[40 + 20 Bonus] Single Subsystem Codes. Suppose we have a quantum system as shown below and wish to define logical qubits in the states with even labels (i.e. the top states). Also, suppose the even states can decay via some decoherence mechanism (e.g. spontaneous emission, amplitude damping, etc.) to the odd states as shown by the arrows. The Kraus operator describing this decay can be written as: K1=n=02p2n+12n a.) [10] Find the no jump operator to complete the Kraus description of the channel. b.) [15] Identify a good logical qubit in the even n space and show that it satisfies Knill-Laflamme conditions. (Hint: It is very simple to satisfy so don't overthink it.) c.) [15] Now suppose that the decays of each of the even states is distinguishable (perhaps because they are no longer degenerate). As a result there are three jump operators: You may assume they each still happen with probability p : K1=p10K2=p32K3=p54 Show that the logical qubit you found in part (b) fails for this quantum channel and explain why. d.) [Bonus 20] Prove that no code is possible for the error model in (c)