Our system Q consists of two qubits: R is initially in the Bell state with one qubit, while the other is in the state |0. This system undergoes an evolution by the superoperator . Calculate the...

Our system Q consists of two qubits: R is initially in the Bell state

with one qubit, while the other is in the state |0. This system undergoes an evolution by the superoperator
. Calculate the coherent information I(Q;R) if

(a) the evolution

swaps the states of the two Q qubits with probability p;

(b) the evolution

applies the X ⊗ X operator to Q with probability p;

(c) the evolution

performs a CNOT operation on Q (with the first system as control) with probability p; and

(d) the evolution

performs a controlled-Z operation on Q with probability p.

In each case, assume that E otherwise does not affect the state of Q. In which cases is perfect quantum error correction possible?