How Does the Trine POVM Relate to a PVM in an Extended Hilbert Space?

[Kreizhn]

Homework Statement

Given the trine POVM, identify the PVM in an extended Hilbert space that reduces to the trine POVM.

Homework Equations

Trine POVM:
Define the projectors $| \psi_i \rangle\langle \psi_i |$ by their action on $\mathbb{C}^2$ as follows
[tex] (\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
$$Tr[(\rho \otimes | \psi \rangle\langle \psi |) P(X) ] = Tr[E(X)\rho]$$

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.

 

[Jhero]

Thank you for your question. In order to identify the PVM in an extended Hilbert space that reduces to the trine POVM, we need to use Neumark/Naimark's Theorem. This theorem states that 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 Tr[(\rho \otimes | \psi \rangle\langle \psi |) P(X) ] = Tr[E(X)\rho].

To apply this theorem to our problem, we need to first define the extended Hilbert space. We can do this by considering the Bloch sphere, which is a representation of the state space for a two-level quantum system. The Bloch sphere has three unit vectors, n_1, n_2, and n_3, that lie 120 degrees relative to each other on a great circle. These vectors are related to the Pauli spin operator, sigma, as described in the trine POVM.

Next, we need to define the unitary operator U and the operator P on the extended Hilbert space. We can choose U to be the identity operator, since it will not affect the final result. For P, we can use the operator E_i from the trine POVM, which is given by E_i = \frac23 | \psi \rangle\langle \psi |. This means that P_i = | \psi \rangle\langle \psi |. Therefore, the PVM {P} of the form U^\dagger (1_A \otimes P) U is simply P_i = | \psi \rangle\langle \psi |.

Finally, we need to find the state | \psi \rangle\langle \psi | in H_B that satisfies the condition given by Neumark/Naimark's Theorem. We can choose | \psi \rangle\langle \psi | to be any state that is orthogonal to the three unit vectors n_1, n_2, and n_3. For example, we can choose | \psi \rangle\langle \psi | to be the state (1/\