18.435/2.111 Homework # 1Due Thursday, September 18.1a: The quantum statecosθ2|0i + eiφsinθ2|1icorresponds to the po int on the Bloch spherej = (sin θ cos φ, sin θ sin φ, cos θ).Show that the operatorσj= jxσx+ jyσy+ jzσzhas eigenvectors which correspond to the points ±j on the Bloch sphere.1b: Show that antipodal points on the Bloch sphere are orthogonal quantum states.1c: Show that if the vectors j and k are perpendicular, then σjand σkanticommute.1d: Show that applying σjto a quantum state |ψi rotates |ψi by the angle π aroundthe j-axis on the Bloch sphere.Hint: one way to do it is use 1a, 1b, 1c. (There are more straightfo rward, althoughcalculation-intensive, ways to do it, so this hint may no t be of much use).2: Suppose you have two orthogonal states of a qubit, |vi and |¯vi, and an observableA on the qubit. Show that if you add the expectation value of the observable A forthe state |vi to the expectation value for the state |¯vi, you obtain TrA.3: Recall that I said (assuming ¯h = 1) that12σzwas an observable for angular mo-mentum in the z direction (similarly for x and y). If you have two qubits, then theobservable fo r total angular momentum in the z direction isJz=12(σz⊗ I + I ⊗ σz)and similarly for x and y.3a: Show that J2x, J2yand J2zall commute.3b: Since they are commuting, they have simultaneous eigenvectors. You already knowone of these eigenvectors: it is the state with 0 angular momentum1√2(|01i − |10i).What are the other three simultaneous eigenvectors?Continued on next page.14: You can work out the behavior of a spin-1 particle by considering the behavior oftwo spin-1/2 particles that are guaranteed not to be in the state1√2(|01i−|10i) (thisstate has Jx= Jy= Jz= 0). If we work in the basis{|↑zi, |0zi, |↓zi} =n|00i,1√2(|01i + |10i), |11iothenJz=1 0 00 0 00 0 −1.4a: Calculate the operators Jxand Jyin this basis and find their eigenvectors.4b: Suppose you have an entangled pair of spin-1 particles in the state1√3(|↑z↓zi − |0z0zi + |↓z↑zi)and you measure the first one in the { |↑xi, |0xi, |↓xi } basis. What are the proba-bilities for each outcome, and what is the state of the second particle after the mea-surement? Here, the states |↑xi, |0xi, |↓xi stand for the eigenvectors with +1, 0, −1spin alo ng the x axis, and similarly for z.4c: What is J2= J2x+J2y+J2z. Use the answer to explain why, if these three operatorsare measured for a spin-1 particle, then J2αwill be 0 in o ne of the three directions, and1 in the other two directions.This is related to the Kochen-Specker theorem. There is a set of 33 directions (thenumber started out much larger, and has also been reduced still further) in threedimensions, which contains a number of triples of orthogonal directions (i.e., bases).This set o f directions cannot be colored red and green so that in every such triple,exactly one is colored green, and such that no pair of orthogonal directions are bothcolored green. This proves that there is no way of assigning ”hidden variables” to thequantum measurements of J2kalong these 33 directions k so that the predictions ofquantum mechanics are deterministically satisfied.For those of you who are interested, these directions are (unnormalized)• (1, 0, 0) and symmetries [3 vectors in total]• (1, ±1, 0) and symmetries [6 vectors in total]• (√2, ±1, 0) and symmetries [12 vectors in total]• (√2, ±1, ±1) and symmetries [12 vectors in