3. Let's generalize the preceding setup a little. Suppose Bob is given a state from m linearlyindependent states |1),..., |bm). Can you construct a POVM {E1,.., Em+1} so that ifoutcome E; occurs with 1

Answered: 3. Let's generalize the preceding setup…

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3. Let's generalize the preceding setup a little. Suppose Bob is given a state from m linearly
independent states |1),..., |bm). Can you construct a POVM {E1,.., Em+1} so that if
outcome E; occurs with 1<i<m then Bob knows for sure that the state is vi)?

Expert Solution

Step 1

Denote by $| ϕ_{i} ⟩$ the vector orthogonal to the subspace spanned by ${| ψ_{j} ⟩}_{j \neq i}$ and define

E_{i} = \sum_{i \neq j} | ϕ j ⟩ ⟨ ϕ j |

E_{m + 1} = 𝕀 - \sum_{j = 1}^{m} E_{j}

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7. Suppose that X is a real valued random variable and : R → R is continuous. Show that if X and (X) are independent, then (X) must be deterministic (i.e. there is a real number a so that P((X) = a) = 1).
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3. Let's generalize the preceding setup a little. Suppose Bob is given a state from m linearly independent states |1), .., [Wm). Can you construct a POVM {E1,…, Em+1} so that if outcome E; occurs with 1