18.435/2.111 Homework # 9Due Tuesday, December 9.1: Show that Alice can teleport two qubits “through” the gateS =111−1in the following manner.Suppose Alice and Bob share four qubits in the state12(| 0000i + | 0101i + | 1010i − | 1111i)with Alice holding the first two qubits and Bob holding the last two. Alice has twoqubits in unknown (to her) states | φi and | ψi. She makes a joint measurement onthe qubit in state | φi and her first entangled qubit using the Bell basis, and she alsomeasures her second qubit | ψi and her second entangled qubit using the Bell basis.Alice sends the results of both measurements to Bob over a classical channel, andBob applies, to his halves of the two EPR pairs, Pauli matrices which depend on theclassical bits he received. Show that if he applies the correct Pauli matrices, Bobwill end up with S | φi | ψi. How do the Paulis Bob applies depend on the r esults ofAlice’s measurements?2: Suppose we have two classical linear codes C2⊂ C1. Consider the state1|C2|1/2Xc∈C2| v + ci (−1)c·twhere v ∈ C1and t is an arbitrary vector. Suppose we apply the Hadamard trans-formation on every qubit. What state do we obtain? Show that it is in t he CSScode tr anslated by s, i.e., it is a superposition of codewordsw + C⊥1+ sE=1|C⊥1|1/2Xc∈C⊥1| w + c + siwith w ∈ C⊥2. How does s depend on t?13a: Consider the modification t o Grover’s algorithm where the oracle now performsO | xi = eiφ| xi if x is a target stateO | xi = | xi otherwise.Show that if you use the transformation˜G = H⊗nh(1 − eiφ) | 0ih0 | − IiH⊗nOinstead of the standard Grover iteration, for any state with M/N sufficiently largeyou can choose φ so the algorithm finds a target state with probability 1 after oneiteration.3b: For what values of M/N is there such a φ? How can you combine the results of3a with the standard Grover iteration (using the standard Grover oracle) to obtaina modification of Grover’s alg orithm that finds a marked state with probability 1,given that M/N is known?The next three problems deal with approximate cloning. We have seen that wecannot perfectly clone a quantum state. However, there are approximate clonersthat work moderately well. The optimal cloner for taking one qubit to two qubitsis the transformation| 0i →s23| 00iAB| 0iR+s16(| 01iAB+ | 10iAB) | 1iR| 1i →s23| 11iAB| 1iR+s16(| 01iAB+ | 10iAB) | 0iR,where A and B are clones of the original qubit, and R is a reference system.4: Show that if we call the cloning transfomation C, then for a ny pure state | ψi onone qubit,Ahψ | TrBRC | ψi hψ | C†| ψiA=56so this transformation clones with fidelity 5/6.5: If we trace out the reference system R, find Krauss operators E1and E2so thatthe cloning transformation is expressible in the operator sum for malism asρ →XiEiρE†i.Note tha t the matrices Eiwill not be square.26a: Now, supp ose we change bases by takingFi= U⊗2EiU†whereU = cos θ − sin θsin θ cos θ!.What are the Fi?6b: Show that the cloning operation is invariant under this change of basis byfinding a unitary matrix {uij} such