Quantum key distribution without alternative measurements

Quan um key dis ibu ion wi hou al e na i e measu emen s Ada ´n Cabello* Depa amen o de Fı ´sica Aplicada, Uni e sidad de Se illa, 41012 Se illa, Spain 共Recei ed 29 Oc obe 1999; published 18 Ap il 2000兲 En anglemen swapping be ween Eins ein-Podolsky-Rosen 共EPR兲pai s can be used o gene a e he same sequence o andom bi s in wo emo e places. A quan um key dis ibu ion p o ocol based on his idea is desc ibed. The scheme exhibi s he ollowing ea u es. 共a兲I does no equi e ha Alice and Bob choose be ween al e na i e measu emen s, he e o e imp o ing he a e o gene a ed bi s by ansmi ed qubi . 共b兲I allows Alice and Bob o gene a e a key o a bi a y leng h using a single quan um sys em 共 h ee EPR pai s兲, ins ead o a long sequence o hem. 共c兲De ec ing E e equi es he compa ison o ewe bi s. 共d兲En anglemen is an essen ial ing edien . The scheme assumes eliable measu emen s o he Bell ope a o . PACS numbe 共s兲: 03.67.Dd, 03.67.Hk, 03.65.Bz The wo main goals o c yp og aphy a e o wo dis an pa ies, Alice and Bob, o be able o communica e in a o m ha is unin elligible o a hi d pa y, E e, and o p o e ha he message was no al e ed in ansi . Bo h o hese goals can be accomplished secu ely i bo h Alice and Bob a e in possession o he same sec e andom sequence o bi s, a ‘‘key’’ 关1兴. The e o e, one o he main p oblems o c yp og- aphy is he key dis ibu ion p oblem, ha is, how do Alice and Bob, who ini ially sha e no sec e in o ma ion, come in o he possession o a sec e key, while being su e ha E e canno acqui e e en pa ial in o ma ion abou i . This p ob- lem canno be sol ed by classical means, bu i can be sol ed using quan um mechanics 关2兴. The secu i y o p o ocols o quan um key dis ibu ion 共QKD兲such as he Benne - B assa d 1984 共BB84兲关2兴, E91 关3兴, B92 关4兴, and o he p o- ocols 关5,6兴, is assu ed by he ac ha while in o ma ion s o ed in classical o m can be examined and copied wi hou al e ing i in any de ec able way, i is impossible o do ha when in o ma ion is s o ed in unknown quan um s a es, be- cause an unknown quan um s a e canno be eliably cloned 共‘‘no-cloning’’ heo em 关7兴兲. In hese p o ocols secu i y is assu ed by he ac ha bo h Alice and Bob mus choose andomly be ween wo possible measu emen s. In his pape I in oduce a QKD scheme which does no equi e ha Alice and Bob choose be ween al e na i e measu emen s. This scheme is based on ‘‘en anglemen swapping’’ 关8–10兴be- ween wo pai s o ‘‘qubi s’’ 共quan um wo-le el sys ems兲, induced by a Bell ope a o measu emen 关11兴. The Bell op- e a o is a nondegene a e ope a o which ac s on a pai o qubi s iand j, and p ojec s hei combined s a e on o one o he ou Bell s a es 兩 00 典 ij⫽1 冑 2共 兩 0 典 i丢 兩 0 典 j⫹ 兩 1 典 i丢 兩 1 典 j), 共1兲 兩 01 典 ij⫽1 冑 2共 兩 0 典 i丢 兩 0 典 j⫺ 兩 1 典 i丢 兩 1 典 j), 共2兲 兩 10 典 ij⫽1 冑 2共 兩 0 典 i丢 兩 1 典 j⫹ 兩 1 典 i丢 兩 0 典 j), 共3兲 兩 11 典 ij⫽1 冑 2共 兩 0 典 i丢 兩 1 典 j⫺ 兩 1 典 i丢 兩 0 典 j). 共4兲 En anglemen swapping wo ks as ollows. Conside a pai o qubi s, iand j, p epa ed in one o he ou Bell s a es, o ins ance, 兩 11 典 ij. Conside a second pai o qubi s kand l p epa ed in ano he Bell s a e, o ins ance, 兩 01 典 kl . I a Bell ope a o measu emen is pe o med on iand k, hen he ou possible esul s ‘‘00,’’ ‘‘01,’’ ‘‘10,’’ and ‘‘11’’ ha e he same p obabili y o occu . In ac , he ou come o each mea- su emen is pu ely andom. Suppose ha he esul ‘‘00’’ is ob ained, consequen ly he s a e o he pai iand ka e he measu emen is 兩 00 典 ik . Mo eo e , he s a e o jand lis p o- jec ed on o s a e 兩 10 典 jl . The e o e, he s a e o jand lbe- comes en angled al hough hey ha e ne e in e ac ed. I will deno e he ini ial s a e o he pai s i,jand k,l,in he p e ious example by 兩 11 典 ij丢 兩 01 典 kl , and he inal s a e o he pai s i,kand j,lby 兩 00 典 ik丢 兩 10 典 jl . Suppose ha he ini ial s a e o he pai s i,jand k,lis a p oduc o wo Bell s a es and, as in he p e ious example, a Bell ope a o measu e- men is execu ed on wo qubi s, one o each pai ; hen, a e he measu emen he s a e o he pai s i,kand j,lbecomes a p oduc o wo Bell s a es. All possibili ies a e collec ed in Table I. The p oposed scheme o QKD is illus a ed in Fig. 1 and i is desc ibed as ollows. 共i兲Conside six qubi s numbe ed 1 o 6. Alice p epa es qubi s 1 and 2 in he Bell s a e 兩 11 典 12 , and qubi s 3 and 5 in he Bell s a e 兩 10 典 35 . In a emo e place, Bob p epa es qubi s 4 and 6 in he Bell s a e 兩 10 典 46 . All his in o ma ion is pub- lic. 2 and 6 will be he only ansmi ed qubi s du ing he p ocess. Alice will always e ain qubi s 1, 3, and 5; and Bob will always e ain qubi 4. 共ii兲Alice ansmi s qubi 2 o Bob using a public channel. This channel mus be a ansmission medium ha isola es he s a e o he qubi om in e ac ions wi h he en i onmen . 共iii兲Alice sec e ly measu es he Bell ope a o on qubi s 1 and 3, and Bob sec e ly measu es he Bell ope a o on qubi s *Elec onic add ess: [email p o ec ed] PHYSICAL REVIEW A, VOLUME 61, 052312 1050-2947/2000/61共5兲/052312共4兲/$15.00 ©2000 The Ame ican Physical Socie y61 052312-1 2 and 4. The esul s o bo h expe imen s a e co ela ed, al- hough Alice and Bob do no know how as ye . The pu pose o he nex s ep is o elucida e how he esul s a e co ela ed wi hou publicly e ealing ei he o hem. 共i 兲Bob ansmi s qubi 6 o Alice using a public channel. Then Alice measu es he Bell ope a o on qubi s 5 and 6, and publicly announces he esul . Suppose ha Alice has ob- ained ‘‘11’’ in he sec e measu emen on qubi s 1 and 3. Then, since he ini ial s a e o 1, 2, 3, and 5 was 兩 11 典 12 丢 兩 10 典 35 , by using Table I Alice knows ha he s a e o 2 and5is 兩 10 典 25 . In addi ion, suppose ha Alice ob ains ‘‘00’’ in he public measu emen on 5 and 6. Then, since she knows ha he p e ious s a e o 2, 4, 5, and 6 was 兩 10 典 25丢 兩 10 典 46 , by using Table I Alice knows ha Bob has ob ained ‘‘00’’ in his sec e measu emen on 2 and 4. Following a simila ea- soning, Bob can know ha Alice has ob ained ‘‘11’’ in he sec e measu emen on 1 and 3. P e iously, Alice and Bob ha e ag eed o choose he sequence o esul s o Alice’s sec e measu emen s o o m he key. The wo ini ial bi s o he key a e he e o e ‘‘11.’’ The public in o ma ion sha ed by Alice and Bob is no enough o E e o acqui e any knowledge o he esul ob ained by one o he pa s. Using his in o ma ion E e only knows ha one o he ollowing ou possible combina ions o esul s o Alice and Bob’s sec e measu emen s ha e occu ed: ‘‘00’’ o Alice’s esul and ‘‘11’’ o Bob’s, ‘‘01’’ and ‘‘10,’’ ‘‘10’’ and ‘‘01,’’ and ‘‘11’’ and ‘‘00.’’ One Bell s a e can be ans o med in o ano he jus by o a ing one o he qubi s. Using his p ope y, Alice 共Bob兲 can change he Bell s a e o qubi s 1 and 3 共2 and 4兲 o a p e iously ag eed public s a e. Then he si ua ion is simila o 共i兲and he nex s age o he p ocess can be s a ed. This scheme o QKD has he ollowing ea u es. 共a兲I imp o es he a e o gene a ed bi s by ansmi ed qubi . In BB84 and in B92 共and in E91兲, Bob 共and Alice兲 mus choose be ween wo al e na i e measu emen s in o de o p ese e secu i y. This implies ha he numbe o use ul andom bi s sha ed by Alice and Bob by ansmi ed qubi , be o e checking o ea esd opping, is 0.5 bi s by ansmi ed qubi , bo h in BB84 and B92 共and 0.25 in E91兲,o a he mos , i can be made o app oach 1 in Re . 关6兴. In ou scheme he a e is 1 bi by ansmi ed qubi . This is so because Alice and Bob always pe o m he same kind o measu emen , a Bell ope a o measu emen , and he e o e, each o hem ac- qui es wo co ela ed andom bi s a e each s age o he p ocess. In each o hese s ages, only wo qubi s a e ans- mi ed 共one om Alice o Bob and ano he om Bob o Alice兲. This imp o emen is e y use ul since a key mus be as la ge as he message o be ansmi ed 共w i en as a se- quence o bi s兲, and canno be eused o subsequen mes- sages 关1兴. 共b兲I only equi es a single quan um sys em 共 h ee EPR pai s兲ins ead o a long sequence o quan um sys ems, o gene a e a key o a bi a y leng h. By con as wi h p e ious schemes, in he one p esen ed he e no sou ce o qubi s is needed. The same wo qubi s 共qubi s 2 and 6兲a e ansmi ed o and om Alice and Bob o e and o e again 关12兴. 共c兲The de ec ion o E e equi es he compa ison o ewe bi s. The ansmi ed qubi s do no encode he bi s ha o m he key, bu only he ype o co ela ion be ween he esul s o he expe imen s ha allow Alice and Bob o sec e ly gen- e a e he key. The e o e, in e cep ing and copying hem does no allow E e o acqui e any in o ma ion abou he key. In TABLE I. All possible esul s o a Bell ope a o measu emen on qubi s iand k. Fo example, i he ini ial s a e is 兩 11 典 ij 丢 兩 01 典 kl , you mus loca e 1101 on he le hal o he able. Then, a e a Bell ope a o measu emen on iand k, he ou possible inal s a es a e ep esen ed on he igh hal o he able by 0010, 0111, 1000, and 1101; whe e, o ins ance, 0010 means 兩 00 典 ik丢 兩 10 典 jl . Ini ial s a e 兩 ijkl 典 Possible inal s a es 兩 ikjl 典 0000 0101 1010 1111 0000 0101 1010 1111 0001 0100 1011 1110 0001 0100 1011 1110 0010 0111 1000 1101 0010 0111 1000 1101 0011 0110 1001 1100 0011 0110 1001 1100 FIG. 1. QKD scheme based on en anglemen swapping. The bold lines connec qubi s in Bell s a es, he dashed lines connec qubi s on which a Bell ope a o measu emen is made, and he poin ed lines connec qubi s in Bell s a es induced by en anglemen swapping. ‘‘00’’ means ha he Bell s a e 兩 00 典 is public knowledge, 共00兲means ha i is only known o Alice, 关00兴means ha i is only known o Bob, 兩 00 兩 means ha i is unknown o all he pa s, 关(00)兴 means ha i is only known o Alice and Bob, e c. ADA ´N CABELLO PHYSICAL REVIEW A 61 052312 052312-2 ac , he s a e o he ansmi ed qubi s is public. Howe e , E e can use a s a egy—also based on en anglemen swapping— o lea n Alice’s sequence o sec e esul s. This s a egy is illus a ed in Fig. 2 and is desc ibed as ollows. 共1a兲Conside he same scena io as in 共i兲bu suppose E e has wo addi ional qubi s 7 and 8, ini ially p epa ed in a Bell s a e, o ins ance, 兩 00 典 78 . 共1b兲E e in e cep s qubi 2 ha Alice send o Bob and makes a Bell ope a o measu emen on qubi s 2 and 8. Then qubi s 1 and 7 become en angled in a known 共 o E e兲Bell s a e. Fo ins ance, i a e E e’s measu emen he s a e o 2 and8is 兩 00 典 28 , hen he s a e o 1 and 7 becomes 兩 11 典 17 . 共2兲The e o e, a e E e’s in e en ion he eal si ua ion is no ha desc ibed in 共ii兲. Now qubi 1 is en angled wi h E e’s qubi 7, and 2 is en angled wi h E e’s 8. 共3a兲In his new scena io, a e Alice’s 共Bob’s兲measu e- men on qubi s 1 and 3 共2 and 4兲, he s a e o qubi s 5 and 7 共6 and 8兲becomes a Bell s a e. Fo ins ance, i Alice 共Bob兲 ob ains ‘‘11’’ 共‘‘00’’兲, he s a e o qubi s 5 and 7 共6 and 8兲 would be 兩 10 典 57 ( 兩 10 典 68). Howe e , hese s a es a e unknown o E e, because she 共s ill兲does no know he esul s o Al- ice’s and Bob’s measu emen s. 共3b兲E e in e cep s qubi 6 ha Bob sends o Alice and makes a Bell ope a o measu emen on qubi s 6 and 8. This e eals he s a e hey we e in. Then E e can know Bob’s esul . Fo ins ance, in ou example, E e would ind ‘‘10’’ and would know ha Bob’s esul was ‘‘00.’’ 共3c兲E e makes a Bell ope a o measu emen on qubi s 7 and 8. Then qubi s 5 and 6 becomes en angled in a Bell s a e 共s ill兲unknown o E e, because she does no know Alice’s sec e esul . Fo ins ance, i E e ob ains ‘‘01,’’ hen qubi s 5 and 6 would be in he s a e 兩 01 典 56 . 共4兲E e gi es qubi 6 o Alice. Alice makes a measu e- men on 5 and 6 and announces he esul . Then E e can know he p e ious s a e o 5 and 7 ( 兩 10 典 57 , in ou example兲 and he esul o Alice’s measu emen on 1 and 3 共‘‘11,’’ in ou example兲. Howe e , E e’s in e en ion changes he co ela ion ha Alice and Bob expec be ween hei sec e esul s. Fo in- s ance, in ou example, Bob, using his esul and he esul publicly announced by Alice, hinks ha he wo ini ial bi s o he key a e ‘‘10.’’ As in p e ious QKD p o ocols, in ou scheme Alice and Bob can de ec E e’s in e en ion by publicly compa ing a su icien ly la ge andom subse o hei sequences o bi s, which hey subsequen ly disca d. I hey ind ha he es ed subse is iden ical, hey can in e ha he emaining un es ed subse is also iden ical, and he e o e can o m a key. In BB84, o each bi es ed by Alice and Bob, he p obabili y o ha es e ealing he p esence o E e 共gi en ha E e is indeed p esen 兲is 1 4. Thus, i Nbi s a e es ed, he p obabil- i y o de ec ing E e 共gi en ha she is p esen 兲is 1⫺(3 4)N.In ou scheme i Alice and Bob compa e a pai o bi s gene - a ed in he same s ep, he p obabili y o ha es o e eal E e is 3 4. Thus i npai s (N⫽2nbi s兲a e es ed, he p ob- abili y o E e’s de ec ion is 1⫺(1 2)N. This imp o emen in he e iciency o he de ec ion o ea esd opping has been poin ed ou o a pa icula ea esd opping a ack, i would be in e es ing o in es iga e whe he mo e gene al a acks exis and whe he he imp o emen in e iciency is also p esen in hese cases. 共d兲I uses en anglemen as an essen ial ool. QKD was he i s p ac ical applica ion o quan um en anglemen 关3兴. Howe e , as shown in Re . 关13兴, en anglemen was no an essen ial ing edien , in he sense ha almos he same goals can be achie ed wi hou en anglemen . Howe e , subsequen s iking applica ions o quan um mechanics such as quan um dense coding 关14,15兴, elepo a ion o quan um s a es 关8,16,17兴, en anglemen swapping 关8,9兴, and quan um com- FIG. 2. E e’s s a egy o ob ain Alice’s sec e esul . 兵 00 其 means ha he Bell s a e 兩 00 典 is only known o E e. The emaining no a ion is he same as in Fig. 1. QUANTUM KEY DISTRIBUTION WITHOUT... PHYSICAL REVIEW A 61 052312 052312-3 pu a ion 关18兴, a e s ongly based on quan um en anglemen . The scheme desc ibed he e elies on en anglemen in he sense ha i pe o ms a ask—QKD wi h p ope ies 共a兲,共b兲, and 共c兲— ha canno be accessible wi hou en anglemen . The p ac ical easibili y o he scheme desc ibed in his pape hinges on he easibili y o a eliable 共i.e., wi h 100% heo e ical p obabili y o success兲Bell ope a o measu e- men . Bell ope a o measu emen s a e also equi ed o eli- able double densi y quan um coding and elepo a ion. As a as I know, he i s p oposals o a eliable Bell ope a o measu emen a e hose which disc imina e be ween he ou pola iza ion-en angled wo-pho on Bell s a es using en- anglemen in addi ional deg ees o eedom 关19兴o using a omic cohe ence 关20兴. I is no expec ed ha he p o ocol o QKD in oduced in his pape will be able o imp o e exis ing expe imen s 关21兴 o eal quan um c yp og aphy in p ac ice. I s main impo - ance is concep ual: i p o ides a di e en quan um solu ion o a p oblem al eady sol ed by quan um mechanics. The au ho hanks J. L. Ce eceda, O. Cohen, A. K. Eke , C. Fuchs, T. Mo , and B. O ila o help ul commen s. This wo k was suppo ed by he Uni e sidad de Se illa 共G an No. OGICYT-191-97兲and he Jun a de Andalucı ´a共G an No. FQM-239兲. 关1兴The Ve nam ciphe o ‘‘one- ime pad’’ 关G.S. Ve nam, J. Am. Ins . Elec . Eng. 45, 109 共1926兲兴 is he only known absolu ely secu e me hod o enc yp ing a message 关C.E. Shannon, Bell Sys . Tech. J. 28, 657 共1949兲兴. Alice w i es he message as sequence o bi s, and adds he key o i , bi by bi , modulo 2. The esul is he ‘‘ciphe ex ,’’ which is publicly ansmi ed. Bob can eco e he message by adding he key o he ciphe - ex , modulo 2. I he same key is used o a second message, E e can ob ain he addi ion modulo 2 o bo h messages jus by adding hei chipe ex s. I only a ini e ocabula y is used in he messages, his is enough o e eal bo h o hem. 关2兴C. H. Benne and G. B assa d, in P oceedings o IEEE In e - na ional Con e ence on Compu e s, Sys ems, and Signal P o- cessing, Bangalo e, India 共IEEE, New Yo k, 1984兲, p. 175. 关3兴A.K. 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