TELEBROADCASTING OF ENTANGLED TWO-SPIN-1/2 STATES

Transcription

1 TELEBRODCSTING OF ENTNGLED TWO-SPIN-/ STTES IULI GHIU Department of Physics, University of Bucharest, P.O. Box MG-, R-775, Bucharest-Mãgurele, Romania Receive December, 4 quantum telebroacasting process combining the many-to-many teleportation an asymmetric broacasting of entanglement from one pair of observers to two spatially separate pairs of observers is presente. By applying the Peres-Horoecki criterion we analyze the inseparability of the final states an show that this epens on the parameter, which characterizes the quantum channel use in the process. The final inseparable states represent the output states generate in the broacasting of entanglement via local cloning.. INTRODUCTION Quantum teleportation is the basic ingreient for many communication processes. It performs the transmission an reconstruction of an unknown quantum state over arbitrary istances with the help of entangle states. In the stanar teleportation scheme introuce by Bennett et al. [], the state is transfere from one sener, lice, to one receiver, Bob. In this paper we review two generalizations of quantum teleportation: oneto-many an many-to-many teleportation, where the information of a quantum system is istribute from one sener to M receivers, an from N seners to M receivers, respectively (Section ). Then, in Section 3 we present the Peres- Horoecki criterion of separability of mixe two-spin-/ particles. summary of broacasting of entanglement using local optimal universal asymmetric cloners is given in Section 4.. In Section 4. we present the telebroacasting of two-spin-/ entangle states to two istant pairs of observers.. QUNTUM TELEPORTTION.. ONE-TO-ONE ND ONE-TO-MNY TELEPORTTION Let us start by reviewing the original teleportation protocol an its generalization, one-to-many teleportation. In the stanar teleportation scheme aress: iughiu@barutu.fizica.unibuc.ro Rom. Journ. Phys., Vol. 5, Nos., P. 7 5, Bucharest, 5

2 8 Iulia Ghiu an unknown quit (a state of a -level system) is faithfully transmitte from one observer, lice, to another observer, Bob, while the initial lice s state is estroye. Let the initial unknown state we wish to teleport be ψ = α k, where α, k= k = an { } k= k is the computational basis. The quantum channel require in this process is a maximally entangle state share by lice an Bob j j B ξ =. (.) The state of the whole system of the three particles is: mn, m= n= k= πikn ( ) (.) ψ ξ = Φ exp α k k+ m, where k+ m= k+ m moulo, an k= π ( ) Φ exp ikn mn, = k k + m, (.3) is the generalize Bell basis [, ]. lice performs a Bell-type measurement on her particles an sens the result to Bob. If the outcome of lice s measurement is Φ, then Bob has to apply the unitary operator [] mn, πijn Vmn ; = exp j j+ m (.4) on his particle in orer to retrieve the initial state. Having performe the Belltype measurement, lice estroye the information containe in the initial unknown state, as it must be conformable to the no-cloning theorem. The protocol introuce by Bennett et al. [] is calle one-to-one teleportation since the information is transmitte from one sener to one receiver. We now briefly present the one-to-many teleportation protocol propose by Murao et al. [3], where the information is istribute from one sener, lice, to M istant receivers, B, B,, B M, using multiparty entanglement. The information of a -level system encoe in an N-particle state is: k k (.5) k= ψ = α ψ, k

3 3 Telebroacasting of entangle two-spin-/ states 9 with, k k an { k } = α = ψ represents a basis in the -imensional space. The quantum channel is a maximally entangle state of the lice s N particles an receivers M particles: where { π j } an { j } j j BB B (.6) M ξ = π φ, φ are bases in the -level spaces of lice s an receivers particles, respectively. lice performs a joint measurement on her particles in the generalize Bell basis. Depening on the result communicate by lice, the receivers apply a local recovery unitary operation (RUO) [3]. If the result of lice s measurement is Φ mn ;, then the receivers perform the unitary operation that satisfies the conition: V πijn = V V... V = exp φ φ. (.7) mn ; B B BM j j+ m Therefore, the information of the initial unknown state of Eq. (.5) has been istribute to several istant parties: j j (.8) BB... BM φ = α φ... MNY-TO-MNY TELEPORTTION Now we present the many-to-many teleportation of a -level system propose by us in Ref. [4]. In this protocol the information of a -level system initially share by N istant observers is transmitte to M istant receivers (with M > N). The initial entangle state of the observers,,, N is given by with k k k k (.9) N k= ψ = α ψ ψ ψ, α, k= k = an { k j } ψ represents a basis in the -imensional space of the jth sener. We efine the quantum channel as a maximally entangle (N + M)-particle state share between seners an receivers:

4 Iulia Ghiu 4 ξ = πj π j πj φj, (.) N BB... BM where we have enote by B the particles that belong to the receivers. The π represent a -imensional basis for the ith sener. The joint state states { j } of the initial system an the channel is k k j k j k j N N k= ψ ξ = α ψ π ψ π ψ π φ j B... B M = N+ Φmn, Φ mn, Φ mn, N m n, n,..., nn exp πik ( n n n N) αk φ. k+ m k (.) The protocol for many-to-many teleportation is the following: a) Each sener performs a measurement of his particles in the generalize Bell basis. b) The seners communicate the result of the measurement to the M receivers. c) Let us analyze the case when the outcome of the seners Bell measurement is: Φ Φ Φ. (.) mn, mn, mn, N Then, the receivers have to apply a local recovery unitary operation that fulfills: V exp ik mn ; (, n,..., n φ ) N k = π n n n N φ. (.3) k m Therefore, the many-to-many teleportation istributes the information of the initial N-particle state (.9) into the M-particle state: j j j j j j N BB... B (.4) ψ = α ψ ψ ψ φ = α φ. M 3. THE PERES-HORODECKI CRITERION OF SEPRBILITY In this section we present the necessary an sufficient conition for the separability of mixe two-spin-/ particles. Let us recall the efinition of the

5 5 Telebroacasting of entangle two-spin-/ states separability of a bipartite system: state is separable if the ensity operator escribing this state can be written as a convex combination of prouct states [5]: () () pi i i (3.5) i ρ= ρ ρ, () where ρ i an ρ () i are ensity operators of the first subsystem, an secon subsystem, respectively. Theorem (Peres-Horoecki). two-spin-/ state is separable if an only if the partial transposition of the ensity operator is a nonnegative one [6, 7]. While the necessary conition foun by Peres is vali for arbitrary bipartite mixe states, the sufficient one is true only for an 3 systems. 4. TELEBRODCSTING OF BIPRTITE TWO-LEVEL ENTNGLED STTE 4.. PRELIMINRIES The no-cloning theorem forbis the existence of a unitary operation that can prouce two perfect copies of an arbitrary quantum state [8]. Therefore some approximate methos for cloning were propose, where the fielity between the final ientical states an the initial one is less than unity [9,, ]. In the case of asymmetric cloning (when the two final clones are not ientical), it is interesting when the universal cloning machine is optimal, that means a machine that creates the secon clone with maximal fielity for a given fielity of the first one [, ]. Cerf has foun the expression of the optimal universal asymmetric cloning machine of -level states using a reference state []. We have also obtaine an equivalent expression of this cloning machine, by eliminating the reference state [4]: U j = ( j j j + + ( )( p + q) (4.6) + p j j+ r j+ r + q j+ r j j+ r, r= r= where p + q =. n interesting application of quantum cloning is broacasting of entanglement propose by Bužek et al. []. In this process, the entanglement originally share by two observers is broacast into two ientical entangle states by using local optimal universal symmetric cloning machine. We have investigate the broacasting of entanglement using the optimal universal asymmetric cloning machine by employing the formula (4.6) for = :

6 Iulia Ghiu 6 U( p) = ( + p + q ) + p + q U( p) = ( + p + q ), + p + q (4.7) with p + q =, where the first two qubits represent the clones an the last one is the ancilla [4]. The initial entanglement share by two observers, lice an Bob is: ψ =α +β. (4.8) The state of the total system, consisting of the two particles an, an another four particles: the blank states 3 an 4, the ancillas 5, 6 is given by Π = U( p) U( p) ψ 35 46, where the particles enote by o number belong to lice, while the even particles belong to Bob. By using the Peres-Horoecki criterion we have evaluate in Ref. [4] the inseparability of the two final states ρ 4 an ρ TELEBRODCSTING OF TWO-SPIN-/ ENTNGLED STTES Consier that two spatially separate observers, an, hol an entangle state an they wish to teleport two copies of this state to two pairs of observers also locate at ifferent places, B B 4, an B B 3, respectively. Suppose that an share an arbitrary two-spin-/ entangle state ψ =α +β. (4.9) Here we use the notation spin up = an spin own =. We propose a new scheme calle telebroacasting of entanglement, which simultaneously copy an transfer the information of the initial entangle state. This protocol combines the many-to-many teleportation an asymmetric broacasting of entanglement. We efine two six-particle states: φ := ( + p + q + + p + q + p + p + pq + (4.) + q + pq + q ; φ := ( + p + q + + p + q + p + p + pq + + q + pq + q. ) ) (4.)

7 7 Telebroacasting of entangle two-spin-/ states 3 We choose the multiparticle quantum channel require in the many-tomany protocol as: ξ = φ + φ, (4.) BBBBBB BBBBBB where B 5, B 6 are two istant observers. The total state is ψ ξ = + + ( ) Φ Φ α Φ +β Φ + +Φ+ Φ αφ +βφ ( ) ( ) ( ) ( ) ( ) ( ) ( ). + +Φ Φ αφ βφ +Φ Φ αφ +βφ Ψ Ψ αφ +βφ +Ψ Ψ αφ βφ + + +Ψ Ψ αφ βφ + +Ψ Ψ αφ +βφ (4.3) The many-to-many protocol consists of three steps, as was shown in Sec..: a) an perform a measurement of the particles available in the Bell basis. b) an communicate the outcomes to the six receivers, B, B, B 3, B 4, B 5, B 6. c) The receivers apply local unitary operations epening on the outcomes of the seners measurements. In Table we have shown the local recovery unitary operations that have to be performe for each outcome of the Bell measurements, which satisfies Eq. (.3). Hence, the information of the inial state (4.9) is encoe in the final state share by the six receivers: µ =α φ +β φ (4.4) BBBBBB BBBBBB We say that the input state ψ has been telebroacast if the following two necessary conitions are satisfie [4]: (i) the local reuce ensity operators ρ BB an ρ 3 BB are separable, 4 (ii) the nonlocal states ρ BB an ρ 4 BB are inseparable. 3 pplying the Peres-Horoecki theorem presente in Section 3, we get the conition for the separability of the local states:

8 4 Iulia Ghiu 8 Table The local recovery unitary operations that have to be applie by the receivers, which epen on the outcomes of the seners measurements The outcome The local recovery unitary operation + + Φ Φ I I I I I I σ I σ I σ I + Φ Φ z z z σ I σ I σ I + Φ Φ z z z Φ Φ I I I I I I + + Ψ Ψ σx σx σx σx σx σx + Ψ Ψ σy σx σy σx σy σx + Ψ Ψ σy σx σy σx σy σx σ σ σ σ σ σ Ψ Ψ x x x x x x or equivalent αβ pq (4.5) 4 ( ) p p 4 p( p) α +. (4.6) The ensity operators of the nonlocal states are: ρ { BB = [ pq +α ( + p + q)] + [ pq + 4 ( + p + q) +β ( + p + q)] + 4pqαβ ( + ) + (4.7) + ( β q4 +β q +α p4 +α p) + +β ( p4 +β p +α q4 +α q), { ρ BB = [ pq +α ( + p + q)] + [ pq + 3 ( + p + q) +β ( + p + q)] + 4pqαβ ( + ) + + ( β p4 +β p +α q4 +α q) + +β ( q4 +β q +α p4 +α p), } } (4.8) gain we use the Peres-Horoecki criterion an fin that the two nonlocal states are inseparable if ( β p +β p +α q +α q )( β q +β q +α p +α p ) 6α β pq (4.9)

9 9 Telebroacasting of entangle two-spin-/ states 5 or equivalent where ( 4 ) ( 4 ) λ α + λ, (4.3) pq pq 4 + pq 4 + pq λ=. pq + pq + pq q q q p p p + 8pq (4.3) The requirements that 4λ has to be positive an the local states are separable when the nonlocal ones are inseparable lea to [4]: 9 + p (4.3) The final states (4.7) an (4.8) obtaine by the four receivers represent the output states generate in broacasting of entanglement using local optimal universal asymmetric cloning machines escribe in Sec. 4.. Our process, telebroacasting of entanglement, performs the teleportation of the final inseparable states obtaine using local broacasting to two pairs of istant observers. In conclusion, we have proven how one can transmit optimal information of an entangle state to two pairs of receivers using only local operations an classical communication. cknowlegment. This work was supporte by the Romanian CNCSIS through a grant for the University of Bucharest. REFERENCES. C. H. Bennett, G. Brassar, C. Crepeau, R. Jozsa,. Peres, an W.K. Wootters, Phys. Rev. Lett., 7, 895 (993).. N. J. Cerf, J. Mo. Opt., 47, 87 (). 3. M. Murao, M. B. Plenio, an V. Veral, Phys. Rev., 6, 33 (). 4. Iulia Ghiu, Phys. Rev., 67, 33 (3). 5. R. F. Werner, Phys. Rev., 4, 477 (989). 6.. Peres, Phys. Rev. Lett., 77, 43 (996). 7. M. Horoecki, P. Horoecki, R. Horoecki, Phys. Lett., 3, (996). 8. W. K. Wootters an W. H. Zurek, Nature, 99, 9 (98). 9. V. Buzek an M. Hillery, Phys. Rev., 54, 844 (996).. D. Bruß, D. P. DiVincenzo,. Ekert, C.. Fuchs, C. Macchiavello, an J.. Smolin, Phys. Rev., 57, 368 (998).. N. J. Cerf, T. Durt, an N. Gisin, J. Mo. Opt., 49, 355 ().. V. Buzek, V. Veral, M. B. Plenio, P. L. Knight, an M. Hillery, Phys. Rev., 55, 337 (997).