Soluton 1 for 018 019 USTC class Physcs of Quantum Informaton Shua Zhao, Xn-Yu Xu and Ka Chen Natonal Laboratory for Physcal Scences at Mcroscale and Department of Modern Physcs, Unversty of Scence and Technology of Chna, Hefe, 3006, P.R. Chna 1. Descrbe and prove the no-clonng theorem. The no-clonng theorem states that t s mpossble to create an dentcal copy of an arbtrary unknown quantum state. The prove can be seen n the Box 1.1 on the page 53 of Quantum computon and quantum nformaton by Nelsen.. Prove that non-orthogonal states can t be relably dstngushed. The proof can be seen n the Box.3 on the page 87 of Quantum computon and quantum nformaton by Nelsen. 3. Fnd the egenvectors, egenvalues, and dagonal representatons of the Paul matrces X, Y and Z. The calculaton s omtted.. Wrte down the commutaton relatons and ant-commutaton relatons for the Paul matrces and prove them. The commutaton relatons: [σ, σ j ] = ɛ jk σ k, the ant-commutaton relatons: {σ, σ j } = δji, where, j, k = 1,, 3. The proof s omtted.
5. Prove the Cauchy Schwarz nequalty that for any two vectors v and w, v w v v w w. The proof can be seen n the Box.1 on the page 68 of Quantum computon and quantum nformaton by Nelsen. 6. Let v be any real, three-dmensonal unt vector and θ a real number. Prove that where v σ = 3 =1 v σ. expθ v σ = cosθi + snθ v σ, Usng the propertes of Paul matrces { σ = I, t can be proved that v σ = I. From Taylor seres expanson σ σ j = ɛ jk σ k, we can get fx = f0 + f 0x +... = n=0 f n 0 x n, n! where A s an operator, A = I. expθa = cosθi + snθa, exp θa = cosθi + snθa, Then, we obtan expθ v σ = cosθi + snθ v σ, f set A = v σ. 7. Prove that for any -dmenson lnear operator A, A = 1 T raid + 1 T raσ k σ k, k=1 n whch σ k k = 1,, 3 are Paul matrces.
Suppose that A = 3 j=0 a jσ j, where σ 0 = Id and σ k k = 1,, 3 are Paul matrces, Aσ k = a j σ j σ k, snce j=0 T rσ j σ k = δ jk, 3 so T raσ k = T ra j σ j σ k = a j δ jk = a k, j=0 j=0 a k = 1 T raσ k, we get that A = 1 T raid + 1 T raσ k σ k, 8. Prove that an operator ρ s the densty operator assocated to some ensemble k=1 {p, ψ } f and only f t satsfes the condtons: a Trace condton ρ has trace equal to one b Postve condton ρ s a postve operator The proof can be seen n the Theorem.5 on the page 101 of Quantum computon and quantum nformaton by Nelsen. 9. Let ρ be a densty operator. 1. Show that ρ can be wrtten as ρ = I + r σ where r s a real three-dmensonal vector such that r 1.. Show that T rρ 1, wth equalty f and only f ρ s a pure state. 3. Show that a state ρ s a pure state f and only f r = 1.
1. Usng the concluson derved n the prevous problem, we get ρ = 1 T rρid + 1 T rρσ σ. =1 By defnng r = T rρσ, = 1,, 3 and usng T rρ = 1, we get. Defne ρ = p φ φ, ρ = I + r σ. ρ = p φ φ p j φ j φ j = p p j φ φ φ j φ j,j = p φ φ j T rρ = T r p φ φ = T r p φ φ = p Snce p = 1, p p = 1, T rρ 1 wth equalty f and only f p j = 1, p j = 0 when states ρ s a pure state, 3. Snce ρ = I + r σ,
5 usng we get that snce T rρ = 1 T ri + r σ + r. T ri =, T rσ = 0, = 1,, 3 T rρ = 1 1 + r T rρ 1 wth equalty f and only f ρ s a pure state, r = 1 f and only f ρ s a pure state. r 1. 10. Consder an experment, n whch we prepare the state 0 wth the probablty C 0,and the state 1 wth the probablty C 1. How to descrbe ths type of quantum state? Compare the dfferences and smlartes between t wth the state C 0 0 + C 1 e θ 1. Ths state s a mxd state, whose densty matrx s ρ = C 0 0 0 + C 1 1 1 = C 0 0 0 C 1 The state ψ = C 0 0 + C 1 e θ 1 s a pure state, whose densty matrx s ρ = C 0 0 0 + C 1 1 1 + C 0 C 1 e θ 0 1 + C 0 C 1 e θ 1 0 C 0 C 0 C 1 e θ = C 0 C 1 e θ C 1 It s easy to see that her densty matrxes are dfferent. When measurng these two states, f { 0, 1 } bass s used, the probabltes we get 0 and 1 are same; f other bass s used, the probabltes are dfferent.
6 11. Please prepare the polarzed optcal quantum state C 0 0 + C 1 e θ 1 from an ntal state 0, wth half wave plate and quarter-wave plate. To mplement arbtrary sngle qubt untary transformaton, how many wave plates are at least needed, and how to perform them? For a rotaton δ around an axs n the Bloch Sphere, the operator s gven by the Jones matrx cosθ snθ 1 0 cosθ snθ U δ θ = snθ cosθ 0 e δ snθ cosθ cos θ + e δ sn θ cosθ snθ e δ cosθ snθ = cosθ snθ e δ cosθ snθ e δ cos θ + sn θ n whch the 0 poston s defned as the poston where horzontal polarzed lght stays horzontal and vertcal polarzed stays vertcal or n other words the rotaton axs concurs wth the z-axs n the Bloch Sphere. A HWP rotates the state vector by δ =. The operator for a HWP reads cosθ snθ U θ = snθ cosθ A QWP rotates the state vector by δ =. The operator for a QWP reads U θ = 1 1 + cosθ snθ snθ 1 cosθ Snce there are three parameters n the untary matrx Û except the global phase, two QWPs and one HWP are needed to mplement arbtrary sngle qubt untary transformaton. Solve the relaton Û = e δ U γu βu α.e.
7 cosα γ cosα β + γ snα + γ snα β + γ Û = e δ cosα β + γ snα γ + cosα + γ snα β + γ cosα + γ snα β + γ cosα β + γ snα γ cosα γ cosα β + γ + snα + γ snα β + γ we can get the angle of the QWPs and HWP. Let the quantum state go through QWP α, HWPβ and QWPγ one after another we can mplement arbtrary sngle qubt untary transformaton Û. For example, the angles for the wave plates for the 3 standard drectons are untary transformaton QWPα HWPβ QWPγ Î 0 0 0 σˆ x ˆσ y 0 ˆσ z 0 To get the state C 0 0 +C 1 e θ 1, perform the untary transformaton Û = eδ U γu βu α on the state 0. By solvng the relaton C 0 = e δ U γu βu α 1 C 1 e θ 0.e. C 0 C 1 e θ = e δ 1 cosα γ cosα β + γ snα + γ snα β + γ cosα β + γ snα γ + cosα + γ snα β + γ we can get the angles of the QWPs and HWP.