{1}$$ By embedding the system in a higher-dimensional space. For eg: A qubit being treated as a qutrit with the third amplitude being zero.

$$|\Psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle )\equiv \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) + 0|2\rangle$$

$$\textbf{2}$$ By attaching an ancilla of a suitable size.

Are these two approaches equivalent?

$$\textbf{Confusion 2:}$$ When the Hilbert space is enlarged by attaching an ancilla, to realise the POVM, is the PVM performed on the ancilla alone or on the total system(original system + ancilla)?