 = I [Ri(20), R,(26')] = -2ieijk sin s in e' + sin(0 - - 0') [Ri(20),oj]=28ijkksin0 16.4 (B): Determine the parameters n, O associated with any unitary transformation U according to the definition: U = exp[i(n - F) Clue: use the generic definition of unitary matrices for U: 16.5 (M): Prove that any 2 x 2 unitary matrix U can be expressed from the two rotation operators Ry, Rz according to the product: where a, B, y, 8 are real numbers (Euler's theorem). 16.6 (M): Considering the quantum-gate circuit ACNOT BCNOT ACNOT show first that the matrix of the intermediate BCNOT gate takes the form BCNOT Quantum bits and quantum gates assuming the computational base {|0>|0), |0)|1), |1)|0), |1)|1)}. Then prove by matrix multiplication that the circuit is equivalent to a CROSSOVER or SWAP gate. 16.7 (M): Analyze the action of a CNOT gate with control qubit |a) in a superposition of states (assume (a) = y/0) + 8|1 and Consider then the two cases y=8=1/v2andy= = = = -8 = 1/2 for the control qubit. What are the possible output states? Show that the same result is obtained with all inputs in pure states and with a Hadamard gate placed on the control input path. 16.8 (T): Prove that the quantum circuit which includes Hadamard gates (H), /4-phase gates (T, T+), and a single /2-phase gate (S), is an equivalent realization of a CCNOT or Toffoli gate. 16.9 (B): Given the Pauli matrices Y, Z, calculate the tensor product YOZ. Given two qubits |a), |b) and their tensor product |a) x lb), show that YOZ(a)by)=Ya)Zb) 16.10 (T): Given the computational base V={|0),|1)} and the Hadamard operator H, show that the two-fold tensor product H = HD. H satisfies where |a) = and are any elements of V2 (with wj==0,1) and Then prove by induction that, in the general case, Quantum measurements This chapter is concerned with the measure of quantum states. This requires one to introduce the subtle notion of quantum measurement, an operation that has no counterpart in the classical domain. To this effect, we first need to develop some new tools, startin