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PVMs from POVMs

  1. Feb 4, 2009 #1
    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 [itex] | \psi_i \rangle\langle \psi_i | [/itex] by their action on [itex] \mathbb{C}^2 [/itex] as follows
    [tex] (\sigma \cdot n_i ) \psi_i = \psi_i [/itex]
    where sigma consists of the Pauli spin operator, and [itex] n_i, i = 1,2,3 [/itex] 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 [itex] E_i = \frac23 | \psi \rangle\langle \psi |[/itex]

    Neumark/Naimark's Theorem:
    For any POVM {E} acting on [itex] H_a [/itex], there exists a PVM {P} of the form [itex] U^\dagger (1_A \otimes P) U [/itex] for unitary U, and P an operator on an extended Hilbert space, and a state [itex] | \psi \rangle\langle \psi |\in H_B[/itex] such that, for any state [itex] \rho \in H_A[/itex] we have
    [tex] Tr[(\rho \otimes | \psi \rangle\langle \psi |) P(X) ] = Tr[E(X)\rho] [/tex]

    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.
     
  2. jcsd
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