Invited Talk by HARI PRAKASH

Transcription

1 QUNTUM TELEPORTTION USING ENTNGLED COHERENT STTES Invited Talk by HRI PRKSH Physics Department, University of llahabad, University of llahabad WORKERS IN THIS FIELD IN THE RESERCH GROUP: Prof. Naresh Chandra, Prof. Ranjana Prakash, mbesh Diit, Dr. Shivani jai Kumar Maurya, jay Yadav, Vikram Verma & Manoj K. Mishra WORKERS IN OTHER FIELDS IN THE RESERCH GROUP: Dr. Devendra K. Singh. Dr. Rakesh Kumar Dr. Pankaj Kumar, Dr. Devendra K. Mishra & Namrata Shukla QIP-0 HRI, LLHBD February 0, 0

2 Classical information Classical bit: 0 or Usually implemented by voltage of either 0 or V across a capacitor. Classical information is encoded as strings of bits Quantum information Unit of Quantum information is Quantum bit or Qubit Qubit: Vector in -dim Hilbert space spanned by basis vector Qubit is physically represented by two-level quantum system state as a superposition of basis vectors in the form,

3 BLOCH SPHERE Bloch sphere representation of qubit, (Left): an arbitrary qubit can be parameterized with two real numbers θ and φ corresponding to polar and azimuthal angles in polar spherical coordinate system as i cos 0 e sin (right): Bloch representations of two frequently used qubit states 0, corresponding to spin-up and spindown eigenvectors along z-ais 0

4 Quantum Entanglement B () H H () B [ 0 B 0 B ] LICE BOB 0 B 0 B 0 B

5 Quantum Entanglement: Concurrence For pure states : C( ) ~ B For Mied states : y ~ B pi i i B y y B y i C( ) {0, B ma 3 } y ; y where { i } are thesquare rootsof eigen of non harmitian matri ~ with, 3 [ W.K Wooters, Phys. Rev. Lett. 80, 5 (998) ] 0, values cos 0 sin 0, cos B B B C

6 Replica T generated 3 LICE Bell state measurement (r) B Results of Bell State Measurement, r=,, 3,, communicated BOB Unitary transformation (r) U : I, σ, σ, σ σ z z Particle Particle Particle 3 Information State, I Entangled State E 3 [ 00 ] 3 Basis States Product states: 00, 0, 0, Bell States ( r) B : [ 00 ], [ 00 ], [ 0 0 ], [ 0 0 ]

7 N EXMPLE Information state I a 0 b Entangled state E [ 0,0, ],3,3,3 ( r) ( r) ( s) ( r) ( r) Bell States B : B B, r rs B B Combined State ( r) ( r) ( r) I B B B If result of BSM is r=3, for eample state of Bob is,,3 r,3,3,,3 (3) B [ 0,, 0 ],3,3,3 (3) (3) Bob B a b 3 3 3,,,3 0 (3) Unitary transformation U then gives (3) (3) (3) T U Bob I Quality of Teleportation: Fidelity, F= IT

8 Eperimental Quantum Teleportation (Institut fu r Eperimentalphysik, Universita t Innsbruck) The eperimental set-up: pulse of ultraviolet radiation passing through a nonlinear crystal creates the ancillary pair of photons and 3. fter retroflection during its second passage through the crystal the ultraviolet pulse creates another pair of photons, one of which will be prepared in the initial state of photon to be teleported, the other one serving as a trigger indicating that a photon to be teleported is under way. lice then looks for coincidences after a beam splitter BS where the initial photon and one of the ancillaries are superposed. Bob, after receiving classical Information that lice obtained coincidence count in detectors f and f identifying, Bell state, knows that his photon 3 is in the initial state of photon which he then can check using polarization analysis with polarizing beam splitter PBS and detectors d and d. The detector p provides Information that photon is under way. 3 [ H [ Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter & nton Zeilinger, Nature 390,575 (997)] V 3 V H 3 ]

9 Eperimental Quantum Teleportation with a Complete Bell State Measurement using Nonlinear Interaction Principle schematic of quantum teleportation with a complete BSM. Nonlinear interactions (SFG) are used to perform the BSM. and represent the respective horizontal and vertical orientations of the optic aes of the crystals. [Yoon-Ho Kim, Sergei P. Kulik, and Yanhua Shih, Phys. Rev. Lett. 86, 370 (00)]

10 BSM is based on nonlinear interactions: optical sum frequency generation (SFG) (or up-conversion ). First type-i SFG crystal: V V H Second type-i SFG crystal: H H V First type-ii SFG crystal: V H H Second type-ii SFG crystal: H V V, [ H H V V Type I SFG crystals ] V H 5 polarization projector G. 5 0 DI, DII, [ H V V H Type II SFG crystals ] V H 5 polarization projector G. 5 0 DIII, DIV

11 Van Enk & Hirota Scheme of QT Entangled coherent states are stronger than standard bi-photonic states against possible photon transfer to reservoir modes QT of one qubit : superposed coherent state, Glauber's coherent states are defined as eigenstate of annihilation operator with eigenvalue. I ε α ε α o Entangled coherent state, o EVEN,α ODD,α EVEN,α E α α (,,, [,, ] ( ) α α α, ODD,α, e ) ( )

12 E,, I P.S.I : α 0 - i α PSI 3 5 BS PSII 6 ( β + i γ ) ( i β + γ ) B S : β, γ 0,3, P.S.I I : η i η To BOB, 5 Figure shows our scheme of quantum teleportation. Entangled state contains two modes and, one of which (mode ) goes to Bob directly We let lice pass her part of the entangled state (mode ) to pass through a phase shifter P.S.I which converts state in mode to state in mode 3. Now lice mies state 3 with the state to be teleported (mode 0) by using a 50:50 beam splitter, modifies one of the two outputs (mode 5) by passing it through a phase-shifter P.S.II which changes the state 5 to 6, and then performs photon counting in the two final outputs and 6. This result is then passed to Bob, which helps him to retrieve the information state by performing unitary transformation on the state. Van Enk & Hirota noted that One of the counts is always 0. If the other counts are odd, teleportation is successful. If the other counts are even, teleportation fails. Success of teleportation is ½.

13 Our Scheme differs from the above We divide counts in to: Zero, nonzero even and odd Zero counts gives failure Odd counts gives perfect QT Nonzero even counts gives almost perfect QT For α² =, or 5 minimum average fidelity = 0.73, , respectively

14 For photon counting modes & 6 and Bob s mode, 6, / ( ) One of the modes and 6 is For count in the other mode Van Enk & Hirota wrote 0 6 and Hence one of the count is always zero We write where α α NZE, EVEN, α ODD, α 0 ( ) NZE, α ( 0 / ) ODD, α,

15 U I,6, 0 0 ODD, 6 In the callouts: + EVEN,α - ODD,α U U II II NZE, α 0 6 ODD,α EVEN,α U III U + III NZE, ODD, EVEN, 6 ODD, 0 EVEN, ODD, 6 U IV 0 ODD, EVEN, ODD, 6 U V

16 These lead to T ODD, I T T ~ EVEN, ODD, II III T T EVEN, ODD, IV V ~ denotes unnormalized state Fidelity defined by F I T or Tr [ ] I T where I I I and T T T F F =F = F I II III + - IV F V

17 Information ( i.e., and ) - arbitrary Consider MSFI (Minimum ssured Fidelity), the minimum of F for variation of (MSFI) I 0 (MSFI) II, III (MSFI) IV, V If P i is probability for occurrence of case I, we will define average fidelity as F av. V P ii i F i ( ) This has minimum value, which we call minimum average fidelity (MVFI), MVFI -( ) - for For α, or 5, MVFI 0.73, , ,respectively

18 Teleportation with non-maimally entangled coherent state Recently H. Prakash and Manoj K. Mishra noted a very interesting thing that MVFI increases on decreasing entanglement. Use of non-maimally entangled coherent state, E = [ α,α + -α,-α ], (+ ) in place of maimally entangled coherent state, E = [ α,α -α,-α ], (- ) gives, MSFI =0 for both zero count, F= for one nonzero even count and F for one odd count. Minimum average fidelity in this case is greater then that when maimally entangled state used by an amount, D (3 )( ) ( )( ) Fig. Curve and shows minimum average fidelity when non-maimally and maimally entangled coherent states respectively, are used as quantum channel. Curve 3 shows variation difference D with α² It is seen that this is important as there is a marked increase in MVFI at low α, and it is difficult to generate superposed coherent states with large α.

19 Wang s Teleportation of Bipartite State Phys. Rev. 6(00)030 Information: Φ =ε α,α +ε -α,-α + - Entangled State 8 (- ) [ α,α,α - - α,-α,-α ] Wang followed the van Enk Hirota scheme and reported success probability /. We modified the Wang's scheme and got near perfect teleportation. H. Prakash, N. Chandra, R. Prakash & Shivani, Phys. Rev. 75 (007); also published in Virtual J. Q. Inf.

20 Our Scheme for QT of Bipartite State I = α,α + -α,-α, +, -, = EVEN,α,α + ODD,α,α +, -, ( α,α + -α-α ) EVEN,α,α =, (+ ) ( α,α - -α-α ) ODD,α,α =, (- ) = ( ± ) ± + - ( ± ) + - = ± ± (+ ) (- ) E 3,,5 [,, 3,,5,, 8 ( ) 3,,5

21 Entangled state 3 5 To Bob 6 PS-I BS-I PS-II BS-II PS-III 8 Figure. Numerals,,..., 6 refers to modes. Entangled states of modes and are to be teleported to bob. Out of E, state in mode 3 goes to lice while states in modes and 5 3,,5 go to Bob. lice (i) converts state to state 6 by using phase shifter PS-I, (ii) mies state 6 with state using a beam splitter BS-I, (iii) modifies output in 7 to state 9 using phase shifter PS-II, (iv) mies state 9 with state 3 using beam splitter BS-II (v) modifies output in 0 to state using phase shifter PS-III, and (vi) performs photon counting in mode and. The results of photon counting, conveyed to Bob by a classical channel helps him construct the entangled state by making unitary transformation on state of mode and 5 PSI : i, PSII : i, PSIII : i BSI :, ( i ), ( i ), BSII :, ( i ), ( i ),6 7,8 9, 3 0,

22 Possible cases of counts in mod es & are I zero,zero II nonzero even,zero III zero,nonzero even, IV odd,zero, V zero,odd Unitary Transformations are U =U = U = EVEN,α,α ODD,α,α ± ODD,α,α EVEN,α,α II,III U = EVEN,α,α v EVEN,α,α - ODD,α,α ODD,α,α I IV Teleported States are T T T I II IV ODD,,, T III T V X X EVEN,, X X EVEN,, ODD,, ODD,,

23 ) ) ( ( F F, F,F This gives F 8 V IV V IV - I av,min av,min av F, For ) ( F ) ( ) ( F

24 TELEPORTTION OF -QUBIT INFORMTION ENCRYPTED IN SINGLE MODE SUPERPOSED COHERENT STTE Information: I=ε α +ε iα +ε -α +ε -iα 0 3 =a α +a α +a α +a α α 0,,,3 are superposed coherent states with n, n+, n+, n+3 photons respectively. Entangled state: E=α,α + iα,iα + -α,-α + -iα,-iα Photon counting is done in four modes. lmost perfect fidelity is obtained if three counts are nonzero and one count zero Minimum average fidelity is 0.99 for α 3.

25 SCHEME FOR TELEPORTTION OF -QUBIT INFORMTION

26 Taking Decoherence into ccount Following van Enk & Hirota, we also assume that 0 R R, R is reservoir Information R0 0 0 R ~ I ~ 0,R0, k ~ k, is reservoir coupled to the mode 0. 0 R0 0 k R0 If R0 contains more than one modes k R 0 i ki R0i, k i ki ( ) En tangled State E K,,R R ( i k i Ri [ ) k i Ri, K i R k i i k i K R ] ( )

27 The calculations are long and difficult but somewhat straight forward and give complicate d results which we show graphicall y. It is interesting that even in the case of zero counts in both modes the minimum assured fidelity is nozero. The fidelity is zero only if the noise is zero and the information is an even state. For nonzero counts, Minimum ssured Fidelity decreases with increase in for low noise. For high noise however the Fidelity increases attains a maimum value and then decreases. The average Fidelity depends appreciabl y on Information for low values of only. Let us put cos( / ), sin( / )

28 Minimum ssured Fidelity Case with Nonzero Even Counts Figure : Variation of MF with η > 0.7: F decreases uniformly for different values of η < 0.7: F has a Maimum, which decreases as noise increases

29 Case with Odd Counts Figure 6: Variation of MF with for different values of Critical value of = 0.738

30 Figure7: Variation of average fidelity with at = 0.9. for different values of

31 Entanglement Diversion E =N [ α,α - -α,-α ] To Clair PS-I E =N 3 3[ α,α - -α,-α ] BS To Bob PS-II 8 Figure. Numerals,,..., 8 refers to modes. States in modes and are with lice while state 3 and are with Bob and Clair respectively. lice (i) converts state to state 5 using phase shifter PS-I, (ii) mies state with state 5 using a beam splitter BS, (iii) modifies output in 7 to state 8 using phase shifter PS-II, and (iv) performs photon counting in modes 6 and 8. The result, conveyed to Bob helps him to perform a unitary transformation on state 3 and and generate an eact replica of entangled state.

32 LONG DISTNCE QT WITH PERFECT FIDELITY ND S GOOD SUCCESS S DESIREED USING REPETED TTEMPTS For QT using ECS Photon-counting in two modes one count always zero Fidelity F = 0 when the other count is zero F= when other counts are odd MSFI when other counts is nonzero-even and α is appreciable In photon-counting information is destroyed and measurements cannot be repeated to increase success Repeated attempts can be possible only if the information is not destroyed in measurements We present another scheme of QT of state of a qubit with lice on to a qubit with Bob using α (i) a light pulse in even coherent state, N[ α α ], N [( e )] (ii) two light pulses in coherent state (iii) Beam splitter assemblies and light detectors. This suits long distance Quantum Teleportation serving as a link between two quantum processors at long distances the photons fly and the quantum state of qubit is teleported. /

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