Lecture Notes PH 411/511 ECE 598 A. La Rosa Portland State University INTRODUCTION TO QUANTUM MECHANICS

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1 Lecture Notes PH 4/5 ECE 598. L Ros Portlnd Stte University INTRODUCTION TO QUNTUM MECHNICS Underlying subject of the PROJECT ssignment: QUNTUM ENTNGLEMENT Fundmentls: EPR s view on the completeness of the Quntum Theory, ell inequlities Implementtion: Opticl experiments using down-conversion nonliner processes ppliction: Quntum sttes teleporttion References:. Chrles H. ennett, Gilles rssrd, Clude Crépeu, Richrd Jozs, sher Peres, nd Willim K. Wootters. Teleporting n unknown quntum stte vi dul clssicl nd Einstein-Podolsky-Rosen chnnels. Phys. Rev. Lett. 7, 895 (993).. D. ouwmeester, J. Pn, K. Mttle, M. Eibl, H. Weinfurter nd. Zeilinger. Experimentl quntum teleporttion. Phil. Trns. R. Soc. Lond. 356, (998). 3. Chrles H. ennett. Quntum Informtion nd Computtion. Physics Tody 48, 4-3 (October 995). 4. Mrk eck, Quntum Mechnics, Theory nd Experiments, Oxford University Press (). 5. EPR Prdox Timeline Fig. Principle of quntum teleporttion proposed by ennett [Ref nd 3]. Fundmentls. Entnglement between quntum systems is pure quntum effect describing correltions between systems tht re much stronger nd richer thn ny clssicl correltions cn be. Originlly this property ws introduced by Einstein, Podolsky nd Rosen, nd by Schrodinger nd ohr, in the discussion on the completeness of quntum mechnics nd by von

2 Neumnn in his description of the mesurement process. Since then entnglement hs been seen s just one of the fetures which mkes quntum mechnics so counterintuitive. pplictions: However, recently the new field of quntum informtion theory hs shown the tremendous importnce of quntum entnglement lso for the formultion of new methods of informtion trnsfer nd for lgorithms exploiting the cpbility of quntum computers. (See Fig. ). Opticl implementtion: While quntum computers need entnglement between number of quntum systems, bsic quntum communiction schemes only rely on entnglement between the members of pir of prticles, directly pointing t possible reliztion of such schemes by mens of correlted photon pirs s produced by opticl prmetric down-conversion processes (see Fig. ). Fig. Implementtion of quntum teleporttion vi opticl mens [Ref ].

3 The following re exerts from References 3 nd 4, which outlines in better detil the quntum teleporttion ppliction. More specific ppers, ddressing the different spects of the teleporttion strtegy shown in the figure bove, re provided in Projects section of this course. Those ppers re intended to serve the seed topics for your project ssignment. You cn choose the one tht fits better your quntum curiosity nd interest. Fundmentl Properties of QUNTUM SYSTEMS: Qubits: Two-stte quntum system Pir of Qubits String of n-qubits

4 DELIVERY OF QUNTUM STTES: The frgility of nonorthogonl entngled sttes

5 LTERNTIVE: TELEPORTTION X Following Ref. 4 Let s cll For ech of the three qubits we will hve, The digrm bove indictes tht lice nd ob hve qubit, nd tht these qubits re entngled. There re four two-qubit entngled sttes tht re possible:

6 () They re known s the ell sttes. The ell sttes form bsis, i.e. ny two-qubit stte cn be expressed s liner combintion of the four ell sttes. pir of entngled qubits, shred by seprte prties, is known s ebit. t the beginning: lice hs qubit hs qubit in stte, The stte lice wnts to teleport () lice nd ob hve shred ebit in stte, (3) The totl stte of the system is then, (4)

7 which cn be written s, - (5) If lice performs mesurement nd let s ssume she finds her qubits re in the stte,, then, ccording to (5), ob s qubit is projected into the stte, which is the desired teleported stte. Thus, by simply performing ell mesurement, lice successfully teleport the stte to ob, with no further ction on ob s prt. If lice s msurement projects her qubits into the stte, then, ccording to (5), ob s qubit is projected into the stte. Subsequently, ob cn trnsform this stte into the desired stte by pplying shift to the component of his qubit. For tht purpose, he uses device ( polrizer) whose mtrix representtion is, For, ~ T

8 In short, ~ T Thus, upon receiving informtion from lice (tht her qubits hve collpsed to stte ), ob knows which trnsformtion to pply to his entngled qubit in order to obtined the desired stte. is is possible to teleport n unknown quntum stte by mking it to become prt of sttionry stte of compound system. The quntum teleporttion is ctully done through the lice-ob entngled system. mzingly, mesurement on mde by lice ffects the entngled system own by ob. Once lice tell ob wht to do, then ob cn perform the pproprite mesurement to ttin (construct) PPENDIX Teleporting n Unknown Quntum Stte vi Dul Clssicl nd Einstein-Podolsky-Rosen Chnnels Chrles H. ennett, Gilles rssrd, Clude Crepeu, Richrd Jozs, sher Peres, nd Willim K. Wootters Phys. Rev. Lett. 7, 895 (993). n unknown quntum stte cn be disssembled into, then lter reconstructed from, purely clssicl informtion nd purely nonclssicl Einstein-Podolsky- Rosen (EPR) correltions. To do so the sender, "lice," nd the receiver, "ob," must prerrnge the shring of n EPR-correlted pir of prticles. lice mkes joint mesurement on her EPR prticle nd the unknown quntum system, nd sends ob the clssicl result of this mesurement. Knowing this, ob cn convert the stte of his EPR prticle into n exct replic of the unknown stte which lice destroyed.