C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned by the other party? Why? Answer: Let ψ v v, where v v We wll provde a scheme equvalent to the one presented n class we wll show that Alce can teleport the frst qubt of ψ wthout destroyng ts entanglement wth the rest of the system By repeatng the argument, Alce can therefore teleport all her qubts to Bob wthout destroyng ther entanglement Alce Bob share a Bell state, so the overall state of the system s ψ v v v v Here, Bob owns the frst qubt Alce the rest In the above, we smply exped out the state of the system, rearranged the ket symbols for example,, ths s just notaton Now Alce measures the second thrd qubts n the Bell state bass: Ψ ± ±, Φ ± ± For each of the four measurement outcomes, we compute the state of the unmeasured part of the system: Ψ : v v ψ Φ : v v X I ψ Ψ : v v Z I ψ Φ : v v ZX I ψ For example, the unnormalzed state remanng f Alce measures Ψ s Ψ Ψ v Ψ Ψ v v v, usng Ψ Ψ, Ψ Ψ Ths normalzes to v v, as clamed The leadng factor of tells us that the probablty of ths measurement outcome s 4 Ths probablty s ndependent of ψ! So f Alce measures Ψ, then the qubt has been correctly teleported, wth all entanglement preserved If she measures Ψ then she needs to tell Bob to apply a phase flp If she measures Φ then Bob needs to apply a NOT gate If she measures Φ then Bob needs to apply a phase flp, then a NOT gate These correctons are the same as for stard teleportaton Create a clonng crcut that correctly clones the states e t takes as nput two qubts ψ, n the case that ψ s ether or t outputs ψ ψ Answer: We can use the clonng crcut for the stard bass a CNOT Frst we have to change the frst qubt from the Hadamard bass to the stard bass, so we apply a Hadamard on the frst qubt Then we apply a CNOT based on the frst qubt e flp the second ff the frst one s Then we want to go back to the Hadamard bass for both qubts, so we apply a Hadamard on each one C/CS/Phys 9, Fall 8
3 Wrte down the observable A that measures the qubt n the stard bass, wth measurement outcome f the qubt s n state, f t s n state Also wrte the observable B that measures n the Fourer bass, wth outcome f the qubt s n state, f t s n state Answer: a A I Z b A X 4 a Fnd the egenvectors, egenvalues, dagonal representatons of the Paul matrces I, X, Y, Z, where I σ X σ x Y σ y Z σ z Show that X Y Z I so that calculatng X n, Y n or Z n becomes really smple b Fnd the ponts on the Bloch sphere that correspond to the normalzed egenvectors of the Paul matrces c Fnd the acton of the Z operator on a general qubt state Ψ α β descrbe ths acton on the Bloch sphere e how does the vector representng Ψ get rotated on the Bloch sphere? d Form the matrx representaton of the exponentated operator e Z/ show how ths exponentated operator acts on the Bloch sphere vector for Ψ e Smlarly form the matrx representatons of the exponentated operators e X/ e Y / Show explctly how these act on the vector at the North pole of the Bloch sphere, e on the qubt state Ψ, specfyng the nature of the resultng rotaton Soluton: a Use stard lnear algebra methods to fnd the egenvalues egenvectors of the matrces as they are gven consult any stard textbook on lnear algebra If you are havng trouble wth ths need help, please let us know, snce a good understng of the concepts of egenvalues egenvectors s essental to ths course Snce the dentty matrx σ I leaves all vectors unchanged, all vectors are egenvectors of ths matrx, they all have egenvalue All the other Paul matrces have two egenvalues, For σ x the correspondng egenvectors are,/ for egenvalue,, / for egenvalue the factor of / s ncluded so that the egenvectors are normalzed to A vector does not need to be normalzed ths way just to be an egenvector, but f we want to use t as a quantum state vector, t needs to be normalzed f the probabltes are to come out correctly For σ y the egenvectors are,/ for egenvalue, / for egenvalue For σ z the egenvectors are, for egenvalue, for egenvalue The dagonal representaton of a matrx s defned as the matrx wrtten n the bass n whch t s dagonal e all off-dagonal elements are zero Ths turns out to be smply wrtng the egenvalues along the dagonal settng all the off-dagonal elements to zero, the correspondng bass s smply usng the egenvectors as bass vectors one can prove that the egenvectors of these matrces can always be made nto an orthonormal bass Please ask us f you have any questons about ths C/CS/Phys 9, Fall 8
b To fnd out whch pont on the Bloch sphere corresponds to a gven state vector ψ α β, wrte the state as ψ cos θ eφ sn θ dentfy whch angles θ φ are needed to gve the rght coeffcents α β These then correspond to the regular angular coordnates on the sphere where θ should le n the nterval [,π] wth θ beng the North pole, θ π the South pole, φ should le n ether the nterval [ π,π] or the nterval [,π], be the angle made wth the x-axs when gong around the z-axs n a counterclockwse drecton The egenvectors of σ x are σx σ x Ths corresponds to θ π/, φ for σ x θ π/, φ π for σ x, e the egenvectors le along the postve negatve x-axs respectvely The egenvectors of σ y are σ y σy 3 Ths corresponds to θ π/, φ π/ for σ y θ π/, φ 3π/ for σy, e the egenvectors le along the postve negatve y-axs respectvely Fnally, for the egenvectors of σ z we have σz σz 4 Ths corresponds to θ for σ z θ π for σ z, e the postve negatve z-axs, or the north the south pole respectvely for these ponts, the φ-coordnate s rrelevant/undefned c In ths problem the next, I wll mostly use Drac bra-ket notaton The problems can of course be done usng stard vector matrx notaton as well, but the Drac notaton s often smpler, t s useful to be able to use t translate between the two notatons In problem d I wll do much of the calculatons wth Drac notaton then translate the results nto matrx notaton to get the answer that the problem asks for The Z operator does nothng to the vector flps the sgn on the vector, e takes t to f you don t see ths mmedately, remember that s, s,, see how the matrx representaton of σ z acts on these vectors Now for a general state vector Ψ α β, represent t n terms of Bloch sphere coordnates by wrtng α cosθ/ β e φ snθ/ The acton of Z on ths s then Z Ψ cos θ eφ sn θ cos θ eφπ sn θ 5 C/CS/Phys 9, Fall 8 3
where we have used the fact that e π Hence, the acton of Z on the Bloch sphere vector representng Ψ s to add π to the angle φ, e rotate the vector by π around the z-axs d In general, evaluatng a functon of an operator can be defned by expng that functon n a power seres In other words, f we have a functon f x wth x a regular real or complex valued varable whch can be exped n a power seres f x c c x c x n c n x 6 then usng an operator A as an argument to that functon s defned as f A c n A n 7 whch s well-defned snce we know how to multply a matrx or a lnear operator n tmes wth tself how to add up the results we leave asde questons of convergence for now Presumably you know the seres expanson for the exponental functon could evaluate e Z/ that way f you remember that Z I remember the seres expanson of the sne cosne functons, you can also smplfy the resultng sum dramatcally However, for an operator that has a complete set of egenvectors e a set of egenvectors that span the whole vector space form a bass, there s another, usually smpler way of dong t: you can use the fact that when f A s appled to an egenvector a of A wth egenvalue a, then we have f A a f a a 8 Convnce yourself that ths s mpled by the defnton n eq 7 Come ask us f you don t see ths Now f you want to wrte out f A as an operator, f A has a complete set of egenvectors, then you can make use of the fact that the sum of the projectors that project onto the egenvectors of A s equal to the dentty, e a a I 9 We can then wrte f A f A I f A a a f a a a In stard vector notaton, ths becomes: f A f a v v C/CS/Phys 9, Fall 8 4
where v now are the egenvectors of A, v s the complex adjont transposton complex conjugaton of v Also, f A has a complete set of egenvectors, then any vector Ψ can be exped usng the egenvectors of A as bass vectors, e Ψ c a wth approprate coeffcents c In that case, f A appled to Ψ becomes smply f A Ψ a c a 3 In our case, the operator that we are nterested n s Z, whose egenvectors are smply the stard bass vectors, wth egenvalues respectvely, so f we wrte Ψ as cosθ/ e φ snθ/ t s already exped n egenvectors of Z Eq 3 then gves us: e Z Ψ e θ cos e e φ sn θ e θ cos eφ sn θ 4 Ths doesn t have qute the stard Bloch sphere form because of the complex exponental on the frst term However, we can factor ths out wrte the state as: e Z [ Ψ e cos θ e φ sn θ ] 5 Now multplyng the whole state by a number wth absolute value has no observable consequences, snce t does not change any probabltes that we mght calculate for measurement outcomes Hence we may as well take the expresson nsde the brackets to be our new state, we therefor see that the effect of the operator e Z/ s to ncrease the φ coordnate by, e t rotates the Bloch sphere vector by an angle around the z-axs We were however also asked to fnd explctly the matrx representaton of e Z/, so let s do ths In bra-ket notaton, we wrte t out as accordng to eq e Z e e e e 6 e e e e whch clearly has the effect of addng to the phase of the second component of the state vector, e to ncrease the Bloch angle φ by, e to rotate the Bloch vector by around the z-axs e We can use much the same procedure here as n 3d, but we now have the added complcaton that the egenvectors of X Y are not just the bass vectors of the stard bass The calculatons n ths problem wll actually look cleaner n stard matrx notaton than n the Drac notaton, so I wll swtch back C/CS/Phys 9, Fall 8 5
For X, the egenvectors are,/, / for egenvalues respectvely Eq then gves us e X e e e e e e cos e e e e sn 7 sn cos Now we apply ths to the vector at the North pole of the Bloch sphere: e X cos sn 8 Ths s a Bloch sphere vector wth coordnates θ φ π Ths means that what the operaton has done s to rotate the vector from the North pole by an angle counterclockwse around the x-axs For Y, the egenvectors are,/ for egenvalue, / for egenvalue We therefore get e Y e e e e e e e e e e Ths does the followng to the vector at the North pole: cos sn 9 sn cos e Y cos sn Ths s a Bloch vector wth coordnates θ φ, n other words the operator e Y / rotates the state vector from the North pole by an angle counterclockwse around the y-axs C/CS/Phys 9, Fall 8 6