Dilation for several POVMs

Question

Let $E$, $F$ be two finite sets. Let $\mathcal{H}$ be a finite-dimensional Hilbert space, and let $(A^{f}_e)_{e,f}$ be a family of positive operators on $\mathcal{H}$ such that for all $e$, $\sum_{f} A^{f}_e = \mathbf{Id}$.

I would like to know if there exists a finite-dimensional Hilbert space $\mathcal{K}$, a family of projections $(P^{f}_e)_{e,f}$ on $\mathcal{K}$ such that for all $e$, $\sum_{f} P^{f}_e = \mathbf{Id}$, and a linear isometry $V : \mathcal{H} \rightarrow \mathcal{K}$ such that for all $e,f$, we have $A^{f}_e = V^* P^{f}_e V$.

For $E$ of cardinal $1$, this is the content of Naimark’s dilation theorem for POVM (positive-operator-valued measure). The proof I know is described in the Wikipedia page and I don’t know how to make it work for several POVMs.

I have also found a reference saying that (when dropping the requirement that $\mathcal{K}$ is finite-dimensional) it follows from a theorem called Arveson’s extension theorem. But I have not had the time to look at the details to find out if one can make $\mathcal{K}$ finite-dimensional.

Can you help me find a proof or give me a reference?

Answer 1

Such an isometry does exist. This construction is sometimes known as the simultaneous Stinespring dilation for a collection of POVMs. One essentially just iteratively constructs the Stinespring isometry for each POVM until a single isometry is constructed which works. The result was likely known to experts for a while but is mentioned explicitly in https://arxiv.org/pdf/1407.6918.pdf (in the proof of Theorem 5.3). A complete proof and explanation are given in the set of notes https://www.math.uwaterloo.ca/~vpaulsen/EntanglementAndNonlocality_LectureNotes_7.pdf so I will not repeat the details, in particular, see Theorem 9.8 and the proof.

Answer 2

The answer by Condo is definitely correct and useful. What I would like to add here is that sometimes people want to have a 'minimal' dilation (see this for what 'minimal' means), and the construction you can find in Wikipedia page is indeed a minimal one (while the Paulsen's construction is not minimal). So, if you want a generalisation of the Wikipedia's construction for multiple POVMs, you can do the following:

1. Let $\{A_e^f\}$ be the POVM elements, and $m_e$ be the number of POVM elements with index $e$, i.e., $\sum_{f=1}^{m_e}A_e^f=\text{Id}$.
2. Let $m$=$\max_e\{m_e\}$, and let $A_e^f=0$ for $f\in[m_e+1,...,m]$. This step is simply padding $0$ operators to make sure that each POVM has the same number of measurement outputs.
3. Assume without loss of generality that each $A_e^f$ is rank-1, i.e., $A_e^f=x_{e,f}x_{e,f}^*$ for some sub-normalised vectors $x_{e,f}$. If you really don't want to make this assumption, you need to take $m=\max_e\{\sum_f\text{rank}(A^f_e)\}$ in step 2.
4. The matrices $M_e=[x_{e,1},...,x_{e,m}]$ now are all $d\times m$ co-isometries, where $d$ is the dimension of $A_e^f$. You can always find $(m-d)\times m$ matrices $N_e$ such that $$U=\begin{bmatrix}M_e\\N_e\end{bmatrix}$$ are unitaries. Then you take the projections onto the column vectors of $U$ to be the PVMs.
5. It is clear that $V=[\text{Id},0]^\intercal$ is the isometry satisfying $V^*P^f_eV=A^f_e$ for all $f,e$.

Of course we cannot say the above construction is minimal because we need to first define the minimal dilation for multiple POVMs. But I believe that it's dimension is the smallest possible.