# PVMs from POVMs

1. Feb 4, 2009

### Kreizhn

1. The problem statement, all variables and given/known data
Given the trine POVM, identify the PVM in an extended Hilbert space that reduces to the trine POVM.

2. Relevant equations
Trine POVM:
Define the projectors $| \psi_i \rangle\langle \psi_i |$ by their action on $\mathbb{C}^2$ as follows
$$(\sigma \cdot n_i ) \psi_i = \psi_i [/itex] where sigma consists of the Pauli spin operator, and $n_i, i = 1,2,3$ are the set of unit vectors that lie 120 degrees relative to each other on a great circle of the Bloch sphere. The trine POVM is then given by {E}, where $E_i = \frac23 | \psi \rangle\langle \psi |$ Neumark/Naimark's Theorem: For any POVM {E} acting on $H_a$, there exists a PVM {P} of the form $U^\dagger (1_A \otimes P) U$ for unitary U, and P an operator on an extended Hilbert space, and a state $| \psi \rangle\langle \psi |\in H_B$ such that, for any state $\rho \in H_A$ we have [tex] Tr[(\rho \otimes | \psi \rangle\langle \psi |) P(X) ] = Tr[E(X)\rho]$$

3. The attempt at a solution

I have tried expanding this out, but cannot find any explicit restrictions that would clearly define P. Any ideas would be useful.