Efficient many-party controlled teleportation of multi-qubit quantum information via entanglement

Transcription

1 Effcent many-party controlled teleportaton of mult-qut quantum nformaton va entanglement Chu-Png Yang, Shh-I Chu, Syuan Han Physcal Revew A, 24 Presentng: Vctora Tchoudakov

2 Motvaton Teleportaton va the control of agents s a way to create a teleportaton network. Can e used for quantum secret sharng. Mult-qut teleportaton allows to teleport (complcated) states. Teleport a whole system (e.g. quantum computer). 2

3 Outlne Introducton Sngle qut teleportaton usng Bell states Sngle qut teleportaton usng GHZ Prevous work Sngle qut teleportaton va the control of n agents (usng GHZ) Extenson to mult-qut teleportaton va the control of n agents (usng GHZ) Presentng a more effcent method Sngle qut teleportaton va the control of one agent (usng Bell states) Extenson to mult-qut teleportaton va the control of one agent (usng Bell states) Extenson to mult-qut teleportaton va the control of n agents (usng Bell states and GHZ)

4 Remarks All normalzaton factors are omtted for smplcty. Throughout the presentaton I wll use the followng to represent the Bell states: ± ± φ j = ± ψ j = ± j j All one qut measurements are performed n the, computatonal ass. I wll refer to untary rotaton y π =,,2, ( I, σ x, σ z, σ xσ z ) respectvely, as smple rotatons. j j 4

5 Teleportaton usng two-partcle entanglement Suppose Alce wants to send the (unknown) quantum state Ψ A = α β to Bo. She prepares an entangled Bell state, and shares t wth Bo. The state of the system now can e rewrtten as: Ψ A ( 2 2 ) = φ ψ 2 2 ( α ( α β β ) φ ) ψ 2 2 ( α ( α β β ) ) Then she measures (n the Bell measurement ase) the two partcles ± ± she possesses and gets one of the states φ 2, ψ 2 She sends Bo two classcal ts, accordng to the state she measured, and he performs a smple rotaton to retreve the orgnal state = α β. 5 Ψ A

6 Teleportaton usng three-partcle entanglement (va the control of one agent) Suppose Alce wants to send Clff the (unknown) state va the control of Bo. Alce uses a three-partcle entangled GHZ state, whch she dvdes etween herself (2) Clff (4) and Bo (). The ntal state of the system can e rewrtten as: Ψ A = α β Ψ A ( ) = φ φ ψ ψ ( α ( α ( α ( α β β β β ) ) ) ) 4 6

7 Teleportaton usng three-partcle entanglement (va the control of one agent)- 2 The algorthm:.. v. Alce performs a Bell- state measurement on her quts (,2) and ± ± gets one of the states φ 2, ψ 2. Then she sends Clff a 2-t classcal message ndcatng whch of the Bell states she measured. Bo performs a Hadamard transformaton on hs qut (), and then measures t and sends the result (one classcal t) to Clff. Once Clff has got all the nformaton, he can reconstruct the orgnal state y performng a smple rotaton on hs qut. Ψ A 7

8 Teleportaton usng three-partcle entanglement (va the control of one agent) - For example: If Alce measured φ 2, then Bo and Clff are left sharng α β After Bo performs Hadamard transformaton ther shared state ecomes α β = α α β β When Bo measures hs qut and sends α the α result to Clff, the latter knows n what state hs qut s - β or β, and whether he should perform a smple rotaton on hs qut or not, respectvely. 8

9 Teleportaton usng three-partcle entanglement (va the control of one agent)- 4 Note that wthout Bo s cooperaton Clff cannot fully restore the orgnal state. Ψ A The densty matrx of Clff s partcle wthout Bo s nformaton s: ρ or 4 = α β 2 ρ 4 (dependng on Alce s measurement 4 4 = α β 4 outcome). 4 Hence Clff has ampltude nformaton aout Alce s qut, ut knows nothng aout ts phase. 9

10 Sngle qut teleportaton va the control of n agents (usng GHZ state) It s possle to use (n2)-qut GHZ state to teleport Alce s state to Bo. The GHZ state s dvded etween Alce (a), Bo () and the n agents. The ntal state of the system can e rewrtten as: n a n a Ψ A ) ( ) ( ) ( ) ( 2 n n Aa n n Aa n n Aa n n Aa n A GHZ = Ψ β α ψ β α ψ β α φ β α φ

11 Sngle qut teleportaton va the control of n The algorthm: agents usng GHZ state - 2 Alce performs a Bell-state measurement on her quts (A, a), gets ± ± one of the states φ Aa,ψ Aa, and sends the result (2-t classcal message) to Bo. Each of the agents performs a Hadamard transformaton on hs qut, measures t, and sends one classcal t to Bo. Bo can reconstruct the orgnal state Ψ A y performng a smple rotaton accordng to Alce s and the agents results.

12 Sngle qut teleportaton va the control of n agents usng GHZ state - Example for n = 2: For example: If Alce measured φ Aa, then Bo and the 2 agents are left sharng α β. After the agents perform Hadamard transformaton the shared state ecomes α α α α β = β β β α When the 2 agents measure ther quts and send the result to Bo, he knows n what state hs qut s - α or α, and whether he β β should perform a smple rotaton on hs qut or not, respectvely. β 2

13 Sngle qut teleportaton va the control of n agents usng GHZ state - 4 After Alce s Bell state measurement, Bo and the agents share a (n)-qut state f the form: n n or n α ± β dependng on Alce s measurement outcome. α ± β n Hence even f only one of the agents doesn t cooperate (and the rest do), after tracng 2 out all the 2 agents quts, 2Bo s qut 2 ρ densty wll = α β ρ = α β e or Ψ A - nsuffcent to reconstruct the orgnal state. (No nformaton aout the phase).

14 4 Mult-qut teleportaton va the control of n!agents usng GHZ state neffcent It s possle to extend the aove method to teleport m quts, y preparng m copes of the (n2)-qut GHZ state and then performng the aove protocol for each of the orgnal m quts. Such a procedure requres for each agent: m GHZ quts m Hadamard transformatons m sngle-qut measurements m-ts classcal message sent to Bo (y each agent) Thus, the descred algorthm requres consderale resources and classcal communcaton for teleportaton of a large numer of quts (large m). Note: Any agent can e chosen to e the recever n ths algorthm. (There s notng specal aout Bo).

15 Mult-qut teleportaton va the control of n agents effcent method The artcle [] presents a more effcent way to teleport m quts va the control of n agents. For each agent t wll requre: GHZ qut Hadamard transformaton sngle-qut measurement t classcal message to Bo How s that acheved? Two-qut entanglement (Bell states) s used for communcaton etween Alce and Bo, One copy of the (n)-qut entangled state (Bell for n=, GHZ for n>) s dstruted among Alce and the n agents for control. Thus, preventng copyng the controllng GHZ state for each teleported qut, as t was n the method descred efore. 5

16 Entanglng entanglement Suppose you have two systems A and B, each has four states: and. a, a2, a, a4, 2,, 4 a a2 2 One can uld an entangled state (for nstance a = =, a2 = 2 = for we wll get. a, a a 2 Now, f are Bell states, the state 2 wll e entangled twce we are entanglng the already entangled Bell states, entanglng entanglement. φ a 6

17 7 Sngle qut teleportaton va the control of (one agent (usng Bell states Suppose Alce wants to send Bo the unknown state va the control of Carol. Alce prepares the followng entangled state: whch s dvded etween herself (2,4), Bo () and Carol (5). Bts (2,) are used for the communcaton, and ts (4,5) are used for control. Notce that ths s an entanglng entanglement state, where the communcaton and control Bell pars are entangled wth each other. The entre system state can e rewrtten as φ ( ) ( ) 2 α β φ2 α β φ ψ ( ) ( ) 2 α β ψ 2 α β φ ( ) ( ) 2 α β φ 2 α β φ ψ ( ) ( ) 2 α β ψ 2 α β φ Ψ A = α β 2 φ45 φ2 φ

18 Sngle qut teleportaton va the control of one agent (usng Bell states) - 2 The algorthm: II. Alce performs a Bell state measurement on quts (,2) and sends the results to Bo lke n smple teleportaton. Then the system state ecomes φ45 ψ ' φ where ψ, ψ 45 s the state of Bo s qut (). If Alce measured If Alce measured ± φ ' ± ψ ' ψ ' ψ = α ψ ' = α ψ = α ψ ' = α ± β β ψ, ψ ' In order to know n whch of hs qut () s, Bo needs nformaton aout quts (4,5). ± β β 8

19 Sngle qut teleportaton va the control of one agent (usng Bell states) - I. Alce and Carol perform Hadamard transformaton on quts (4,5) respectvely, then H H. φ φ45 φ ψ They measure ther respectve quts and send the result to Bo. He can determne now n whch state hs qut s: If he got () from oth Alce and Carol then hs qut s n state ψ ψ ' If he got () from Alce, and () from Carol then hs qut s n state. Now Bo can reconstruct Alce s orgnal state y performng a smple rotaton on hs qut. 9

20 2 Mult-qut teleportaton va the control of one agent (usng Bell states) Suppose Alce wants to send Bo m quts, va the control of one agent (Carol). Alce prepares the followng entangled state: m = m ( ) ( ) ( ) ( ) ' " '" ac ac '" '" ac = whch s dvded etween herself (a, ) Bo ( ) and Carol (c). Then the whole system state can e rewrtten as φ ψ ( α β ) φ ( ) ' α β " " " " ( β α ) ψ ( α β ) m ' = ' " " ' " " m = φ ψ ' ' Ψ m ( α β ) = = A ( ) ( α β ) φ ( ) ' α β " " " " ( β α ) ψ ( β α ) " " ' " ac " ac ac ( ) ac ac

21 Mult-qut teleportaton va the control of one agent(usng Bell states) - 2 The algorthm: II. Alce performs Bell-state measurements for quts. Then the system state s m where ψ = ψ and ' = ψ ', whle " " = are the states of the Bo s quts. ψ (, )' ( ) ψ '( ) = ac ac ac m ψ ψ, ψ ' " " ac Alce measured Alce measured ± φ ' ± ψ ' Bo gets Bo gets ψ = α ψ ' = α ψ ψ = α ' = α " " ± β β " " " ± β " " β " 2

22 Mult-qut teleportaton va the control of one agent (usng Bell states) - II. III. Bo can recover the orgnal state Ψ A y performng a smple rotaton on hs quts. But n order to know whch rotaton to perform, he needs nformaton aout the phase of Alce s orgnal state. To provde Bo wth that nformaton, Alce and Carol oth perform a Hadamard transformaton on quts (a,c) respectvely. The system state now s ψ ( ) ψ '( ). Alce and Carol measure the quts (a, c) respectvely and send the results (-t classcal message) to Bo. Now he has enough nformaton to recover Alce s orgnal state: If oth Alce and Carol sent hm (or ) then he knows hs quts are n the state ψ. If Alce sent () and Carol (), then Bo knows hs quts are n the state ψ '. ac ac ac ac 22

23 Mult-qut teleportaton va the control of one agent(usng Bell states) - 4 Let us show, that wthout Carol s collaoraton Bo cannot recover Alce s orgnal state: If only Alce performs the Hadamard transformaton the system s state ecomes [( ψ ψ ') ( ψ ψ ') ] ( ψ ψ ') ( ψ ψ ) [ ' ] c c a c c a After tracng out qut c, Bo s m quts densty operator s: ρ = ( ψ ψ ')( ψ ψ ') ( ψ ψ ')( ψ ψ ') 2

24 Mult-qut teleportaton va the control of one agent(usng Bell states) - 5 Then, after some tedous math, one can show that the densty operator for any qut (elongng to Bo), after tracng out the other m- quts s: 2 ρ " = α β 2, f Alce measured ± φ ' and 2 ρ " = α β 2, f Alce measured ± ψ ' Wthout Carol s cooperaton Bo only has the ampltude nformaton aout each qut n Alce s orgnal state, ut knows nothng aout t s phase. 24

25 Comparng the methods mult-qut teleportaton va the control of one agent Yang, Chu, and Han GHZ - only method Alce Bo Alce Bo Carol Carol message controller anclla entanglement 25 twce entanglement target

26 Comparng the methods mult-qut teleportaton va the control of one agent - 2 Yang, Chu and Han s method requres 2(m) quts to prepare the Bell states qut for the agent sngle-qut Hadamard transformaton and sngle-qut measurement performed y the agent t classcal message sent y the agent to the recever Usng only GHZ entanglement (as descred earler) requres: m qut to prepare the entangled GHZ state m quts for the agent m sngle-qut Hadamard transformatons and m sngle-qut measurements performed y the agent m t classcal message sent y the agent to the recever Yang et al method s more effectve for m 2 26

27 Mult-qut teleportaton va the control of many agents (y Yang, Chu, and Han) We wll expand the prevous method to n> agents control, y dvdng a (n)-qut entangled GHZ state etween Alce and the n agents. Alce n = Bo Ths way Bo s alty to fully reconstruct Alce s quts wll depend on the collaoraton of all n agents, yet the reconstructon process wll reman very smlar to oneagent controlled teleportaton Carol Dana Eve message controller anclla entanglement 27 twce entanglement target

28 28 Mult-qut teleportaton va the control of many agents(y Yang, Chu, and Han) - 2 Decomposton of GHZ states: When performng a Hadamard transform on each of the GHZ state s quts, we get: Where and, And x { x } l y l { y l l s a sum over all possle ass states each contanng an even (odd) numer of s. For example, n=4: GHZ GHZ { x } { } { } { } x l y { x } l y l { } { } l { } x l y l y l { x l } = xx2... xn { y l } = y y2... y x, {, } n l y l ( ) { } { } } { x l } { x } =... l l

29 Mult-qut teleportaton va the control of many agents(y Yang, Chu, and Han) - Suppose Alce wants to send Bo m quts, ( A, A2,..., A va the control of n agents. n Alce prepares the followng entangled state: m = whch s dvded etween herself ( ), Bo ( ) and the agents (the (n)-qut GHZ states are dvded etween Alce and the agents). The state of the whole system now s: ) Ψ m ( α β ) = = A m ( ) GHZ ( ) GHZ ' " '" '" '" m = m = = [ ( α β ) ( ) ] '" '" [ ( α β ) ( ) ] GHZ '" '" GHZ 29

30 Mult-qut teleportaton va the control of many agents(y Yang, Chu, and Han) - 4 The algorthm: II. Alce performs two-qut Bell state measurements on her m qut pars (, )'. Then the system state s: ψ GHZ ψ ' GHZ where ψ and ψ ' are the states of Bo s m quts ( ). IV. Alce and the n agents perform a Hadamard transformaton on ther GHZ quts. Then, the state of the system ecomes: ψ { xl } { yl } ψ ' { xl } { yl } a a a a { xl } { y l } { xl } { y l } Alce and the n agents measure ther GHZ quts and send the results (-t classcal message each) to Bo. He can reconstruct the orgnal state Ψ from state ψ (or ψ ' A ) usng a smple rotaton.

31 Mult-qut teleportaton va the control of many agents(y Yang, Chu, and Han) - 5 Bo wll determne n whch of the states ψ or ψ ' hs quts are y the results of Alce s and the agents measurement on ther GHZ quts. If the n agents results contan an even (odd) numer of s and Alce measured (), then Bo s quts are n state ψ. If the n agents results contan an odd (even) numer of s and Alce measured (), then Bo s quts are n state ψ '. If Alce and all the agents collaorate (perform Hadamard and measure), Bo can reconstruct the orgnal state, and the teleportaton succeeds. It s possle to show, wth more tedous math, that even f one agent does not collaorate, Bo s densty matrx wll not e I he wll not e ale to reconstruct Alce s orgnal state.

32 References [] Effcent many-party controlled teleportaton of multqut quantum nformaton va entanglement. C.P. Yang, S.I. Chu, and S. Han, Physcal Revew A 7, 2229 (24). [2] Quantum teleportaton usng three-partcle entanglement. A. Karlsson and M. Bourennane, Physcal Revew A 58, 494. [] Quantum Secret Sharng. M. Hllery, V. Buzek, and A. Berthaume, Physcal Revew A 59, 829 (999). 2