Soluton 1 for 017 018 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. Wrte down the commutaton relatons and ant-commutaton relatons for the Paul matrces and prove them. The commutaton relatons: [σ, σ j ] = ϵ jk σ k, the ant-commutaton relatons: {σ, σ j } = δji, where, j, k = 1,, 3. The proof s omtted. 4. 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.
5. Prove that for any -dmenson lnear operator A, A = 1 T raid + 1 3 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, then 3 Aσ k = a j σ j σ k, snce j=0 T rσ j σ k = δ jk, then so T raσ k = 3 3 T ra j σ j σ k = a j δ jk = a k, j=0 j=0 a k = 1 T raσ k, then we get that A = 1 T raid + 1 3 T raσ k σ k, k=1 6. Prove that an operator ρ s the densty operator assocated to some ensemble {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. 7. Let ρ be a densty operator.
3 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 3 T rρσ σ. By defnng r = T rρσ, = 1,, 3 and usng T rρ = 1, we get =1 ρ = I + r σ.. Defne ρ = p ϕ ϕ, then ρ = p ϕ ϕ p j ϕ j ϕ j = p p j ϕ ϕ ϕ j ϕ j,j = p ϕ ϕ j then T rρ = T r p ϕ ϕ = T r p ϕ ϕ = p Snce p = 1, p > 0, then p p = 1, then T rρ 1
4 wth equalty f and only f p = 1, p = p 3 =... = p n = 0 whch states ρ s a pure state, 3. Snce then usng we get that snce ρ = I + r σ, T rρ = 1 4 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, then r 1. r = 1 f and only f ρ s a pure state. 8. 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
5 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 and 1 bass s used, the probabltes we get 0 and 1 are same; f other bass s used, the probabltes are dfferent. 9. 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 = π. The operator for a HWP reads cosθ snθ U π θ = snθ cosθ A QWP rotates the state vector by δ = π. The operator for a QWP reads U π θ = 1 1 + cosθ snθ snθ 1 cosθ
6 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 π γu πβu π α.e. 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 π 4 π 4 π 4 ˆσ y π 4 0 π 4 ˆσ z π 4 0 π 4 To get the state C 0 0 +C 1 e θ 1, perform the untary transformaton Û = eδ U π γu πβu π α on the state 0. By solvng the relaton.e. C 0 = e δ U π γu πβu π α 1 C 1 e θ 0 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.
10. Show that r σ has egenvalues ±1, r = 1,and that the projectors onto the correspondng egenspaces are gven by P ± = I±r σ. Suppose that the normalzed egenvectors of r σ are ϕ + and ϕ,then So, then, and P + = 1 I + r σ r σ ϕ + = + ϕ + r σ ϕ = ϕ I = ϕ + ϕ + + ϕ ϕ = r σ ϕ + ϕ + ϕ ϕ r σ = ϕ + ϕ + ϕ ϕ = I ϕ ϕ ϕ ϕ = 1 I r σ = P 11. Suppose ψ and ϕ are two pure states of a composte quantum system wth components A and B, wth dentcal Schmdt coeffcents. Show that there are untary transformatons U on system A and V on system B such that ψ = U V ϕ. Snce ψ and ϕ has dentcal Schmdt coeffcents, then they can be wrte as ψ = λ A B Let U and V satsfy ϕ = λ a b j A = a a j A = U j a j B = b b j B = V j b 7
8 so U and V are untary transformatons on system A and on system B such that ψ = U V ϕ. 1. Prove that suppose ψ s a pure state of a composte system,ab. Then there exst orthonormal states A for system A,and orthonormal states B for system B such that ψ = λ A B, whereλ are non-negatve real numbers satsfyng λ = 1known as Schmdt coeffcents. The proof can be seen n the Theorem.7 on the page 109 of Quantum computon and quantum nformaton by Nelsen. 13. Suppose ABC s a three component quantum system. Show by example that there are quantum states ψ of such systems whch can not be wrtten n the form ψ = λ A B C where λ are real numbers, and A, B, C are orthonormal bases of the respectve systems. Consder the quantum state ψ = 1 3 001 + 010 + 100, presume that t can be wrtten n the form ψ can be rewrtten as ψ = λ 0 0 0 + 1 λ 1 1 1. ψ = 3 0 A 1 1 01 + 10 BC + 3 1 A 00 BC. = λ 0 A µ BC + 1 λ 1 A ν BC
9 n whch, µ BC = 1 01 + 10 BC ν BC = 00 BC so there exsts a untary matrx U such that 0 A = u 00 0 A + u 01 1 A 1 A = u 10 0 A + u 11 1 A then we get that µ BC = u 00 λ 0 0 + u 01 1 λ 1 1 ν BC = u 10 λ 0 0 + u 11 1 λ 1 1 whch means that µ BC and ν BC can be Schmdt decomposed smultaneously and have the same Schmdt coeffcents. Snce µ BC s the maxmally entangled state and ν BC s a product state, t s mpossble that they have the same Schmdt coeffcents. So ψ can not be wrtten n the form descrbed as above. 14. The ampltude dampng channel can be seen as a POVM performed to the system by the envroment. It s operators are [ ] [ 1 0 E 0 = 0 0 ] γ, E 1 =, 1 γ 0 0 where γ can be seen as a parameter to descrbe the dampng strength. Suppose the ntal state s [ ] a b ρ =, b c please gve the state through the channel.
10 ϵρ = E 0 ρe 0 + E 1ρE 1 1 0 = 0 a b 1 0 1 γ b c 0 0 γ a b 0 γ + 1 γ 0 0 b c 0 0 a + cγ 1 γb = 1 γb 1 γc See page 380 of Quantum computon and quantum nformaton by Nelsen for reference 15. Suppose a two partcle pure state s of the form Φ = j a j j. By defnng A j = a j, calculate the reduced densty matrces ρ A and ρ B. ρ AB = ϕ ϕ = a j a kl j kl jkl ρ A = T r B ρ AB = a j a kl k A l j B jkl = a j a kl k δ lj jkl = a j a kj k jk ρ B = T r A ρ AB = a j a kl j l B k A jkl = a j a kl j l δ k jkl = a j a l j l jl
11 Let A j = a j, then ρ A k = A j A kj = A j A jk = AA k j ρ B jl = A j A l = A T j A l = AT A jl so ρ A = AA ρ B = A T A j