Teleportation of qubit states through dissipative channels: Conditions for surpassing the no-cloning limit

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1 Teleporttion of quit sttes through dissiptive chnnels: Conditions for surpssing the no-cloning limit Şhin Ky Özdemir,,2,3 Krol Brtkiewicz, 4 Yu-xi Liu, 5,6 nd Adm Mirnowicz,3,4 SORST Reserch Tem for Intercting Crrier Electronics, 4--8 Honmchi, Kwguchi, Sitm 33-2, Jpn 2 CREST Reserch Tem for Photonic Quntum Informtion, 4--8 Honmchi, Kwguchi, Sitm 33-2, Jpn 3 Grdute School of Engineering Science, Osk University, Toyonk, Osk , Jpn 4 Institute of Physics, Adm Mickiewicz University, 6-64 Poznń, Polnd 5 Frontier Reserch System, Institute of Physicl nd Chemicl Reserch (RIKEN), Wko-shi 35-98, Jpn 6 CREST, Jpn Science nd Technology Agency (JST), Kwguchi, Sitm 332-2, Jpn Received 8 April 27; pulished 9 Octoer 27 We investigte quntum teleporttion through dissiptive chnnels nd clculte teleporttion fidelity s function of dmping rtes. It is found tht the verge fidelity of teleporttion nd the rnge of sttes to e teleported depend on the type nd rte of the dmping in the chnnel. Using the fully entngled frction, we derive two ounds on the dmping rtes of the chnnels: one is to et the clssicl limit nd the second is to gurntee the nonexistence of ny other copy with etter fidelity. The effect of the initilly distriuted mximlly entngled stte on the process is presented; the concurrence nd the fully entngled frction of the shred sttes re discussed. We intend to show tht prior informtion on the dissiptive chnnel nd the rnge of quit sttes to e teleported is helpful for the evlution of the success of teleporttion, where success is defined s surpssing the fidelity limit imposed y the fidelity of the -to-2 optiml cloning mchine for the specific rnge of quits. DOI:.3/PhysRevA PACS numer s : 3.67.Hk, 3.65.Ud, 3.65.T I. INTRODUCTION The quntum stte of system cn e trnsmitted from loction to distnt one using only clssicl informtion provided tht quntum chnnel exists etween the sender nd the receiver. Shring entngled sttes etween the two prties opens the necessry quntum chnnel. Reserch in quntum stte trnsfer 2, especilly the quntum teleporttion 3, hs emerged s one of the mjor reserch res of theoreticl nd experimentl quntum mechnics. Vrious discussions nd criteri hve ppered out the evlution of the stte trnsfers under idel nd imperfect conditions 4. In perfect scheme, the shred entngled stte is mximlly entngled stte MES enling perfect quntum stte trnsfer. However, in prctice, entnglement is susceptile to locl interctions with the environment, which cn result in loss of coherence. In this pper, we study the teleporttion of quits through dmping chnnels. We consider quntum stte trnsfer s n opertion, such s cloning nd teleporttion, which ets the clssicl limits on mesurement nd trnsmission. The resemlnce of two quntum sttes nd the properties of quntum stte trnsfer teleporttion nd cloning re quntified y the fidelity F in = in ˆ out in, which mesures the overlp of the sttes in to e teleported cloned nd the output stte with the density opertor ˆ out. A quit stte to e teleported in = + with = cn e represented on Bloch sphere s in = cos /2 e i + sin /2, where nd re the polr nd zimuthl ngles, respectively. Since this stte is generlly unknown, it is more pproprite to clculte the verge of the fidelity F in over ll possile sttes in to quntify the process. This verge fidelity F= in ˆ out in 5 cn e clculted s F = 4 2 d d F, sin, where 4 is the solid ngle. The reltion etween the teleporttion fidelity nd the degree of entnglement shred y the prties hs een studied y mny reserchers e.g., in 3 nd others cited in nd it hs een shown tht i less entngled quntum chnnel reduces the fidelity nd the rnge of sttes, which cn e teleported 3, ii for the stndrd teleporttion scheme, the mximum ttinle verge fidelity is simply relted to the fully entngled frction of iprtite entngled stte 6, nd iii some mixed sttes, which do not violte the Bell inequlities, cn still e used for teleporttion 5. On the other hnd, only few studies re directed to the reltion etween the fidelity of teleporttion nd the type nd strength of the dmping in the quntum chnnel. Tht is the topic of the present study. According to the definition of teleporttion s stted y Bennett et l. 3, in the process of quntum teleporttion, one cn construct n exct replic of the originl unknown quntum stte with the cost of destroying the originl stte. Therefore to cll quntum stte trnsfer opertion s quntum teleporttion, the process should not only generte output sttes with etter qulities thn wht cn e done clssiclly ut lso oey the no-cloning theorem. Defining teleporttion opertor Û tel, which cn e implemented in stndrd quntum circuit see, e.g., 2 with n input stte ˆ in = ˆ nd shred entngled stte ˆ ent = ˆ,c, the output stte ˆ out is written s /27/76 4 / The Americn Physicl Society

2 ÖZDEMIR et l. ˆ out =Tr in, Û tel ˆ in ˆ ent Û tel. 3 If the teleporttion process is idel then ˆ out = ˆ in implying fidelity vlue of unity. However, in prcticl pplictions, this is not the cse due to the presence of noise which my e due to i noisy sources of ˆ in nd ˆ ent, ii noisy entnglement distriution chnnel, iii noisy mesurements nd unitry opertors, nd iv n evesdropper who ttempts to clone ˆ in. Since, in generl one cnnot e sure of which of the ove is the reson, ll the noise in the process should e ttriuted to n evesdropper in order to ssess the security whenever quntum teleporttion is to e used s mens of secure communiction. This ssessment to quntify the process should e done ccording to the definition of the teleporttion given ove. Tht is, one should check to see whether F in Eq. 2 stisfies the conditions of i eting the clssicl limit, nd ii oeying the no-cloning. The linerity of quntum mechnics forids the exct cloning of n unknown quntum stte, however, if one llows discrepncies etween the originl quntum stte nd its copy, then it is possile to devise scheme tht cn produce clones nd copies of given unknown stte with the highest resemlnce to the originl one 3 6 for reviews see 2. This is known s the optiml cloning, where with the incresing numer of clones copies, the resemlnce to the originl stte decreses. It hs een shown tht for stte-independent universl cloning mchine the reltion etween the optimum fidelity F of ech copy nd the numer M of copies is given y F= 2M + / 3M. In clssicl situtions, one cn mke n infinite numer of copies M of given stte resulting in fidelity F=, which is the est one cn do with clssicl opertions. On the other hnd, when M =2, the universl cloning mchine hs n optimum fidelity of F= 3 6. Comining the ove informtion on teleporttion nd cloning, one cn infer tht teleporttion process ets the clssicl limit if F, nd oeys the no-cloning requirement if F 3 6. If this is ssured, then there is not ny other copy of the output stte with etter fidelity, therefore the teleporttion process is secure. It is noteworthy tht this is true if nd only if the quntum stte ˆ in is completely unknown to the evesdropper. In some cses, ˆ in my e prepred in stte tht is selected from known ensemle of sttes. If the evesdropper hs this priori knowledge out ˆ in, stte dependent cloner which cn perform etter thn the optiml universl one cn e constructed. Thus the fidelity constrint imposed on teleporttion due to the no-cloning condition will ecome much stricter. Qulity of the shred entngled stte is good criterion to quntify the reliility of the quntum teleporttion. Bennett et l. 7 nd, in generl cse, Horodecki et l. 6,8 for review see hve shown tht for shred iprtite entngled stte ˆ ent to e useful for quntum teleporttion, its fully entngled frction f ent, defined y 9 f ent = mx ˆ ent, 4 must e greter thn /2. In Eq. 4, mximum is tken over ll MES. It hs lso een shown tht the mximum SCENARIO SCENARIO 2 Alice Alice Clssicl Communiction Quntum Chnnel Clire Quntum Chnnel Bo Clssicl Communiction Bo FIG.. Teleporttion scenrio, where oth quits of ˆ ent re ffected y the chnnel, nd scenrio 2, where only one of the quits is ffected. A quntum chnnel is formed y the shred entngled stte. chievle teleporttion fidelity F is relted to f ent y 6 F = 2f ent Sttes with f ent /2 cnnot e used directly for teleporttion unless they re enhnced through filtering to stisfy f ent /2. Choosing the oundry vlue of f ent =/2 gives teleporttion fidelity of F = which is the oundry etween clssicl nd quntum stte trnsfer. Tht is if f ent /2 nd hence F, then the sme opertion cn e done clssiclly. According to this definition, if in process F is chieved then it cn e clled quntum teleporttion. On the other hnd, the discussion on cloning in the former prgrphs implies tht one cn mke n infinite numer of copies of quit with fidelity of which violtes the originl definition of quntum teleporttion given y Bennett et l. 3. Here gin rises the question of chievle teleporttion fidelity, which gurntees etter thn clssicl teleporttion nd surpsses the no-cloning limit. The prolem studied in this pper cn e formulted s follows: Alice nd Bo re fr from ech other, nd they shre n entngled quntum stte ˆ en. The entngled stte is prepred either y third prty, sy Clire, nd delivered to Alice nd Bo scenrio : two-quit ffected scenrio or prepred y Alice nd one of the quits is sent to Bo nd the other is kept with her scenrio 2: one-quit ffected scenrio s shown in Fig.. The only mnipultions tht Alice nd Bo re llowed to do re locl quntum opertions nd clssicl communictions. Now suppose tht Alice wnts to trnsfer the quntum stte represented y the quit stte in to Bo nd the entngled stte is distriuted through dissipting chnnel. Then, how does the dissiption of the chnnel ffect the entnglement properties of the distriuted entngled stte, nd hence wht is its effect on the trnsferred quntum stte? Wht is the llowle mount of dissiption tht does not ffect the security of quntum stte trnsfer?

3 TELEPORTATION OF QUBIT STATES THROUGH In this pper, we derive the dmping rtes of quntum chnnels t which quntum stte trnsfer tht overcomes the clssicl counterprt cn e relized. In the sme wy, conditions, which gurntee secure quntum teleporttion, re lso derived. We study the effect of noise on the rnge of quits tht cn e teleported ccurtely. The noisy chnnels, including mplitude dmping chnnel, phse dmping chnnel, nd depolrizing chnnel, nd the effects of these noisy chnnels on the distriuted entnglement nd teleporttion process re studied in Secs. II nd III. Finlly, Sec. IV includes rief summry nd conclusion of this study. II. EFFECT OF DAMPING CHANNELS ON ENTANGLEMENT AND TELEPORTATION We consider the two scenrios shown in Fig.. In the first scenrio, the quits of the initil MES re distriuted through two chnnels, which my or my not hve the sme dmping properties. On the other hnd, in the second scenrio, only one of the quits of the initil MES is distriuted through the dmping chnnel. In the following, we give nlyticl expressions, which show how given stte is ffected when trnsmitted through noisy chnnels cusing mplitude dmping, phse dmping, or depolriztion. Initil MESs tht re considered in this study re the Bell sttes =, 2 =. 6 2 We derive the ounds for the dmping rte of the chnnel to stisfy the quntum teleporttion conditions discussed in the previous section. In the following, we ssume tht there is no priori informtion on ˆ in, therefore the optiml universl cloning mchine which imposes F is considered. A. Amplitude dmping chnnel The evolution of environment denoted y suscript e nd system suscript or, equivlently, with the sttes nd is defined y the following trnsformtion in the presence of the mplitude dmping chnnel ADC 2 : e e, e q e + p e, where q p. This trnsformtion implies tht system with n excited stte mkes trnsition to the ground stte with proility p nd emits photon to the environment which mkes trnsition to the excited stte. When the system is initilly in the ground stte, there is no trnsition.. Input Bell sttes If oth of the quits in the Bell sttes re trnsmitted through n ADC scenrio, then using Eq. 7 we cn write the stte t the output of the chnnel s 7 e e 2 = 2 q q e e 2 + p e e 2 p e e 2, 8 where we ssumed tht chnnels hve different dmping rtes denoted y p, p nd, for simplicity, we denote q p, q p. If we ssume p = p = p nd the environment is not monitored unwtched chnnel, the shred stte etween Alice nd Bo t the outputs of the chnnels cn e found y trcing out the environment vriles resulting in ˆ =q + p. It is seen tht the MES survives with proility of q. On the other hnd, if the environment is monitored, Alice nd Bo proceed with the protocol if no photon is detected in the environment implying they hve MES, nd they do nothing when photon is detected in the environment. If only one of the quits sy, tht for Bo is sent through the chnnel scenrio 2, the dmping in the chnnel ffects only tht prt. If the chnnel is wtched nd no photon is detected in the environment, the stte tht is shred etween Alice nd Bo ecomes = q +. 2 p For n unwtched chnnel, the shred stte is given s 9 ˆ = 2 2 p + p. It is clerly seen tht if only one quit of the initil MES is sent through the ADC, the shred stte etween the prties is no longer MES. Using Eq. 8, one cn find tht for scenrio, the fully entngled frction is 8 f ent, = 4 q + q 2 if p 2 q + +2p 3p 2 is stisfied, otherwise it ecomes f ent,2 = 4 p + p. 2 If we ssume tht oth chnnels hve the sme dmping properties, tht is p = p = p, the fully entngled frction is found s = f ent q if p ; p/2 if p. For scenrio 2, where p =, f ent 3 ecomes f ent = 4 + q 2 for ll p. Imposing the condition f ent /2, which ssures tht quntum stte opertion ets the clssicl limit, gives the reltion q + q 2 for scenrio 8. Tking p s vrile, it cn e found tht p 2 2 nd p p q must e stisfied simultneously 8. Similrly for the cse p = p = p, one cn find tht the clssicl limit cn e eten only when p /2. For scenrio 2, it cn esily e shown tht p 2 2 must e stisfied. If the chnnels

4 ÖZDEMIR et l. re wtched ut no photon is detected, then f ent /2 cn lwys e stisfied provided tht p p nd p for scenrios nd 2, respectively. If the conditions given in the ove prgrph for unwtched chnnels re stisfied, one cn only e sure tht the opertion is quntum one with fidelity F, however, one cnnot e sure out the security of the process, which requires F ccording to the 2 cloning condition. Then solving Eq. 5 for f ent to stisfy F, we find tht f ent must e stisfied for the fully entngled frction. Imposing this condition on the shred entngled stte etween the two prties results in much tighter condition on the chnnel dmping rtes, which cn e summrized s follows: f ent if p 4, scenrio 3 for p = p = p, 4 p p 3+2 3q, scenrio p or vice vers, for p p, p 2 3 3, 2. Input Bell sttes scenrio 2. 5 In the scenrio, when oth quits of re sent through the dmping chnnels, the shred stte etween Alice nd Bo for the wtched nd unwtched chnnels re found s nd = q q +q q ˆ = 2 +q q + q p 6 + p q + p p, 7 respectively, where we hve considered tht no photon is detected in the environment for the wtched chnnel cse. From these equtions, it is seen tht MES survives with nonzero proility iff p = p =. When only one of the quits sy gin, Bo s quit of the MES is propgted through the ADC, the shred stte etween Alice nd Bo is not mximlly entngled unless p = for oth wtched nd unwtched chnnels s cn e seen in the following expressions given, respectively, for wtched nd unwtched chnnels: nd = 2 p q 8 ˆ = 2 2 p + p. 9 Then the fully entngled frction of the shred stte, when the chnnel is not wtched, is found s which reduces to f ent = 4 p p + + q q 2, 2 4 f ent = 2 p2 2p +2, scenrio, for p = p p, + q 2, scenrio 2. 2 It cn esily e found from Eq. 2 tht the condition f ent /2 is stisfied for ny p in the rnge p when oth chnnels hve the sme dmping rtes in scenrio ; nd for p 2 2 in scenrio 2. When the chnnels hve different dmping rtes, we cn write using Eq. 2 tht p p + + q q 2 2 must e stisfied to et the clssicl limit. The nlyticl solution for this is very lengthy to give here. Insted, to give n ide on the reltion etween p nd p to stisfy the condition f ent /2, we give some numericl vlues: when p =/2, p 7/8 nd when p =/4, p must stisfy p /2 to et the clssicl limit. As it hs een pointed out y Bndyopdhyy 8, scenrio cn e mde to hve higherf ent thn scenrio 2 such tht f ent /2 is stisfied. This, in turn, implies tht for the stte, one cn let one of the quits undergo controlled dissiption if the informtion on the dissiption of the other quit in the other chnnel is ville. Looking t the condition f ent 3/4 for quntum teleporttion to surpss the no-cloning limit, we find the following constrints on the dmping rtes of the ADC: 3 4 if p f ent 2 2, scenrio for p = p p, p 2 3 3, scenrio p g p or vice vers, for p p, p 2 3 3, scenrio 2, 22 where g x = 2x 2 3+x 3+2x +2 x 2x 2 6x+3. Contrry to the ove cse, controlled dissiption cnnot increse f ent ove 3/4. B. Phse dmping chnnel A phse dmping chnnel PDC ffects n input stte with the following trnsformtions 2 : e q e + p e,

5 TELEPORTATION OF QUBIT STATES THROUGH e q e + p 2 e. 23 In this chnnel, the energy of the informtion crrier is conserved no losses to environment, however, the stte of the crrier is decohered. Bell sttes evolve into. Input Bell sttes ˆ = 2 q q + + q q, 24 when oth quits re sent through the unwtched chnnel. For the limiting cse, p = p =, off-digonl components of the density mtrix vnish resulting in mixed stte. For wtched chnnel with no photon detected in the environment, there is proility of q q tht the stte oserved is. On the other hnd, when only one quit is sent scenrio 2, the proility tht the MES survives ecomes q when the chnnel is wtched. When the chnnel is not wtched, then the output stte, which is mixed nd not MES, cn e found from Eq. 24 y sustituting p =. When the f ent of the output stte t the end of the unwtched chnnels re clculted it is seen tht scenrio f ent = 2 +qq, for p p, p 2 2p +2, scenrio, 25 for p = p p, 2 p, scenrio 2. Then we find tht f ent is lwys greter thn /2 provided tht p p nd p re stisfied for oth scenrios. Moreover, we find scenrio cnnot e mde to hve f ent lrger thn scenrio 2. The no-cloning limit imposes the following conditions on the llowle PDC rte: 2 if p 2, scenrio f ent 3 for p = p p, 4 p 2p / 2q, scenrio p /2 or vice vers, for p p, p /2, scenrio Input Bell sttes When the input is, then the stte t the output of the chnnels ecomes ˆ = 2 q q + + q q 27 for n unwtched chnnel for scenrio. A comprison of this output stte with Eq. 24 revels tht the sme discussions nd the conditions on the chnnel dmping properties re vlid here, too. C. Depolrizing chnnel When quit is sent through depolrizing chnnel DC with proility q p it is intct, while with proility p n error it flip error, phse flip error, or oth occurs. The trnsformtion tht chrcterizes this chnnel is 2 e 3p 4 e + p 4 e + i 2 e + 3 e, e 3p 4 e + p 4 e i 2 e 3 e. 28 In the DC, ny given stte evolves to n ensemle of the four sttes, ˆ x, ˆ y, nd ˆ z where k is the Puli opertor. p = corresponds to complete depolriztion where ech of the four sttes occur with equl proilities. If the input stte to the chnnel is or oth quits re sent through the chnnel then with proility of 4 3p 4 3p /6, this stte is conserved t the output of the chnnels if the chnnel is wtched nd no photon is detected. For n unwtched chnnel with n input,2 =, for indices nd 2, respectively, the output stte cn e written s = q q 4,2,2 + 2, 2, + + 2, + 2, ˆ + +3q q 4,2,2, 29 which ecomes mixture of Bell sttes with equl proility /4 when p=. The effect of this chnnel on the input stte when only one of the quits is sent through cn e found simply y sustituting p =. Imposing the criteri f ent /2 nd f ent 3/4 on the stte t the output of the chnnel for oth scenrios, we find the following rnges for dmping rte of the DC: if p 3/3, scenrio for p = p = p, f ent 2 p 2 3p ; p, scenrio 3q for p p, p, scenrio 2 3 nd

6 ÖZDEMIR et l. if p 6/3, scenrio for p = p p, 3 4 p 3p ; p /3, scenrio 3q for p p, p /3, scenrio 2. 3 f ent f ent nd Concurrence (i) Dmping prmeter p (ii) Dmping prmeter p D. Concurrence nd fully entngled frction The fully entngled frction f ent, given y Eq. 4, cn e regrded s mesure of entnglement when the quntum chnnel is in pure stte nd it is relted to the Wootters concurrence C 2 through the reltion f ent = +C /2. However, when the quntum chnnel is in mixed stte, f ent cn no longer e used s mesure of entnglement. This is due to the fct tht entnglement cnnot e incresed y locl quntum opertions nd clssicl communictions, ut the fully entngled frction f ent cn e incresed s shown y Bndyopdhyy 8 nd Bdziąg et l. 8. In the following, we present the dependence of concurrence on the properties of the unwtched dmping chnnels in oth scenrios introduced previously nd discuss the reltion etween f ent nd concurrence so tht we cn ssure quntum stte trnsfer nd secure quntum teleporttion. As defined y Wootters 2, concurrence of mixed stte ϱ is given y C=mx, 2 3 4, where i re the squre roots of the eigenvlues, in decresing order, of the non-hermitin mtrix ϱϱ with ϱ = ˆ y ˆ yϱ * ˆ y ˆ y, where Puli mtrices ct on Alice nd Bo quits, respectively, nd the sterisk * stnds for complex conjugtion. For the ADC, we oserve tht, in the first scenrio, the reltions etween the concurrence nd the dmping prmeter re different for the initil entngled sttes nd. The concurrence of the shred stte t the output of the chnnels for the input stte is given s C = q q ; nd the concurrence for ecomes C = p p C. When oth chnnels hve the sme dmping rte p = p p. It is seen tht while C decreses linerly with p, C decreses with p 2. On the other hnd, for scenrio 2, oth initil sttes show the sme tendency, which is given s C =C = q. In the cses of the PDC nd DC, C C =C. For the PDC, concurrence is found s C=q q for the first scenrio. The expression for the second scenrio cn e found y tking p = nd p = p. Although the expressions found for concurrence for the ADC nd PDC re vlid for ll vlues of p nd p in the rnge of,, the expressions for concurrence in the cse of the DC re vlid only for limited rnge of dmping rtes. For exmple, for the second scenrio, concurrence is found s C= 3p/2 provided tht p, otherwise it is zero implying seprle stte. For the first scenrio when oth DCs hve the sme dmping rte, concurrence is given s C=+3p p 2 /2 when p 3/3, otherwise C=. When the dmping rtes re different, we find tht C= 3q q /2 provided tht p nd p / 3q re stisfied simultneously, otherwise C=. f ent nd Concurrence (iii) It is seen from Fig. 2 tht f ent is lwys /2 for C=; nd even very smll mount of entnglement shifts the process from clssicl to quntum regime. III. RANGE OF QUBITS FOR ACCURATE TELEPORTATION In this section, we nlyze the effect of the noise in the system on the rnge of quits tht cn e teleported with desired fidelity vlue. In order to show how the priori informtion on the ensemle from which ˆ in is prepred ffects the fidelity criterion on secure teleporttion, we will consider n optiml one-to-two phse-covrint cloning mchine 22,23 in comprison with the universl cloning mchine. We will ssume tht the sttes to e teleported re chosen from the whole set of quit sttes with fixed nd specified polr ngle in the Bloch sphere. An evesdropper, who knows, cn use the optiml one-totwo for which the cloning fidelity is given y 23,24 F = 2 Dmping prmeter p + sin2 + cos sin2 = cos + 2 cos 2, 32 where =[ 2 / ], i.e., = for 2 nd = for 2. Note tht fidelity F for ny is greter thn the fidelity of the optiml universl cloning mchine 3, given y F=. For the quit sttes on the equtor of the Bloch sphere = /2, the optiml prepres clones with fidelity F /2 = On the other hnd, when the sttes re close to the poles, tht is in the neighorhood of or in the Bloch sphere, i.e., for fixed ngle = or =+ with, Eq. 32 simplifies to (iv) Dmping prmeter p FIG. 2. Comprison of concurrence solid curves nd fully entngled frction dshed curves for the ADC, when the initil entngled stte is i nd ii, s well s the PDC iii nd the DC iv for scenrios curves nd 2 curves. Note tht for the PDC nd DC results re independent of the initil entngled stte

7 TELEPORTATION OF QUBIT STATES THROUGH F = cos 2 cos 2 = O In teleporttion process, mesurement of Alice results in four possile outcomes m i where i=,, 2, nd 3 with m =, m =, m 2 =, nd m 3 =. Then the stte t Bo s side conditioned on Alice s mesurement cn e written s ˆ m i. In the stndrd teleporttion protocol with shred MES, upon receiving the clssicl informtion i, Bo cn mke the pproprite unitry opertions on his quit ˆ m i to otin the teleported stte ˆ out = ˆ in.we discuss how the mesurement result ffects this process in the presence of noise. Since the entnglement distriution chnnel is noisy, the stte t the output of the teleporttion process given in Eq. s 3 cn e rewritten s ˆ out =Tr in, Û tel ˆ in ˆ ent where s ˆ ent Û tel is the noisy entngled stte. We cn sy tht the fidelity is function of, nd the noise introduced into the system, nd we cn represent it s F, F in. We oserve tht F, is independent of, s denoted y F F,. A. Amplitude dmping chnnel. Input Bell sttes For the ADC, in scenrio, let us ssume tht p = p = p nd Alice mde mesurement, otined the outcome m, nd then sent the clssicl informtion k= to Bo. The output density opertor conditioned on m ecomes ˆ m =N q ˆ in +2p sin 2 /2 with N eing the renormliztion constnt defined s N = p cos nd ˆ in is the density opertor of the stte to e teleported. Bo cnnot rotte this ˆ m to the desired stte without the prior knowledge of nd. Since in is supposed to e unknown, stndrd teleporttion protocol fils to reproduce the desired stte t Bo s side. This conclusion is vlid for ll m i. Interestingly, the output stte t Bo s side cn e grouped into two s = ˆ m, ˆ m 2 nd = ˆ m, ˆ m 3. Although the output sttes in one of these groups cn e rotted into ech other y using Z gte or first X then Z gte, sttes elonging to different groups cnnot e rotted to ech other. This prolem is cused y the ADC, which reduces the degree of entnglement nd introduces the dditionl terms 2p cos 2 /2 nd 2p sin 2 /2, respectively, for nd. We oserved tht for teleporttion in the presence of this ADC, if Alice s mesurement yields m, Bo does not need to do nything. For other mesurement results m, m 2, nd m 3, Bo should pply ˆ x, ˆ y, nd ˆ z, respectively. In this wy, he rottes his quit into the output stte ˆ out m k =N k q ˆ in + p + k cos k k with N k eing the renormliztion constnt defined s N k =+ k p cos nd stnds for ddition modulo 2. Then the sttedependent fidelity ecomes F mk = p + k cos k p cos, 34 where k=,, 2, nd 3. When p, the limiting vlues re clculted s F m,3 =cos 2 /2 nd F m,2 =sin 2 /2. For p / nd p /5, ll sttes cn e teleported, respectively, with F nd F, independent of Alice s mesurement result. For the equtoril quits = /2, we find tht s fr s p 2/2, teleporttion fidelity will surpss tht of the regrdless of the mesurement outcome. On the other hnd, if the quits re chosen t the neighorhood of, then for even k the chnnel dmping rte should e ounded s p.62. Although stte-dependent teleporttion fidelity is minly determined y Alice s mesurement result, the verge fidelity clculted using Eq. 2 is the sme for ll mesurement results nd given s F = 4p 2 2p + q 2 q ln 35 +p, which tkes the minimum nd mximum vlues of /2 nd for p= nd, respectively. For the second scenrio, the output density opertor elements jl out m k re found in terms of the input density jl opertor elements in s follows: out m k =q in + k p/2, out m k = q in, out m k = q in, nd out m k =q in + + k p/2. When Alice mesures m k nd pplies the pproprite unitry trnsformtion to get the highest fidelity for the process, i.e., when Bo receives the informtion tht k=3 or k= for + nd, respectively, he cn use Z gte to rotte the stte on his side to otin the ove stte. Then stte-dependent fidelity cn e found s F mk = 2 p + k cos x, 36 where x= q q sin2. In the limit of p, we get the sme functions s those otined from Eq. 34. It is clerly seen from the ove results tht the rnge of quits tht cn e teleported correctly depends not only on the strength of the ADC ut lso on the mesurement result of Alice. Results imply tht some of the sttes cn e teleported with much etter fidelity thn others depending on m k. This enles Alice nd Bo, in communiction protocol, to decide to choose their quits rndomly from rnge of sttes with higher fidelity when certin mesurement result, sy m,m 2, is otined. As seen in Figs. 3 nd 4, some sttes give etter fidelity thn others depending on m k. In the figure we hve shded regions where ll the sttes cn e teleported with F regrdless of Alice s outcome. Note tht the sttes with = /2 cn tolerte much higher dmping rtes thn the ones locted round the poles of the Bloch sphere. It is lso seen tht in Scenrio it is dvntgeous to rotte the initil entngled stte into ecuse it is more immune to the ADC nd therefore provides lrger prmeter spce for F teleporttion. Let us ssume tht quits chosen from rnge defined y hve higher fidelity when Alice mesures m,m 2,onthe other hnd quits chosen from hve higher fidelity when Alice mesures m,m 3. Then in teleporttion protocol, Alice first mixes her stte chosen from with her prt of the entngled stte nd mkes mesurement, when she otins m,m 2, she sends the other quit of entngled stte to Bo

8 ÖZDEMIR et l. Dmping prmeter p Dmping prmeter p π/4 π/2 3π/4 π Stte ngle δ π/4 π/2 3π/4 π Stte ngle δ Dmping prmeter p π/4 π/2 3π/4 π together with the clssicl informtion, then Bo pplies unitry trnsformtion to get the desired stte. When she gets m,m 3, either she sends nothing or dummy stte. In this wy, they cn increse the fidelity of the process. If they decide to ort the protocol whenever Alice mesures m,m 3 then the efficiency of the process is low. If Alice nd Bo decide to keep ll mesurement results then the fidelity of the process cn e written s 3 F = p mk F mk = q x 2, k= Stte ngle δ FIG. 3. Color online Optiml stte-dependent fidelity in the presence of the ADC when the initil MES is top nd ottom nd oth quits re ffected y dmping. Contours correspond to F=, F= nd the optiml fidelities when Alice s mesurement result is or, solid curves, nd when Alice s mesurement result is or, dotted curves. 37 FIG. 4. Color online Sme s in Fig. 3 top for ut for the cse when only one of the quits is ffected y the ADC. For, the mening of curves is reversed. where x is defined s in Eq. 36, F mk is the fidelity of the output stte to the teleported stte when Alice otins m k, nd p mk is the proility of otining this result. Moreover, if they do tht for ny,, they end up with F= + 2 q p /6. From Eq. 36, it cn e seen tht for fixed p of the chnnel, if the stte to e teleported is chosen such tht /2, then the set m,m 3 gives higher teleporttion fidelity for tht stte; otherwise, the set m,m 2 yields higher fidelity. Let us ssume tht Alice rndomly chooses stte to e teleported from the upper hemisphere of the Bloch sphere /2 therefore their preferred mesurement set is m,m 3, which occurs with proility of /2. When the mesurement result is m,m 2, she sends nothing ccording to the protocol descried ove. In this wy, the fidelity of the process increses to F =F m =F m3 nd the verge fidelity ecomes F=+ 4 q+ p /2. In the sme wy, if the entngled stte is distriuted y third prty nd oth quits undergo dmping, Alice proceeds s explined ove. If Alice nd Bo decide to keep ll mesurement results then the fidelity of the process ecomes 3 F = k= p mk F mk = 2 p + cos2 2 p 2 cos 2, 38 where F mk is the fidelity of the output stte to the teleported stte when Alice otins m k, nd p mk is the proility of otining this result. Moreover, if they do tht for ny,, they end up with F= 4p 2 2p+q 2 ln q +p. From Eq. 34, it cn e seen tht for fixed p of the chnnel, if the stte to e teleported is chosen such tht /2, then the set m,m 2 gives higher fidelity; otherwise, the set m,m 3 does. Let us ssume tht Alice rndomly chooses the stte to e teleported from the lower hemisphere of the Bloch sphere, /2, therefore their preferred mesurement set is m,m Input sttes The output density opertors for Alice s outcomes m k=,,2,3 cn e written s ˆ out m k = N q ˆ in + p + k p cos ˆ xk ˆ xk + k pq cos ˆ xk ˆ xk 39 with N =+ k p cos from which the stte dependent fidelity is derived s

9 TELEPORTATION OF QUBIT STATES THROUGH p 3 2p 2p cos 2 F mk = 4 + k 4 p cos with the limiting vlues of F m,2 =cos 2 /2 nd F m,3 =sin 2 /2 for p pproching. It is esy to see tht s p pproches, F mk. For p /6, ll sttes cn e teleported with F regrdless of the outcome. Averge vlues of teleporttion fidelity for these two cses re the sme s given in Eq. 35. For the second scenrio, contrry to the first scenrio, the output stte tht Bo gets fter the proper ppliction of the quntum gtes nd its fidelity to the desired stte is the sme s tht of the cse when initil MES is. When only one of the quits of the MES goes through the ADC, distriuting either or does not give ny dvntge to the prties. B. Phse dmping chnnel When the chnnel is the PDC, given y Eq. 23, only the off-digonl elements re ffected y the dmping. The fidelity of the teleporttion process when the initil MES is sujected to the PDC is independent of Alice s mesurement result ecuse, contrry to the ADC cse, Bo cn use X nd Z gtes to rotte ll the possile outcomes to ech other nd to the stte with the highest fidelity to the input one. The elements of the density mtrix cn e written s out = in, out =q 2 in, out =q 2 in, nd out = in for the first scenrio. In the cse of the second scenrio, the elements of the density mtrix re the sme s ove with q 2 replced y q. Then the fidelity of the teleporttion process for scenrio cn e written s nd for scenrio 2 s F = 2 p 2 p sin2, F = 3 p 2 p, F = 2 p sin2, F = 3 p The effect of the PDC on the teleporttion fidelity is the sme for oth initil MES nd. The quit sttes with = nd which re locted t the poles of the Bloch sphere re lwys teleported with F= ecuse these sttes correspond to nd, which do not crry reltive phse informtion nd hence re not ffected y the PDC. Indeed, these results show tht if Alice chooses the sttes to e teleported round the poles then they cn hve etter teleporttion fidelity see Fig. 5. On the other hnd, sttes with = /2, which correspond to ll the sttes lying on the equtor of the Bloch sphere, re the most ffected sttes. If Eve does not hve the informtion on the region from which the quits re chosen, then the est she cn do is to Dmping prmeter p π/4 π/2 3π/4 π Stte ngle δ FIG. 5. Stte-dependent fidelity in the presence of the PDC when the initil MES is ny of the Bell sttes nd only one of the quits dshed curves nd oth quits solid curves re ffected y dmping. Contours show F=, F= nd the optiml fidelities. Horizontl dshed lines correspond to the vlues of dmping rte for f ent =3/4 for scenrios lower nd 2 upper. use the optiml universl quntum cloning mchine of Bužek et l. 3. Then we find tht ny quit stte stisfying sin 2 /3p 2 p nd sin 2 /3p, respectively, for the first nd second scenrios, cn e teleported in the presence of PDC with higher fidelity thn tht of the cloning mchine of the evesdropper. It is pprent tht if there is no evesdropper nd tht prties just wnt to et the clssicl limit, the rnge of quits t fixed p is much lrger. Now, let us ssume tht the sttes to e teleported re chosen with fixed ut vrying, nd the informtion on my e leked to n evesdropper. Since the evesdropper my use the optiml, to spek out secure teleporttion its fidelity should exceed the fidelity given in Eq. 33. Compring Eq. 33 with stte dependent teleporttion fidelities for PDC given in Eqs. 4 nd 42, wefind cos +/x where x = 2 +2p 2 p nd cos /x where x = 2 +2p, respectively, for scenrios nd 2. If is chosen in the neighorhood of, the dmping rte of the chnnel should stisfy p /4 nd p +2 2 /2, respectively, for the first nd second scenrios. On the other hnd, if the sttes to e teleported re chosen from the equtoril quit sttes, the dmping rte of the chnnel should stisfy p /2 /4 nd p / 2, respectively, for the first nd second scenrios. These requirements re oviously stricter thn those for the universl CM. We see in Fig. 5 tht while for the universl cloning mchine the constrint on p relxes s we pproch the poles of the Bloch sphere, for the it ecomes tighter. This is ecuse s we pproch the poles, the fidelity of the clones from the gets closer to one requiring PDC with dmping rtes pproching zero. C. Depolrizing chnnel When the chnnel is the DC, the elements of the density mtrix cn e written s out = in +, out = in,

10 ÖZDEMIR et l. out = in, nd out = in + for the first scenrio, where we hve used = q /2 nd = +q nd =q 2.In cse of the second scenrio, the elements of the density mtrix re the sme s ove with = nd =q. Then the fidelity of the teleporttion process for the first nd second scenrios cn e written, respectively, s F = F = q2, 43 Dmping prmeter p c F = F = q 44 from which we see tht fidelity is independent of in. For DC too, contrry similr to the ADC PDC, Bo cn use quntum gtes to rotte ll possile outcomes to ech other. Therefore the fidelity is independent of the input stte, of Alice s mesurement result, nd of the initilly distriuted MES. The prties in the protocol cn choose their quits from the whole Bloch sphere nd n evesdropper my use universl quntum cloning mchine, in tht cse the dmping rtes of the chnnels should stisfy p 6/3 nd p /3 to surpss the no-cloning limit. In cse of n evesdropper with the, the reltion etween the quits sttes tht cn e teleported securely nd the dmping rte of the chnnel ecomes cos x /2 /2 nd cos p 2 /2 / for the first nd second scenrios. D. Direct trnsmission: Noisy stte + shred MES For Bo, to whom Alice wnts to teleport the unknown stte in, it is difficult to distinguish whether the in is noisy stte or the quntum chnnel is responsile for the noise. The stte to e teleported might e sujected to noise, loose its coherence, nd ecomes mixed stte efore it is teleported. Let us ssume tht Alice nd Bo shre MES, which they hve otined using entnglement distilltion nd purifiction protocols. In this section, we ssume tht the quit is influenced y the ADC, PDC, nd DC, nd discuss the outcome of the teleporttion process. We ssume tht only the quit to e teleported is sujected to noise nd the shred entngled stte is ny of the Bell sttes. Indeed, this is similr to the direct trnsmission scheme where the originl stte in is sent directly to Bo through noisy chnnel. If in is sujected only to the ADC, the elements of the output density mtrix ecome out = in + p in, out = q in, out = q in, nd out =q in where kl in re the elements of the density mtrix of in. Then fidelity is found s F = 2 2p sin2 /2 q q sin Averging this over ll possile input sttes, verge fidelity is found s π/4 π/2 3π/4 π Stte ngle δ FIG. 6. Stte dependent teleporttion fidelity when the quit to e teleported is sujected to the ADC, : p=.2, : p=.5, nd c: p=.9. Horizontl dotted lines denote the limits etween clssicl nd quntum opertions lower nd the secure quntum teleporttion upper. The curve mrked with circles corresponds to the optiml fidelity. F = q p, 46 which is the sme s for scenrio 2, when the entngled stte is distriuted through the ADC. We see tht depending on the dmping prmeter, the rnge of quits tht cn e teleported with desired fidelity chnges see Fig. 6. For exmple, when p=.8, when only in is sujected to noise, the sttes with.5436 nd.42 cn e teleported, respectively, with F nd F. For the in dmped cse, ll the sttes stisfying.2677 cn e teleported with F. In Fig. 6, we hve depicted the fidelity of the from which we see tht when p=/2, the teleporttion fidelity nd the fidelity re equl for the quits /2. In this rnge of quits, secure teleporttion is possile for dmping rtes p /2. As pproches, the dmping rte p pproches zero to chieve secure teleporttion. When the quit is sujected only to the PDC, the output density opertor ecomes ˆ out =q ˆ in + p cos 2 /2 +sin 2 /2, resulting in fidelity F= p sin 2 /2 from which verge fidelity cn e written s F= p/3. Compring these equtions with Eq. 42, it is seen tht when the PDC ffects only the quit to e teleported, the fidelity is the sme s in scenrio 2 when the distriuted entngled sttes undergo the PDC. We oserve the sme similrity if only the quit in is sujected to DC. In this cse the fidelity expression is given s in Eq. 44. In the nlysis of security of dmping prticulr chnnel, the fidelities of optiml cloning mchines were tken s reference: either i the optiml universl cloning mchine if no priori informtion out teleported stte is given or ii the optiml phse-covrint cloning mchine if prior prtil informtion out the stte is ville. Clerly, chnnel is secure if it provides etter fidelity thn the optiml cloning. This is the lowest fidelity ound for security of ny

11 TELEPORTATION OF QUBIT STATES THROUGH chnnel ssuming tht Alice sends her quit through dmping chnnel, while n evesdropper copies quit t Alice s site nd does not send it or sends it through perfect chnnel. Otherwise, the ction of the chnnel will restrict the qulity of the cloning consistent with the chnnel nd, thus, less demnding security conditions cn e given. FIdelity of teleporttion c c IV. CONCLUSION We hve exmined the prolem of teleporttion fidelity in the presence of vrious types of noise during the entnglement distriution of the teleporttion process. Using the fully entngled frction nd concurrence, we derived the ounds on the dmping prmeters of chnnels so tht the verge fidelity i exceeds the clssicl limit, nd ii stisfies the security condition for teleporttion. Moreover, we derived the rnge of sttes tht cn e teleported ccurtely with desired fidelity vlue nd studied how this rnge is ffected y noise. For the security condition, we considered evesdroppers with universl nd phse-covrint cloning mchines where the first evesdropper hs no informtion on the quit to e teleported ut in the ltter he/she knows the ut not the reltive phse. For the ADC, lthough the ounds on p for one-quit ffected cse re the sme for oth nd s the source entngled stte, for the two-quit ffected cse we find tht the ounds re different nd much tighter for. This implies tht if one is given, insted of distriuting this stte directly, it is etter to first loclly convert to nd distriute it. In tht cse the effect of dmping is less pronounced. We oserve tht only for the ADC these ounds chnge with the initil MES to e distriuted. We hve found tht contrry to the cse of the ADC, in the presence of the PDC nd DC, the two-quit ffected cse cnnot e mde to hve higher entngled frction thn the one-quit ffected cse. Hence the verge fidelity cnnot e incresed y sujecting one of the quits to controlled dissiption. As seen in Fig. 7, verge fidelity is dependent on the type nd strength of dmping in the chnnel. For the PDC, fidelity is lwys lrger thn if p, on the other hnd for the ADC nd Dmping prmeter p Dmping prmeter p FIG. 7. Averge fidelity with noisy chnnels: : PDC, : ADC, nd c: DC for scenrios left nd 2 right. Horizontl dshed lines denote the limits etween clssicl nd quntum opertions lower nd the secure quntum teleporttion upper. DC verge fidelity decreses elow down to /2 depending on the dmping rte. We hve discussed the direct trnsmission cse too.we oserve tht the results otined for direct trnsmission nd teleporttion with the one-quit ffected entnglement distriution cse scenrio two re the sme in the cses of the DC nd PDC. However, discrepncies re seen for the cse of the ADC. Averge fidelity for scenrio 2 is more immune to dmping thn the direct trnsmission. This study shows tht informtion on the noise ffecting the teleporttion process during the phses of entnglement distriution nd the quit preprtion cn e helpful in incresing the fidelity. Moreover, it is importnt to note tht if the source of noise in the process is not known then ll should e ttriuted to n evesdropper. Thus the criterion on the teleporttion fidelity should e reformulted tking into ccount the set of sttes from where the teleported stte is chosen nd the optiml cloning mchine for tht set. ACKNOWLEDGMENTS We thnk Professor Nouyuki Imoto nd Professor Msto Koshi for their useful comments. A.M. ws supported y the Polish Ministry of Science nd Higher Eduction under Grnt No. P3B R. Horodecki, P. Horodecki, M. Horodecki, nd K. Horodecki, e-print rxiv: qunt-ph/72225, Rev. Mod. Phys. to e pulished. 2 V. Scrni et l., Rev. Mod. Phys. 77, ; N. J. Cerf nd J. Fiurášek, Prog. Opt. 49, C. H. Bennett, G. Brssrd, C. Crepeu, R. Jozs, A. Peses, nd W. K. Wootters, Phys. Rev. Lett. 7, H. F. Hofmnn, Phys. Rev. A 66, S. Popescu, Phys. Rev. Lett. 72, ; N. Gisin, Phys. Lett. A 2, ; R. Horodecki nd M. Horodecki, Phys. Rev. A 54, M. Horodecki, P. Horodecki, nd R. Horodecki, Phys. Rev. A 6, ; R. Horodecki, M. Horodecki, nd P. Horodecki, Phys. Lett. A 222, F. Grosshns nd P. Grngier, Phys. Rev. A 64, 3 R 2. 8 S. Bndyopdhyy, Phys. Rev. A 65, G. G. Crlo, G. Benenti, G. Csti, nd C. Meji-Monsterio, Phys. Rev. A 69, G. Gordon nd G. Rigolin, Phys. Rev. A 73, W. K. Wootters nd W. H. Zurek, Nture London 299, ; D. Dieks, Phys. Lett. A 92, M. A. Nielsen nd I. L. Chung, Quntum Computtion nd Quntum Informtion Cmridge University Press, Cmridge, Englnd, 2. 3 V. Bužek nd M. Hillery, Phys. Rev. A 54, N. Gisin nd S. Mssr, Phys. Rev. Lett. 79, V. Bužek nd M. Hillery, Phys. Rev. Lett. 8, D. Bruss, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Mcchivello, nd J. A. Smolin, Phys. Rev. A 57,

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