TRIPARTITE ENTANGLEMENT SWAPPING OF CONTINUOUS VARIABLES

TRIPARTITE ENTANGLEMENT SWAPPING OF CONTINUOUS VARIABLES DAO-HUA WU Dpartmnt of Mathmatics and Physics, Anhui Collg of Chins Traditional Mdicin, Wuhu 4000, Popl s Rpublic of China, Email: wudaaohua@sina.com Rcivd April 8, 0 W propos a schm for tripartit ntanglmnt swapping of continuous variabls, whr on EPR pair and on local W-typ ntangld stat ar utilizd. Bcaus of th co-xistnc of both bipartit and tripartit ntanglmnt in a W-typ ntangld stat, on optical bams of th EPR stat will b ntangld with th output mods displacd by Bob and Clair. Actually, th ntanglmnt swapping procssing could b viwd as tlclon of ntangld stat of continuous variabls. Ky words: Continuous variabls, Tripartit ntanglmnt swapping, W-typ ntangld stat, EPR pair. PACS: 0.67.-a, 0.65.Ud, 4.50.Dv.. INTRODUCTION Entanglmnt is on of th most important faturs in quantum mchanics. Assistd with ntangld stat, on can complt many impossibl tasks within th classical world,.g. it can b usd as quantum channl for quantum tlportation [] and quantum dns coding [], and in quantum computation, such as quantum algorithms [], quantum cryptography [4] and quantum rror corrction [5]. Quantum tlportation is a good xampl manifsting th nonlocality of quantum ntanglmnt. It has attractd much attntion right aftr it was proposd and has alrady bn vrifid xprimntally with both photonic [6] and atomic qubits [7, 8]. It has also bn gnralizd to continuous variabls (CV) [9]. Thn a protocol for tlporting of quadratur amplituds of a light fild was proposd in 998 [0]. In th sam yar, tlporting an arbitrary unknown CV stat was ralizd with bipartit ntanglmnt, gnratd from two singl-mod squzd vacuum stats combind by a bam splittr (BS) []. So far, th bst fidlity achivd, for tlporting a cohrnt stat input, is F coh = 0.70 ± 0.0 [], but improvmnts [, 4] ar still on th way. At th sam tim, quantum tlportation in anothr aspct, was gnralizd to tlport an unknown ntangld stat rathr than a singl-qubit suprposition stat, which is also calld ntanglmnt swapping in Rf. [5]. Pan t al. hav xprimntally ralizd th ntanglmnt swapping of th polarization ntanglmnt cratd by RJP Rom. 57(Nos. Journ. Phys., -4), Vol. 555 56 57, Nos. -4, (0) P. 555 56, (c) 0-0 Bucharst, 0

556 Dao-Hua Wu typ-ii paramtric down-convrsion [6]. Continuous-variabl ntanglmnt swapping is also an important application for quantum communication and has bn invstigatd in thory [7 9] and xprimnt [0]. Rcntly, w hav proposd a schm for controlld ntanglmnt swapping via tripartit GHZ-typ ntangld stat []. For th thr-body ntanglmnt, it falls into two inquivalnt classs [], i.., th GHZ- and W-typ ntangld stats. W-typ ntangld stat is a spcial kind of tripartit ntangld stat, which has bipartit and tripartit quantum ntanglmnt simultanously. Consquntly, whn a W-typ ntangld stat is usd in a quantum communication protocol, thr will b som nw rsults [, 4]. Motivatd by th discussions mntiond-abov, w prsnt a schm for tripartit ntanglmnt swapping via CV W-typ ntangld stat. Th schmatic stup is shown in Fig.. In our schm, th W-typ ntangld stat, which has both bipartit (mods A, B and mods A, C) and tripartit (mods A, B, and C) ntanglmnt, and a pair of EPR stat(mods in and rf) ar usd. On of th mods of th EPR bam is combind with th mod A of th W-typ ntangld stat. Thn, th othr two mods of th W-typ ntangld stat that ar not combind will b displacd according to th rcivd classical information of th combind mod. By so doing, th ntanglmnt btwn th two output mods and th mod rf ar stablishd. And th ntanglmnt swapping procssing is accomplishd. It is notabl that if on wants to accomplish th ntanglmnt swapping, th fidlity of tlportation of cohrnt stat should surpass th no-cloning limit [], i.., F coh > /. As th only limiting factor for this fidlity is th squzing dgr of th usd quantum ntangld stats, w will giv th succssful tlportation conditions in trms of ths squzing paramtrs. Actually, in our schm, w tlclon on bam of th EPR ntangld stat to two rcivrs(bob and Clair) via W-typ ntangld stat, and th othr bam of th EPR stat ar both ntangld with th two outputs displacd by Bob and Clair. In a sns, th swapping procssing could b viwd as tlclon of th EPR stat.. THE PROCEDURE FOR TRIPARTITE ENTANGLEMENT SWAPPING In ordr to accomplish th ntanglmnt swapping, a local CV W-typ ntangld stat, involving thr ntangld optical bams, and a local CV EPR stat, involving two ntangld optical bams, should b utilizd. Th EPR stat can b gnratd by combining two squzd vacuum stats with a half BS [5] or from nondgnrat optical paramtric amplifir procss [6]. Th W-typ ntangld stat can b gnratd by first combining two squzd vacuum stats with a 50:50 BS and thn on of th output bams is dividd into two bams B and C by anothr 50:50 BS [7]. In our schm, th local W-typ ntangld stat consists of mods A, B and C, which blong to Alic, Bob and Clair, rspctivly. Th EPR stat consists of mods in and rf; th mod rf is usd as a rfrnc, whil th mod in will b

Tripartit ntanglmnt swapping of continuous variabls 557 combind with mod A and masurd by Alic. First, w tlport th mod in to Bob and Clair via tripartit ntanglmnt swapping, simultanously. Th W-typ ntangld stat and th EPR stat, in Hisnbrg pictur, ar [5, 7, 0], ˆx A = + (a) ˆp A = + +r ˆp (0) (b) ˆx B = +r ˆx (0) r ˆx (0) + ˆx (0) (c) ˆp B = r ˆp (0) +r ˆp (0) + ˆp (0) (d) ˆx C = +r ˆx (0) r ˆx (0) ˆx (0) () ˆp C = r ˆp (0) +r ˆp (0) ˆp (0) (f) and ˆx in = (0) sˆx 5 (a) ˆp in = s ˆp (0) 4 + +s ˆp (0) 5 (b) ˆx rf = (0) +sˆx 4 (0) sˆx 5 (c) ˆp rf = s ˆp (0) 4 +s ˆp (0) 5 (d) whr th suprscript (0) dnots th initial vacuum mods, r and s ar th squzing paramtrs, and oprators ˆx and ˆp ar th lctromagntic fild quadratur amplituds, i.. th ral part and imaginary part of th mod s annihilation oprator. Using th sufficint insparability critria [8, 9] for CV ntanglmnt in Eq. (), on obtains [0] A,B = [ (ˆx A ˆx B )] + [ (ˆp A + ˆp B )] ( ) ( r = + + ) r + 4 < (a)

558 Dao-Hua Wu 4 g x Output g p Output Alic rf x u BS in BS p v g x g p PM Bob AM B BS C AM PM Clair A BS Fig. Schmatic stup of tripartit ntanglmnt swapping of continuous variabls. Mods A, B and C ar initially prpard in W-typ local ntangld stat. Bfor th dtctions, mod in is ntangld with mod rf. Th brokn lin is vacuum mod. BS, AM and PM dnot bam splittr, amplitud modulator and phas modulator, rspctivly. A,C = [ (ˆx A ˆx C )] + [ (ˆp A + ˆp C )] ( ) ( r = + + ) r + 4 < (b) This mans mods A and B and mods A and C ar bipartitly ntangld, whil mods A, B and C ar tripartitly ntangld. Th inquality can b minimizd whn r = +, i.., 7.7 db squzing. By using ths bipartit ntangld mods, w can prform th ntanglmnt swapping among involvd rgions. Hr, w introduc Alic, Bob, Clair for illustrating th whol protocol of tripartit ntanglmnt swapping of continuous variabls. Initially, Alic, Bob and Clair shar th W-typ ntangld stat of mods A, B and C. Alic posssss th mods in and mod rf. In ordr to accomplish th ntanglmnt swapping, Alic should tlport hr mod in to Bob and C Clair simultanously. Thus sh first

5 Tripartit ntanglmnt swapping of continuous variabls 559 combins hr mod in with mod A at a 50/50 BS and obtains th quadraturs [5] ˆx u = ˆx in ˆx A (4a) ˆp v = ˆp in + ˆp A (4b) Manwhil, Bob s mod B and Clair s mod C bcom ˆx B = ˆx in (ˆx A ˆx B ) ˆx u ˆp B = ˆp in + (ˆp A + ˆp B ) ˆp v ˆx C = ˆx in (ˆx A ˆx C ) ˆx u ˆp C = ˆp in + (ˆp A + ˆp C ) ˆp v Thn Alic masurs th x u, p v for oprators ˆx u and ˆp v by homodyn dtctors and informs Bob and Clair of th masurmnt rsults by classical channls. Aftr rciving th classical information from Alic, Bob and Clair can displac his and hr mods as ˆx B ˆx out = ˆx B + g x x u pˆ B ˆp out = ˆp B + g p p v ˆx C ˆx out = ˆx C + g x x u pˆ C ˆp out = ˆp C + g p p v whr g x and g p ar information gains of classical channls from Alic to Bob and Clair, which w assum to b unity, i.., g x = g p =. Thrfor, Bob s and Clair s output mod ar [5, 0] ˆx out = (0) sˆx + + ˆx (0) 5 + (5a) (5b) (5c) (5d) (6a) (6b) (6c) (6d) (7a) ˆp out = s ˆp (0) 4 + +s ˆp (0) r ˆp (0) + ˆp (0) ˆx out = (0) sˆx + ˆx (0) 5 + + 5 + (7b) (7c)

560 Dao-Hua Wu 6 ˆp out = s ˆp (0) 4 + +s ˆp (0) 5 + + r ˆp (0) ˆp (0) Thus, via W-typ ntangld stat, th mod in is tlportd to two outputs at Bob s and Clair s locations and th two output mods ar both ntangld with Alic s mod rf, and mod rf, mod out and mod out shar a tripartit ntanglmnt which could b dscribd as: ˆx rf = (0) +sˆx 4 (0) sˆx 5 (8a) (7d) ˆp rf = s ˆp (0) 4 +s ˆp (0) 5 (8b) ˆx out = (0) sˆx + + ˆx (0) ˆp out = s ˆp (0) 4 + +s ˆp (0) r ˆp (0) + ˆp (0) ˆx out = (0) sˆx + ˆx (0) ˆp out = s ˆp (0) 4 + +s ˆp (0) 5 + 5 + + 5 + 5 + + r ˆp (0) ˆp (0) Nxt, w prov th nwly st ntanglmnt in dtail. If two mods A and B ar ntangld, thy should satisfy [8 0] (8c) (8d) (8) (8f) A,B = [ (ˆx A ˆx B )] + [ (ˆp A + ˆp B )] < (9) According to Eq. (8), w can obtain ˆx rf ˆx out = ˆx (0) + + sˆx (0) 5 (0a)

7 Tripartit ntanglmnt swapping of continuous variabls 56 rf,out 0.8. 0.6 0.5 0.75 s.5 0.8 r 0.6.5 0.4 Fig. rf,out vrsus th squzing paramtrs r and s, which ar of th tripartit ntanglmnt and th EPR stats, rspctivly. + ˆp rf + ˆp out = + r ˆp (0) + ˆp (0) + s ˆp (0) 4 (0b) Assuming all mods hav zro man valus and givn th fact that ( ˆx (0) i ) = ( ˆp (0) i ) = 4 (i =,,,4,5), w obtain rf,out = [ (ˆx A ˆx out )] + [ (ˆp A + ˆp out )] ( ) ( r = + + ) r + s + 4 In Fig., w plot rf,out vrsus th squzing paramtrs r and s. It shows that rf,out gos down with both incrasd squzing of th tripartit ntanglmnt and th EPR stat. W now illustrat it by considring th following two cass for Eq. (): () In th cas of infinit squzing for th EPR stat (s ), rf,out(out) = ( ) ( r + + () ) r + 4 <, rf,out(out) can b minimizd whn r = + (-7.7 db squzing), which is shown in Fig.. Namly, th mod rf and out(out) ar ntangld, and th ntanglmnt swapping procssing is compltd. () If th tripartit ntanglmnt rsourc is th bst ntangld stat (corrsponds to 7.7 db squzing), rf,out(out) = + s. A,out(out) dcrass with th squzing paramtr of th input EPR stat, which is shown in Fig.4. From Fig.4, w find that only in th cas of s < or th squzing paramtr s > ln4, rf,out(out) <. So th ntanglmnt swapping can b ralizd for squzd

56 Dao-Hua Wu 8 rf,out 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0. 0.4 0.6 0.8..4 r Fig. Th rlationship of rf,out with th quantum channl paramtr r. rf,out. 0.8 0.5.5.5 s 0.6 Fig. 4 Th rlationship of rf,out with th squzing paramtr s. paramtr s > ln4. That is to say, w could accomplish tripartit ntanglmnt swapping via W- typ CV ntangld stat. In this cours, mod out and mod out ar tlclond from mod in. In a sns, this procssing could b viwd as tlclon of th CV EPR stat (th ntangld stat mod rf and mod in tlclon to mod rf, mod out and mod rf, mod out).. CONCLUSION In summary, w hav proposd a schm for tripartit ntanglmnt swapping of continuous variabls. If th input EPR stat is th bst ntanglmnt, th swapping procssing will b ralizd. Whil th EPR stat is not prfct, swapping procssing will b ralizd in th condition of s>ln4. Whn th W-typ ntanglmnt is also not prfct, th swapping will b ralizd with conditional paramtrs as plottd in Fig.. Acknowldgmnts. This work is supportd by National Natural Scinc Foundation of China (NSFC) undr Grant Nos: 606780 and 070400, th Spcializd Rsarch Fund for th Doctoral Program of Highr Education undr Grant No. 006057008, Anhui Provincial Natural Scinc Foundation undr Grant No. 0704060, Natural Scinc Foundation of Anhui Collg of Chins Traditional Mdicin undr Grant No ZRKX04. th Talnt Foundation of Anhui Univrsity, and Anhui Ky Laboratory of Information Matrials and Dvics (Anhui Univrsity).

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