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PRL 93, 230503 (20 0 4)

PHYSICA L R EVIEW LET T ERS

week end i ng 3 DECE M BER 20 0 4

Q ut r it Stat e Engi neer i ng wit h Biphotons
Yu. I. Bogda nov
Russia n Cont rol Syst em Agency, ``A ngst rem ,'' Moscow 12 4 460 Russia

M. V. Chek hova , S. P. Kuli k , G. A. Ma slen n i kov, a nd A. A. Z hukov.
Depa r t ment of Physics, Moscow M.V. Lomonosov St at e Un iversit y, 119 9 92 Moscow, Russia*

C. H. Oh a nd M. K. Tey
Depa r t ment of Physics, Facult y of Science , Nat iona l Un iversit y of Si ngapore , 117542 Si ngap ore ( Re ceived 9 June 20 0 4; publ ishe d 1 De cember 20 0 4) T he novel exper i ment a l rea l i zat ion of t h ree - level opt ica l qua nt u m system s is present e d. We use t he p ola r i zat ion st at e of biphot on s t o generat e a spe cific sequence of stat es t hat a re use d i n t he ext ende d version of fou r-st at e QK D prot o col qua nt um key d ist r ibut ion prot o col. We exper i ment a l ly ver i f y t he or t hogona l it y of t he ba sic st at es a nd demon st rat e t he abi l it y t o ea sily swit ch bet ween t hem. T he t omog raphy proce du re is employe d t o re con st r uct t he den sit y mat r ices of generat e d stat es.
DOI: 10.1103/ PhysRevL et t.93. 230503 PACS nu mbers: 03.6 7. Mn , 42. 50. Dv

T he a r t of qua nt um st at e eng i neer i ng, i. e. , t he abi l it y t o generat e , t ra n sm it , a nd mea sure qua nt u m system s is of g reat i mp or t a nce i n t he emerg i ng field of qua nt u m i n format ion t e ch nolog y. A va st major it y of prot o cols rely i ng on t he prop er t ies of t wo- level qua nt u m system s (qubit s) were i nt ro duce d a nd exper i ment a l ly rea l i ze d. But nat ura l ly, t here a rose a quest ion of a n exten sion of d i men sion a l it y of system s used a s i n for mat ion ca r r iers a nd t he new feat ures t hat t h is ext en sion ca n offer. T he si mplest exten sion provokes t he usage of t h ree -stat e qua nt u m system s (qut r it s). Re cent ly new qua nt u m key d ist r ibut ion (QK D) prot o cols were prop ose d t hat dea lt spe cifica l ly wit h qut r it s [1, 2] a nd t he eavesd roppi ng a na lysis showe d t hat t h is system s were more robust aga i n st spe cific cla sses of eavesd roppi ng at t ack s [3, 4]. T he ot her adva ntage of usi ng mult i level system s is t hei r p ossible i mplement at ion i n t he f unda ment a l t est s of qua nt u m me cha n ics [5], g iv i ng more d ivergence f rom cla ssica l t heor y. T he usage of mul t i level system s a lso prov ides a p ossibil it y t o i nt roduce ver y spe cific prot o cols, wh ich ca n not be i mplement e d w it h t he help of qubit s such a s qua nt u m bit com m it ment , for exa mple [6]. Re cent exp er i ment s on rea l i zat ion of qut r it s rely on severa l issues. I n one ca se, t he i nt er fero met r ic proce dure is use d , where ent a ngle d qut r it s a re generat e d by send i ng a n ent a ngle d phot on pa i r t h roug h a mult ia r me d i nt er feromet er [7]. T he number of a r m s defines t he d i men siona l it y of t he system. O t her t e ch n iques rely on t he proper t ies of or bit a l a ng ula r momen t u m of single phot ons [6 ,8] a nd on p ost sele ct ion of qut r it s f rom four-phot on st at es [9]. Un for t unat ely a l l ment ione d t e ch n iques prov ide on ly a pa r t ia l cont rol over a qut r it st at e. For exa mple i n a met hod , ment ione d i n [6 ,8] a spe cific holog ra m should be made for g iven qut r it st at e. T he rea l pa r t s of t he a mpl it udes of a qut r it , generat e d i n [7] a re fixe d by a cha ract er ist ics of a fiber t r it t er, ma k i ng it ha rd t o swit ch bet ween t he stat es. Besides, i n t h is 0031-9 007= 04 =93(23)=230503(4)$22. 50

met hod no t omog raph ic cont rol over generat e d stat e had been yet p er for me d. I n t h is L et t er we repor t t he exper i ment a l rea l i zat ion of a r bit ra r y qut r it st at es t hat exploit s t he p ola r i zat ion st at e of si ngle - mo de biphot on field. T h is field con sists of pa i rs of cor relat e d phot ons, is most ea sily obt a i ne d wit h t he help of sponta neous pa ra met r ic dow n - conversion. By sayi ng ``single - mo de'' we mea n t hat t w i n phot ons for m i ng a biphot on have e qua l f re quencies a nd propagat e a long t he sa me d i re ct ion. A pure p ola r i zat ion st at e of such field ca n be w r it t en a s t he fol lowi ng super p osition of t h ree ba sic stat es. ji c1 j2; 0i c2 j1; 1i c3 j0; 2i c1 ji c2 ji c3 j i;

(1)

where ci jci jeii a re complex probabi l it y a mpl it udes. The st at es j2; 0i and j0; 2i cor resp ond t o t y p e - I pha se mat ch i ng where t wi n phot ons have col l i nea r p ola r i zat ion ve ct ors (for exa mple , st at e j2; 0i cor resp onds t o t wo phot ons bei ng i n hor i zont a l H p ola r i zat ion mo de) , a nd st at e j1; 1i is obt a i ne d v ia t y p e - I I pha se mat ch i ng, where phot ons a re p ola r i ze d or t hogona l ly (say, one of t hem is i n H a nd t he ot her one is i n V mo de). T here exist s a n a lt ernat ive represent at ion of st at e ji t hat maps t he st at e onto ґ t he sur face of t he Poi nca re sphere [10]. T he operat iona l or t hogona l it y cr it er ion for t he pola r i zat ion st at es of single - mo de biphot ons wa s propose d i n [11] a nd exper i ment a l ly ver ifie d i n [12]. Accord i ng t o t he or t hogona l it y cr it er ion for biphot on p ola r i zat ion st at es, t wo p ola r i za t ion st at es a and b a re or t hogona l i f one obser ves zero coi ncidence rat e i n t he Brown -Twiss scheme , prov ide d t hat t he stat e a is at t he i nput , a nd p ola r i zat ion filt ers i n each a r m a re t une d t o a ssure ma x i ma l t ra n sm it t a nce of each phot on for m i ng t he st at e b (set stat e). T he goa l of our work wa s t o demon st rat e t he abi l it y t o prepa re a ny 200 4 T he A mer ica n Physica l Societ y

230503-1


PRL 93, 230503 (20 0 4)
TA BL E I. St at e j j j j j j j j j j j j i i i 0 i 0 i 0 i 00 i 00 i 00 i 000 i 000 i 000 i jc1 j 1 0 0 p 1=p 3 1=p3 1=p 3 1=p3 1=p 3 1=p3 1=p 3 1=p3 1= 3

PHYSICA L R EVIEW LET T ERS
GP HWP1 H

week end i ng 3 DECE M BER 20 0 4

12 st at es use d i n qut r it QK D prot ocol. jc2 j 0 1 0 p 1=p 3 1=p3 1=p 3 1=p3 1=p 3 1=p3 1=p 3 1=p3 1= 3 jc3 j 0 0 1 p 1=p 1=p 1=p 1=p 1=p 1=p 1=p 1=p 1= 1 0 0 0 0 0 0 120 0 0 Ъ120 0 0 2 0 0 0 0 120 Ъ120 0 120 0 0 Ъ120 0
3



12

V comp crystal

PZT driven mirror


2 BBO type I DM QP1 UVM

13

3 3 3 3 3 3 3 3 3

0 0 0 0 Ъ120 120 0 0 120 0 0 Ъ120

HWP2

QP2 BBO 4 QP3 type II

UVM

F IG. 1.

E xp er i ment a l set up ( prepa rat ion pa r t).

g iven p ola r i zat ion st at e ji a nd a s a st ra ig ht for wa rd a nd pract ica l exa mple of g iven st at es, we chose t he spe cific se quence t hat wa s present e d i n [1]. T h is sequence of 12 st at es for ms fou r mut ua l ly unbia sed ba ses wit h t h ree st at es i n each , a nd ca n be used i n a n extende d version of BB84 QK D prot o col for qut r it s. T he 12 st at es a re define d i n Table I. T he prepa rat ion pa r t of our set up ( Fig. 1) is bui lt on t he ba se of a ba la nce d Mach - Z eh nder i nt er feromet er [13]. T he pump pa r t con sist s of f re quency double d ``Coherent Mi ra 9 0 0'' femt osecond la ser, operat e d at cent ra l waveleng t h of 800 n m , 75 MHz rep et it ion rat e a nd wit h a pulse widt h of 10 0 fs, average pump p ower wa s 20 mW. T he Gla n -Tompson pr ism , t ra n sm it t i ng t he hor i zont a l ly p ola r i ze d f ract ion of t he U V pump a nd refle ct i ng t he ver t ica l ly p ola r i ze d f ract ion , ser ves a s a n i nput m i r ror of M ZI. T he refle ct e d pa r t , a f t er pa ssing t he comp en sat ion -barium borat e ( BBO) cr yst a l a nd a ha l f wave plat e ( H WP 2), pumps t wo con se cut ive 1 m m t h ick t y p e - I BBO cr yst a ls whose opt ica l a x is a re or ient e d p er p end icula rly w it h resp e ct t o each ot her. T he biphot ons f rom t hese cr ysta ls pa ss t h rough a 10 m m qua r t z plat e (QP1) t hat ser ves a s a comp en sat or, a nd t he pump is refle ct e d by a n U V m i r ror. T hen t he biphot ons a r r ive at a d ich roic m i r ror ( DM) t hat is desig ne d t o t ra n sm it t hem a nd t o refle ct t he hor i zont a l ly p ola r i ze d comp onent of t he pu mp com i ng f rom t he upper a r m of M ZI. A piezo ele ct r ic t ra n slat or ( PZT ) wa s use d t o cha nge t he pha se sh i f t of t he hor i zont a l comp onent of t he pump w it h resp e ct t o t he one propagat i ng i n t he lower a r m. T he U V bea m , refle ct e d f rom DM ser ves a s a pump for 1 m m t h ick t yp e - I I BBO cr yst a l. Two 1 m m qua r t z plat es (QP 2) ca n be rot at e d a long t he opt ica l a x is t o i nt ro duce a pha se sh i f t bet ween hor i zont a l ly a nd ver t ica l ly p ola r i ze d t y pe - I biphot ons, a nd a set of fou r 1 m m t h ick qua r t z plat es (QP3) ser ves t o comp en sat e t he g roup velo cit y delay bet ween or t hogona l ly p ola r i ze d phot ons dur i ng t hei r propagat ion i n t y p e I I BBO cr yst a l. T he mea surement set up ( Fig. 2) con sists of a Brow n -Twiss scheme wit h a nonp ola r i z i ng 50=50 bea m spl it t er ; each a r m cont a i n s con secut ively place d qua r t er- a nd ha l f wave plat es a nd a n a na lyzer t hat was set t o t ra n sm it t he ver t ica l p ola r i zat ion. T h is sequence of

wave plat es a nd a na lyzer is refer re d t o a s a p ola r i zat ion filt er. I nt er ference filt ers of 5 n m ba ndwidt h , cent ere d at 800 n m a nd pi n holes a re used for spe ct ra l a nd spat ia l mo da l sele ct ion of biphot ons. We use EGG-SPC M- AQR 15 si ngle phot on count i ng mo dules a s our det e ct ors ( D1 a nd D2). We should ment ion , t hat due t o t he low pump p ower, t he st i mulat e d processes i n our set up a re negl igi bly sma l l a nd on ly pa i rs of phot ons have been generat e d. T he mea surement of t he generat e d st at es is done usi ng t he t omog raphy prot o col t hat wa s develop e d for p ola r i zat ion qut r it s [14]. I n order t o re con st r uct t he den sit y mat r i x of t he mea sure d st at e (wh ich is genera l ly m i xed) one ha s t o p er for m n i ne proje ct ive mea surement s of t he fou r t h order moment s of t he field for d i fferent set t i ngs of p ola ri zat ion filt ers. Pola r i zat ion den sit y mat r i x ca n be define d i n t he fol lowi ng way i n t er m s of t hese moment s [14 ,15]. p 2 2 221 hay abi; 211 hay a2 i; p 2 233 hby b2 i; 232 hay by b2 i; (2) 2 22 hay by abi; 231 hay b2 i: T h is config u rat ion of t he mea surement set up ( Fig. 2) a l lows us t o ver i f y t he or t hogona l it y of t he st at es t hat belong t o t he sa me ba sis. Compen sat ion.--I n order t o have t he t h ree t er ms i n super p osit ion (1) i nter fer i ng, one must ach ieve t hei r p erfe ct overlap i n f re quency, moment um , a nd t i me doma i n s. From t he exper i ment a l p oi nt of v iew t h is mea n s t hat t he biphot on wave packet s com i ng f rom t he t wo t yp e I cr yst a ls a nd f rom t he t y p e I I cr ysta l must be overlapp e d. T he overlap i n t he f re quency doma i n is ach ieved by t he usage of 5 n m ba ndwidt h i nt er ference filt ers a nd t he overlap i n moment u m is en sure d by usi ng pi n holes t hat sele ct one spat ia l mo de of t he biphot on field. But t he overlap i n t i me ca n not be ach ieve d ea sily when using a pulse d la ser source , be cause it is ne cessa r y t o comp en sat e for a l l t he g roup delays t hat biphot on wave packet s acqui re du r i ng t hei r propagat ion t h roug h t he optica l element s of t he

F IG. 2.

E xp er i ment a l set up (mea surement pa r t).

230503-2


PRL 93, 230503 (20 0 4)

PHYSICA L R EVIEW LET T ERS

week end i ng 3 DECE M BER 20 0 4

set up [16]. It wa s found t hat i n order t o overlap t yp e - I biphot ons w it h t y pe -II, t he pump pulse f rom t he lower a r m must be delaye d by 50 ps. T h is was ach ieve d by i n ser t i ng a n add it iona l 2 m m BBO cr yst a l i n t he lower a r m. T he overlap bet ween t he st at es j2; 0i and j0; 2i wa s ach ieve d by i n ser t i ng a 10 m m qua r t z plat e d i re ct ly a f t er t he t wo t y p e - I BBO cr yst a ls. A f t er overlappi ng t he bi phot ons w it h t hese t e ch n iques, t he average coi ncidence count rat e t hat we obser ve d wa s of about 1 Hz. T he h ig h v isibi l it y of i nt er ference pat t er n s t hat we obt a i ne d was a cr it er ion for a goo d comp en sat ion. E xp er i ment a l pro ce dure.--I n order t o creat e a g iven qut r it stat e we nee de d t o have i ndep endent cont rol over fou r rea l pa ra met ers -- t wo relat ive a mpl it udes a nd t wo relat ive pha ses. I n t he exp er i ment we used H WP1 t o cont rol t he a mpl it ude of t he st at e j1; 1i, a nd H WP 2 t o cont rol t he relat ive a mpl it udes of t he st at es j2; 0i and j0; 2i. T he relat ive pha se 13 3 Ъ 1 bet ween t he st at es j2; 0i and j0; 2i ca n be cont rol le d wit h t he help of rot at i ng qua r t z plat es (QP 2). T he relat ion of t he pha se 12 2 Ъ 1 bet ween t he st at e j0 i j2; 0i ei13 j0; 2i and j1; 1i t o t he volt age appl ie d t o PZT ca n be found by mon it or i ng t he pump i nt er ference pat t er n i n t he M ZI. We found t hat t he cha nge of volt age by one V result e d i n t he pha se sh i f t of 51:7 and 12 g rew l i nea rly w it h t he appl ie d volt age. Stat es t hat con stit ute t he first ba sis a re t r iv ia l ( Table I ). T hey ca n be produce d w it h t he help of a si ngle cr ysta l , cor resp ondi ng t o t y p e I or t y p e I I i nt eract ion. St at e j2; 0i is generat e d when first =2 ( H WP1) a ngle cor responds t o the ma x i mal refle ct ion of t he pump bea m i nt o t he lower a r m of a Mach - Z eh nder a nd t he a ngle of t he second ha l f la mbda wave plat e ( H WP 2) is e qua l t o 0 . I n order t o generat e st at e j0; 2i, t he H WP 2 must be rot at e d by 45 deg rees f rom 0 , a nd t o generat e st at e j1; 1i t he H WP1 is rot at e d such t hat t he whole pump goes i nto t he upper a r m 0

of t he M ZI. T herefore , i n t he fol lowi ng, we wi l l con sider on ly t he generat ion of t he rest n i ne st at es, i. e., t hose for m i ng t he ot her t h ree ba ses. Accord i ng t o Table I , on ly t he relat ive pha ses bet ween t he ba sic st at es a re t o be va r ie d. T he fol low i ng pro ce dure wa s use d i n order t o ver i f y t he or t hogona l it y of t he st at es t hat for m a cer t a i n ba sis. Fi rst we chose a set stat e t o wh ich we would t une our p ola r i zat ion filt ers. T hen t he va lues of t he a ngles of qua r t er- a nd ha l f -wave plat es ( Fig. 2) (1 ;1 ;2 ;2 ) t hat a ssure t he ma x i ma l proje ct ion of t he p ola r i zat ion stat e of each phot on on t he V d i re ct ion ca n be ca lculat e d ґ by mappi ng t he set st at e on t he Poinca re sphere. Here , t he lower i ndex ``1'' cor resp onds t o t he t ra n sm it t e d a r m , a nd t he i ndex ``2'' to t he refle ct e d a r m of BS. We chose st at es j0 i, j00 i, a nd j000 i t o be our set st at es for each ba sis. T hen , by set t i ng t he pha se 13 fixe d a nd by va r y i ng t he pha se 12 we mea sure d t he number of coi ncidence count s t hat cor resp ond t o t he cer t a i n fou r t h - order moment of t he field. Accord i ng t o t he or t hogona l it y cr it er ion , t he coi n cidence rat e should fa l l t o zero when t he va lues of 13 and 12 cor resp ond t o t he generat ion of t he stat es or t hogona l t o t he set ones. Result s a nd discussion.--L et us con sider t he generat ion of t he st at e j00 i. I n t h is ca se 13 0;12 120. I n Fig. 3 t he mea sure d va lues of t he rea l a nd i mag i na r y pa r t s of t he den sit y mat r i x comp onent s 21 and 32 on pha se 12 a re shown a s f unct ion of t he pha se 12 . T he number of accident a l coi ncidences wa s negl igibly sma l l a nd wa s not subt ract e d i n dat a pro cessi ng. T he pha se 13 0 rema i ne d con st a nt dur i ng t he t omog raphy pro ce du re. A f t er obt a i n i ng t he dep endence of t he moment s 21 and 32 on pha se 12 we fitte d our dat a wit h t heoret ica l dep endencies, using t he lea st -squa res approx i mat ion met hod. T he obt a i ne d va lues of a l l comp onent s were substit ut e d i n E q. (2). T he obt a i ne d den sit y mat r i x for stat e j00 i is given below: (3)



00

1 0:355 Ъ0:054 Ъ 0:210i 0:315 Ъ 0:010i @ Ъ0:054 0:210i 0:340 Ъ0:106 0:262i A: 0:315 0:010i Ъ0:106 Ъ 0:262i 0:305

T he eigenva lues of t h is mat r i x a re 1 0:877;2 0:136;3 Ъ0:013. A cor resp ond i ng set of eigenve ct ors is X 0:587; Ъ0:173 0:521i; 0:594 Ъ 0:071i; Y 0:642; 0:379 Ъ 0:649i; 0:048 0:143i; Z 0:493; Ъ0:287 0:224i; Ъ0:769 Ъ 0:178i. A lt hough t he den sit y mat r i x [ E q. (3)] is Her m it ia n a nd t he cond it ion Tr 1 is sat isfied , it does not cor resp ond t o a ny physica l st at e be cause of t he negat iv it y of one of t he eigenva lues. We wa nt t o p oi nt out t hat a first ma i n com p onent 1 ij Xi Xj of a con sidere d den sit y mat r i x , exp wh ich ha s a weig ht 0:878 is a l ready close t o t he t heoret i ca l st at e ve ct or j00 i a nd t he cor resp ondi ng fidel it y is F Trth 1 0:9903. T he ot her t wo comp onent s cor re exp spond t o t he ``exper i ment a l noise'' t hat is due ma i n ly t o

m isa l ig n ment s of a set up a nd sma l l volu me of col le ct e d dat a. We have obt a i ne d si m i la r eigenva lues for a l l ot her stat es a nd raw fidel it y comput e d for t he ma i n den sit y mat r i x comp onent a s descr ibe d above have va r ie d f rom 0.983 t o 0.9 98. We a lso employe d t he ma x i mu m l i ke l i hoo d met hod of qua nt u m st at e root est i mat ion [14 ,17] t o ma ke a t omog raph ica l ly re con st r uct e d mat r i x sat isf y it s physica l proper t ies, such a s p osit iv it y. T he result s a re present e d i n t he fol lowi ng t able ( Table I I ). T he level of stat ist ica l fluct uat ion s i n fidel it y est i mat ion was det erm i ne d by t he fin it e size of reg ist ere d event s 500. A l l exper i ment a l fidel it y va lues l ie w it h i n t he t heoret ica l ra nge of 5%F 0:9842 and 95%F 0:9991 qua nt i les [14,18].

230503-3


PRL 93, 230503 (20 0 4)

PHYSICA L R EVIEW LET T ERS

week end i ng 3 DECE M BER 20 0 4

Number of coincidences /30s

20 15 10 5 0 -200 0 12 (deg) 200

F IG. 4 (color on l i ne). Dependence of nu mber of coi ncidences on a pha se 12 for a g iven set t i ngs of p ola r i zat ion filt ers. F IG. 3 (color on l i ne). I mag i na r y a nd rea l va lues of nond iagona l den sit y mat r i x comp onent s use d t o re con st r uct st at e j00 i. T heoret ica l dep endence is plot t e d wit h a sol id cu r ve.

T he obt a i ne d fidel it y va lues show t he h ig h qua l it y of t he prepa re d stat es. T he ot her t est of t he qua l it y of st at es is t he f ulfil l ment of t he or t hogona l it y cr it er ion for t he st at es t hat belong t o t he sa me ba sis. For each set st at e we ca lculat e d t he set t i ngs of wave plat es i n our mea surement set up t hat en sure d t he ma x i ma l proje ct ion of each phot on on t he ver t ica l p ola r i zat ion d i re ct ion. I n Fig. 4 we show t he dep endence of t he coi ncidence rat e for t he fol lowi ng set t i ng of wave plat es 1 28:3 ;1 Ъ33:5 ;2 Ъ24 ;2 Ъ2 . T hese va lues cor resp ond t o t he set stat e j000 i. As one ca n see , for t he fixe d va lue 13 0 t he coi ncidence rat e is a l most e qua l t o zero , when pha se 12 Ъ120 . T h is cor resp onds t o t he gen erat ion of t he stat e j000 i, wh ich is or t hogona l t o j000 i.T he v isibi l it y of t h is pat t er n is e qua l t o 93. 2%. For t he ot her ba ses, t he obta i ne d va lues of v isibi l it ies va r ie d f rom 92% t o 95%. Wit h t hese va lues of v isibi l it y, t he lowest va lue of coi ncidence rat e cor resp onds t o t he accident a l ( Poissonia n) coi ncidence level a nd t herefore t he obt a i ne d dat a ver ifies t he or t hogona l it y cr it er ion. Conclusion s.--We rea l i ze d a n i nt er feromet r ic met hod of prepa r i ng t he t h ree - levels opt ica l system s on dema nd. The spe cific sequence of st at es wa s generat e d a nd mea sure d wit h h igh fidel it y va lues. T he or t hogona l it y of t he st at es t hat for m mut ua l ly unbia sed ba ses wa s exper i men ta lly ver ifie d. As a n adva nt age of t h is met hod we not e
TA BL E II. met ho d. St at e j i j0 i j 0 i
0

t hat a l l cont rol of t he a mpl it udes a nd pha ses of each ba sic stat e i n super p osit ion (1) is done usi ng l i nea r opt ica l element s, ma k i ng it ea sy t o swit ch f rom one st at e t o a not her a nd prov idi ng t he f ull cont rol over t he st at e (1). T he ma i n d isadva ntage is t hat we ca n not generat e a n ent a ngle d qut r it s i n t h is config u rat ion. Usef ul d iscussions w it h A. E ker t , B. Engler t , D. Ka z l i kowsk i , C. Ku r t siefer, L. C. Kwek , A. L a ma sLi na res, a nd A. Pen i n a re g rat ef ul ly ack nowle dge d. T h is work wa s suppor t e d i n pa r t by Russia n Foundat ion of Ba sic Resea rch ( P roje ct s No. 03-02-16444 and No. 0202-16843) a nd t he Nat iona l Un iversit y of Singap ore's Ea st er n Eu rope Resea rch Scient ist a nd St udent P rog ra m me.

Fidel it ies est i mat e d wit h ma x i mu m l i kel i ho o d FML Stat e j i j00 i j 00 i
00

E

FML

E

Stat e j i j000 i j 000 i
000

FML

E

0:9989 0:9967 0:9883

0:9967 0:9989 0:9883

0:9883 0:9989 0:9967

*Ele ct ron ic add ress: p ost ma st@qopt.phys. m su.su [1] H. Be ch ma n n - Pa squinucci a nd A. Peres, Phys. Rev. L et t. 85, 3313 (200 0). [2] D. Ka szl i kowsk i et a l. , Phys. Rev. A 67, 012310 (20 03). [3] D. Br uss a nd C. Mach iavel lo , Phys. Rev. L et t. 88, 1279 01 (20 02). [4] T. Dur t et a l. , Phys. Rev. A 67, 012311 (20 03). [5] D. Col l i n s et a l. , Phys. Rev. L et t. 88, 0 40 40 4 (20 02). [6] N. L a ng ford et a l. , Phys. Rev. Let t. 93, 053601 (200 4). [7] R. T. T hew et a l. , Qua nt um I n f. Comput. 4, 93 (20 0 4). [8] A. Va z i r i et a l. , Phys. Rev. L et t. 91, 2279 02 (2003). [9] J. C. Howell et a l. , Phys. Rev. L et t. 88, 030 401 (20 02). [10] A. V. Bu rla kov a nd M. V. Chek hova , J ET P Let t. 75, 432 (20 02). [11] A. V. Bu rla kov a nd M. V. Chek hova , J ET P L et t. 75, 8 (20 02); 75, 432 (20 02). [12] M. V. C hek hova et a l. , J ET P L et t. 76, 596 (20 02). [13] G. A. Maslen n i kov et a l. , J. Opt. B 5, 530 (20 03). [14] Yu. I. Bogda nov et a l. , Phys. Rev. A 70, 0 42303 (20 0 4). [15] D. N. K lysh ko , J ET P 84 , 1065 (1997). [16] Y-H. K i m et a l. , Phys. Rev. A 61, 051803(R) (20 01). [17] Yu. I. Bogda nov et a l. , J ET P L et t. 78, 352 (20 03). [18] T he 5% a nd 95% qua nt i les cut s t he lef t a nd r ig ht t a i l of d ist r ibut ion cor respondi ngly.

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