Probability of error in BB84 if Eve uses Breidbard basis

[demonsrun]

Homework Statement:

Assume that Eve uses the Breidbart basis with eigenvectors |φ0⟩ = cos(π/8)|0⟩ + sin(π/8)|1⟩, |φ1⟩ = − sin(π/8)|0⟩ + cos(π/8)|1⟩. Eve uses as measurement POVM {|φ0⟩⟨φ0|,|φ1⟩⟨φ1|} which implies that: Pr{mE = i} = |⟨φi|ψ⟩|^2, where |ψ⟩ is the transmitted qubit.

1. Compute the probability of error at Eve and Bob of the encoded key bits assuming that Eve is consistently measuring all transmitted qubits of the quantum channel using the Breidbart basis

2. Compare the results to the case in which Eve also measures all qubits but randomly chooses the X or Z basis

3. Finally obtain the probability of error at Eve and Bob if Eve only measures each of the transmitted qubits using the Breidbart basis with a probability p and with probability 1 − p Eve does not perform any measurement on that qubit and flips a fair coin to guess the encoded bit

Relevant Equations:

|φ0⟩ = cos(π/8)|0⟩ + sin(π/8)|1⟩, |φ1⟩ = − sin(π/8)|0⟩ + cos(π/8)|1⟩

Pr{mE = i} = |⟨φi|ψ⟩|^2

For the second question, I'm thinking that if Eve randomly chooses the X or Z basis, then the probability of error at Eve would be 0.5.

And for the third case if p is 1, then the case is the same as the first, and if the probability is 0, then Eve doesn't affect the error rate at all.

Other than that I'm a bit lost as to how to calculate the probability especially in the first exercise.