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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)?

I already posted this on https://quantumcomputing.stackexchange.com/questions/26018/confusion-regarding-neumarks-naimarks-extension-of-povm

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