Algothmc Supactvaton of Asymptotc Quantum Capacty of Zo-Capacty Quantum Channls Laszlo Gyongyos, Sando Im Dpatmnt of Tlcommuncatons Budapst Unvsty of Tchnology and Economcs - Budapst, Magya tudoso t, ungay gyongyos@ht.bm.hu Th supactvaton of zo-capacty quantum channls mas t possbl to us two zo-capacty quantum channls wth a postv jont capacty fo th output. Cuntly, w hav no thotcal bacgound to dscb all possbl combnatons of supactv zo-capacty channls; hnc, th may b many oth possbl combnatons. In pactc, to dscov such supactv zo-capacty channl-pas, w must analyz an xtmly lag st of possbl quantum stats, channl modls, and channl pobablts. Th s stll no xtmly ffcnt algothmc tool fo ths pupos. Ths pap shows an ffcnt algothmcal mthod of fndng such combnatons. Ou mthod can b a vy valuabl tool fo mpovng th sults of fault-tolant quantum computaton and possbl communcaton tchnqus ov vy nosy quantum channls.. Intoducton Ths pap ntoducs an xtmly ffcnt algothmc soluton fo fndng supactv zocapacty channls and povds an algothmc famwo fo analyzng th popts of channl output stats n quantum spac. In 008, Smth and Yad hav found only on possbl combnaton fo supactvaton [5], and th should b many oth possbl combnatons., w confm ths wth ou mthod, whch can b xtndd to dscov oth such combnatons of channls. Th numb of ffcnt appoxmaton algothms fo quantum nfomatonal dstancs s vy small bcaus of th spcal popts of quantum nfomatonal gnato functons and of asymmtc quantum nfomatonal dstancs. If w wsh to analyz
th popts of quantum channls usng today s classcal comput achtctus, an xtmly ffcnt algothm s ndd. Th poposd mthod gvs an algothmc soluton to th supactvaton poblm of zo-capacty channls. W xhbt a fundamntally nw and ffcnt algothmc soluton fo dscovng all possbl supactv channl combnatons. As w wll show, Smth s sult [5] s only on possbl soluton to supactvaton, and many oth solutons can b dscovd by ou algothmc soluton. Ths pap s oganzd as follows. In Scton, w psnt th thotcal bacgound of th poposd appoach. In Sctons 3 and 4, w analyz th gomty of quantum stats and th poblm of supactvaton. In Scton 5, w gv a fundamntally nw gomtcal ntptaton of th supactvaton of asymptotc quantum capacty. In Scton 6, w gv an llustatv xampl. Fnally, n Scton 7, w psnt ou conclusons.. Rlatd Wo Th supactvaton of th quantum capacty of zo-capacty quantum channls was ntoducd by Smth and Yad [5]. Snc th volutonay popts of supactvaton quantum channl capacts w fst potd on, many futh quantum nfomatonal sults hav bn achvd [9,4,53], [58-6]. Rcntly, Duan and Cubtt t al. found a possbl combnaton fo th supactvaton of th classcal zo-o capacty of quantum channls, whch has opnd up a dbat gadng th xstnc of oth possbl channl combnatons [9,4]. Th poblm of supactvaton can b dscussd as pat of a lag poblm th poblm of quantum channl addtvty. Th addtvty poblm s consdd to b a vy mpotant poblm n quantum nfomaton thoy. Addtvty poblm of quantum channls Supactvaton of zo-capacty quantum channls Fg.. Th poblm of supactvaton of zo-capacty quantum channls as a sub doman of a lag poblm st.
Th ntptaton of quantum channl capacty basd on th quantum latv ntopy functon was thotcally psntd by Cots [8], ayash t al. [3,33], and Schumach- Wstmoland [48,49]. owv, th most basc qustons gadng th dffnt typs of capacty of a quantum channl man opn. Most typs of quantum channl capacty hav bn found to b non-addtv [5,,6,8,50,5,55]. In ths pap, w xhbt a fundamntally nw gomtcal appoach to fndng supactv zo-capacty quantum channls. Fo th combnaton of any quantum channl that has som pvat capacty ( P > 0 and of a fxd 50% asu symmtc channl, th followng conncton holds btwn th asymptotc quantum capacty Q of th jont stuctu Ä, and th pvat capacty P ( of : ( Ä Q( Ä ³ P (. ( Th channl combnaton fo th supactvaton of th asymptotc quantum capacty of zocapacty quantum channls s shown n Fg.. Zo-capacty quantum channl wth som pvat capacty P Q P Q 0 0 0 0 n n 50% asu channl Q Asymptotc quantum capacty 0 P Fg.. Th fst channl has som postv pvat capacty, and th scond quantum channl s a 50% asu channl wth zo quantum capacty. In what follows, w wll s that t s possbl to fnd oth combnatons of quantum channls and, whch a, ndvdually, zo-capacty n th sns that Q ( Q( ( = = 0, and yt satsfy
( Q Ä >. (3 0 W us computatonal gomtc mthods snc ths algothmc tools can b mplmntd vy ffcntly [,3,5,4,43]. Th poblm of clustng n quantum spac, usng th quantum nfomatonal dstanc as a dstanc functon, s a compltly nw aa n quantum nfomaton thoy. Rcntly, th possblts of th applcaton of computatonal gomtc mthods n quantum spac hav bn studd by Kato t al. [38] and Nlsn t al. [44] and Noc and Nlsn [45]; howv, th poblm of clustng was not analyzd n th wo. Th cost mthod fo dffnt dstancs has bn studd n th ltatu. Eucldan mthods w studd n [,6,7,3,3,44] and a non-eucldan mtc by Banj [3] and Acmann t al. n []. A vy usful and pactcal appoach to th computaton of quantum channl capacty usng gomtc mthods was also psntd by ayash t al. [3,33]. In ou wo, w wll constuct advancd algothmc appoachs to analyz th supactvaton of th asymptotc quantum capacty of quantum channls [9].. Computatonal Gomty n Quantum Infomaton Pocssng In ou wo, w apply computatonal gomty n quantum spac. Wth th hlp of ffcnt computatonal gomtc mthods, th supactvaton of zo-capacty quantum channls can b analyzd vy ffcntly. W would l to analyz th popts of th quantum channl usng classcal comput achtctus and algothms [4,0,7,4,56] snc, cuntly, w hav no quantum computs. To ths day, th most ffcnt classcal algothms fo ths pupos a computatonal gomtc mthods. W us ths classcal computatonal gomtc tools to dscov th stll unnown supactv zo-o capacty quantum channls. Computatonal gomty was ognally focusd on th constucton of ffcnt algothms and povds a vy valuabl and ffcnt tool fo computng had tass. In many cass, tadtonal lna pogammng mthods a not vy ffcnt. To analyz a quantum channl fo a lag numb of nput quantum stats wth classcal comput achtctus, vy fast and ffcnt algothms a qud. W wll us quantum nfomaton as a dstanc masu nstad of classcal gomtc dstancs. Unl odnay gomtc dstancs, th quantum nfomatonal dstanc s not a mtc and s not
symmtc. W combn th modls of nfomaton gomty and th mthods of computatonal gomty. At psnt, computatonal gomty algothms a an actv, wdly usd, and ntgatd sach fld [,37,40,47,46]. Many dffcult poblms can b solvd by computatonal gomtc mthods f povdd wth wll-dsgnd and ffcnt algothms [,3,0,,,3,4,5,6,7,8,9]... Analyss of Supactvaton To study th gomty of supactvaton, w dfn a nw abstact gomtc objct, th quantum nfomatonal supball. Usng th cost mthod [,5] w can analyz th capacts of quantum channls fo an xtmly lag nput st and fo all possbl channl modls wth xtmly hgh ffcncy. Ths nfomatonal gomtc algothms can b appld to quantum nfomatonal dstancs. Thn th addtvty popts of dffnt quantum channl modls wth vaous channl pobablts can b analyzd xtmly qucly n ou fundamntally nw famwo. Th tatons a mad on th channl nput stats, channl modls, and channl paamts. Th output of th algothm s th adus of th quantum nfomatonal supball, whch dscbs th asymptotc quantum capacty of th channl. 3. Communcaton ov Quantum Channls In th classcal communcaton modl, th snd and cv can b modld by andom { },,, vaabls X = p = P ( x = N and ( Y = { p = P y }, =, N., wh th logathm functon, log s, as usual n ths contxt, to ( In classcal systms, th Shannon ntopy of th dsct andom vaabl X s dfnd as ( X p log( p bas. Th pobablty of a andom vaabl X gvn (o, as on says, condtond on Y s dnotd by N =-å = p( X Y. Th nos n th channl ncass th unctanty n X, gvn Bob s output Y. Th nfomatonal thotc nos of th channl ncass th condtonal Shannon X Y ntopy, dfnd as ( X Y = ååp( x, yj logp( x yj N N = j= nclosng quantum nfomatonal ball wll dcas fo fxd [35,55]. ; thus, th adus of th smallst ( X
Th gnal classcal nfomatonal thotc modl fo a nosy quantum channl s as d d follows. A quantum stat can b dscbd by ts dnsty matx Î, whch s a d d matx wh d s th lvl of th gvn quantum systm. Fo an n qubt systm, th lvl of th quantum systm s d = n. W us th fact that patcl stat dstbutons can b analyzd pobablstcally by mans of dnsty matcs. A two-lvl quantum systm can b dfnd by ts dnsty matcs as follows: æ + z x - yö =, x y z, ç + + x y z çè + - ø (4 wh dnots a squa oot of. Th dnsty matx = ( xyz,, can b dntfd wth a pont ( xyz,, n th-dmnsonal spac, and th ball B fomd by such ponts, {( xyz x y z B =,, + + psntd n sphcal coodnats = (, q, }, s calld th Bloch ball. A quantum stat can also b j, wh s th dstanc fom th quantum stat to th ogn and q and j psnts th lattud and longtud otaton angls, spctvly. A quantum channl can b dscbd by an affn map, whch maps quantum stats to oth quantum stats. Th psnc of nos n a quantum channl mans that th mappng s no long a noslss on-to-on latonshp. Th mag of th quantum channl s tansfom s an llpsod [5,8,48]. Gomtcally, maps th Bloch ball to a dfomd ball contand nsd th Bloch ball [38]. To psv th popty of bng a dnsty matx, th map that modls th quantum channl must b tac-psvng,.., T ( = T (, wh T ( s th tac opaton, and t must b compltly postv;.., fo th dntty map I, Ä I maps a sm-postv mtan matx to a sm-postv mtan matx [,39,5]. In Fg. 3, Alc s pu stat s dnotd by A, and Bob s mxd nput stat s dnotd by ( =. Fo andom vaabls X and Y, as abov, ( X, Y = ( X + ( Y X. W A s B wll gv a gomtc psntaton of th nfomaton that can b tansmttd n th psnc of nos n a quantum channl.
X XY X X Y Alc s pu quantum stat A Quantum channl Bob s mxd nput stat B Fg. 3. Th classcal communcaton modl. In classcal nfomaton thoy, nfomaton that can b tansmttd though a nosy channl can b masud n a gomtc way by th adus of th smallst nfomatonal ball that ncloss channl output stats. W s to maxmz ( X and mnmz ( X Y to maxmz th adus of th smallst nclosng ball snc th adus nfomatonal ball can b computd as: = max - X Y. classcal all possbl x { } ( X ( classcal of th classcal (5 In quantum nfomaton thoy, nstad of a classcal ball, w hav a quantum nfomatonal ball that contans all channl output quantum stats. Th ntops btwn mxd quantum stats a masud by th von Numann ntopy [9,36,57] nstad of th classcal Shannon ntopy and th adus of th ball dscbs th quantum capacty of th quantum channl [,39,5]. ( Th classcal capacty C of a quantum channl s masud by th numb of classcal bts that can b tansmttd p channl us. Although, n ths cas, th quantum mutual nfomaton s usd to dscb th channl capacty, n th cas of th quantum capacty Q ( ( of th sam quantum channl, w hav to us th quantum cohnt nfomaton Icoh = I ( : A A, whos maxmum s qual to S s - S s, wh th nput of th channl s dnotd by, whl S(s s th von Numann ntopy of A B max ( ( ( A B E Bob s stat s B = ( A and th nfomaton lad to th nvonmnt s masud by S( s E. Th von Numann ntopy of a dnsty matx s dfnd as S( =-T ( log. ( Q dscbs th quantum capacty of th channl as th numb of quantum bts p channl us that can b tansmttd n a cohnt stat though a nosy quantum channl.
3. Quantum Capacty of a Quantum Channl olvo ntoducd th concpt of th olvo quantty of nfomaton [8,35,48,49]. A vy mpotant but not wll-nown fact was shown by Schumach and Wstmoland n [48]: th quantum cohnt nfomaton can b computd as th dffnc btwn two olvo quantts olvo nfomaton AB, whch masus th olvo quantty btwn Alc and Bob [48], and olvo nfomaton AE, whch masus th olvo quantty btwn Alc and th nvonmnt dung th tansmsson of th quantum stat [48]. In Fg. 4, w summaz a vy mpotant conncton btwn th classcal olvo nfomaton and th quantum cohnt nfomaton. Classcal nfomaton btwn Alc and Bob Classcal nfomaton passd to th nvonmnt Classcal nfomaton AB AE Quantum cohnt nfomaton AB AE Fg. 4. Expsson of th quantum cohnt nfomaton n tms of olvo quantts. As follows, th quantum capacty can also b xpssd as th dffnc btwn th von Numann ntops of two channl output stats. Th fst stat s cvd by Bob, whl th scond on s cvd by a non-vald cv th nvonmnt [48]. Quantum cohnt nfomaton plays a fundamntal ol n th supactvaton of th quantum channl and, l th olvo quantty n th classcal SW (olvo-schumach-wstmoland capacty [8, 49] of a quantum channl, th quantum cohnt nfomaton plays a cucal ol n th asymptotc LSD (Lloyd-Sho-Dvta capacty of a quantum channl [,39,5]. To dfn th asymptotc LSD quantum capacty, w hav to gulaz th maxmum of th quantum cohnt nfomaton I ( : (, mployng th paalll us of n cops of channl as follows: A A Q ( = = ( Än lm Q ( n Än lm max I ( A : ( A, n n n A (6
wh Q ( ( = max I ( : ( A A s th sngl us quantum capacty of th quantum A channl. In ou pap, w us th fact that th asymptotc (.. not th sngl us quantum capacty can b computd by th olvo quantts as follows: Q n A AB -AE, (7 n Ä n A, x ( = lm max ( wh and ( ( s - å p S( ( = S (8 AB AB AB AB ( ( s - å p S( ( = S (9 AE AE AE AE masu th olvo quantts btwn Alc and Bob and btwn Alc and nvonmnt E, wh s = å p and s AE = å p a th avagd stats [48,49]. AB 3. Quantum Rlatv Entopy Intptaton of Quantum Capacty W us th sults of Schumach and Wstmoland [48,49] to dscb gomtcally th channl capacty of quantum channls n tms of th quantum latv ntopy. Thy hav shown that, fo a gvn quantum channl, th olvo quantty fo vy optmal output stat and th s avag stat can b xpssd as: ( = D ( s, (0 wh s = å p s th optmal avag output stat, and D, th latv ntopy functon of two dnsty matcs, s dfnd as: D ( = T é log( - log( s s ù ë û. ( Fo non-optmal output stats d and optmal avag output stat ( = D( d s D( s s = å p, w hav. Schumach and Wstmoland [49] also showd that th s at last on optmal output stat {, } p that achvs th optmum olvo quantty ( = D ( s. Th gomtc ntptaton of quantum channl capacty was
ntoducd n [48] as th adus of th smallst ball nsd th Bloch sph that contans all channl output stats, wh smallst mans, whn usng th quantum latv ntopy functon as a dstanc masu: ( ( ( ( = = mn { } max D s { }. ( s If w dnot th convx hull of possbl channl output stats [8] fo channl by th convx hull of th st of stats by, thn fo Î and Î s : and ( ( = = mn { } max D s { }. (3 s Schumach and Wstmoland [49] also povd that th s an optmum output stat { p, } fo vy s that satsfs th maxmzaton such that s = å p, and that th avag output stat s = å p, whch maxmzs th capacty fo any optmal st of output stats { p, } = å, s unqu. W analyz th supactvaton of th quantum channl by clustng and convx hull calculatons basd on th quantum latv ntopy. If w dnot th optmal output stats that achv th olvo capacty ( of channl by { ( p } and th avag s s = å p, thn th quantum channl capacty can y =, b dvd n tms of th quantum latv ntopy as follows [8,49]: å ( s = å( é log( ù - é log( s ù pd pt ë û pt ë û é ù = é ù - ë û ê ú û = - å( pt log( T å( p log( s ë å( pté log( ù Téslog( s ù ë û ë û S( s å ps(. = - = (4 W wll us th gomtc ntptaton of th olvo quantty to xpss th asymptotc quantum capacty of a quantum channl. Th gomtc ntptaton of th olvo quantty has bn studd by Cots [8], who also xtndd ths sults to gnal qudt (.., hgh dmnsonal channls. 3.3 Th Asymptotc Quantum Capacty In ou wo, w us th sults of Lloyd-Sho-Dvta [,39,5] to analyz th supactvaton popty of quantum channls. Accodng to th LSD thom and th sult of Schumach
and Wstmoland [49], th sngl us quantum capacty ( Q ( of a quantum channl can b dfnd as th adus of th smallst quantum nfomatonal ball: ( ( max{ all possbl and } = Q = - æ æ n öö n = max S p - p,, pn,,, n çè ç å è øø æ æ öö - S p + çè ç å è øø p AB AE ( åp S( ( AB AB ç= = n n ( åp S( (, AE AE ç= = (5 wh s th olvo quantty of Bob s output, s th nfomaton lad to th AB nvonmnt dung th tansmsson, and ( AE psnts th -th output dnsty matx obtand fom th quantum channl nput dnsty matx. Usng th sultng quantum latv ntopy functon and th LSD thom, th asymptotc quantum channl can b xpssd wth th hlp of th ad of th smallst quantum nfomatonal balls as follows: ( ( æ n ö Än ( Q = lm Q = lm n n n nç å çè = ø = lm max I = lm max n n n n n n = lm å mn max - n n n n n AB-AE AB-AE = lm å( mns max D (, n n n s n ( - coh AB AE n p,, p,,, n p,, p,,, n AB AB AE AE ( s D( s D( s (6 AB AE wh s th sngl us capacty of th -th channl us of quantum channl, s - th optmal output channl stat, and AB AE s - s th avag stat. Th supscpt AB-AE dnots th nfomaton that s tansmttd fom Alc to Bob mnus th nfomaton that s lad to th nvonmnt dung th tansmsson. Usng ths sult, th supactvaton of th asymptotc quantum capacty [5] can b studd by usng th quantum latv ntopy functon as a dstanc masu [4,0,7,4,56]. 4. Gomtcal Intptaton of Quantum Capacty Th latv ntopy n classcal systms s a masu of how clos a pobablty dstbuton p D p q p = å p, whl th q s to a modl pobablty dstbuton q [5,35] and s gvn by ( log latv ntopy btwn quantum stats s masud by
D ( s = T é ( - s ù ë log log û. (7 Th quantum nfomatonal dstanc has som dstanc-l popts: D ( s ³ 0 ff ¹ s, and D ( s = 0 ff = s. owv, t s not symmtc [8,48]. Th quantum latv ntopy btwn a gnal quantum stat = ( xyz,, and a mxd stat s = ( xyz,, wth ad = + y + z y + z x, and s = x + s gvn by: ( ( - s ( s ( - + + D( s = log ( - + log - log ( - - log s,, 4 4 s s (8 wh s, = ( xx + yy + zz s = ( xyz,, = ( 0, 0, 0. Fo a maxmally mxd stat and s = 0, th quantum latv ntopy can b xpssd as: ( + ( - D( s = log ( - + log - log. 4 4 (9 Th latv ntopy of quantum stats can b dscbd by a stctly convx and dffntabl gnato functon F : ( =- ( = T ( F S log, (0 wh - S s th ngatv ntopy of quantum stats. Th quantum latv ntopy fo dnsty matcs and s s gvn by F as follows: ( = ( - ( - -, ( D ( s D s F F s s F s, ( wh s, = T ( s s th nn poduct of quantum stats and F s th gadnt. In Fg. 5, th quantum nfomatonal dstanc, D ( s, s th vtcal dstanc btwn th gnato functon F and ( s (, th hypplan tangnt to F at s [44,45]. Th pont of ntscton of quantum stat on s dnotd by. ( s ( s
Fg. 5. Dpcton of gnato functon as a ngatv von Numann ntopy. Fo th quantum nfomatonal dstanc functon, th gnato functon s th ngatv von Numann ntopy functon -S, wh ( ( =- ( = T ( F S log, F S. Th quantum nfomatonal dstanc functon : d D F ( s ( wth gnato functon F ( =-S( s llustatd n Fg. 6. Th gnato functon of th quantum nfomatonal dstanc s th ngatv von Numann ntopy functon. Fg. 6. Ngatv von Numann gnato functon. Th quantum nfomatonal dstanc functon s lna; thus, fo convx functons " F Î " F Î, D F + lf ( s = D F ( s + l D F ( s fo any l ³ 0. Th gomtc stuctu of quantum nfomatonal balls dffs fom th gomtc stuctu of odnay Eucldan balls. In Fg. 7, w hav llustatd th ccumcnt c of fo th Eucldan dstanc and fo and
quantum latv ntopy [44,45]. Fo a tangl, th ccumcnt s dfnd as th cnt of a ccl that ccumscbs th tangl [,3,7]. Fg. 7. Ccumcnt fo Eucldan dstanc and quantum latv ntopy. In Fg. 8, w compa th smallst quantum nfomatonal ball and th odnay Eucldan ball. Fg. 8. Th smallst balls dff fo th quantum nfomatonal dstanc and Eucldan dstanc. Thus, th quantum stats, and 3, whch dtmn th smallst nclosng ball n, Eucldan gomty, dff fom th stats that would do th sam fo th quantum nfomatonal ball. 4. Asymptotc Quantum Capacty and Supball Rpsntaton In ths scton, w dfn a nw gomtc objct to dscb th asymptotc quantum capacty of th quantum channl. Accodng to th LSD thom [,39,5], usng th optmal output stat and channl avag stat of n uss of th quantum channl, th adus of th smallst nclosng quantum nfomatonal supball can b xpssd as: AB-AE sup AB AE s -
sup ( = ( = lm n ( Än ( Q Q n AB-AE AB-AE = lm å( mns max D (. n n s n n n (3 In ou gomtc psntaton, w analyz th supactvaton popty of th quantum AB-AE AB AE s - sup channl usng th mn-max cton fo stats and. Th adus of th supball, masud usng th latv ntopy functon as th dstanc masu [48], s qual to th asymptotc quantum capacty. Usng th adus of th quantum nfomatonal ball to xpss th sngl us channl capacty, th asymptotc quantum capacty Q ( n = lm Q ( of a quantum channl can b xpssd as: ( Ä n n sup æ n ö lm = = n n ç å, (4 çè = ø ( Q( wh s th adus of th smallst quantum nfomatonal ball, whch dscbs th sngl us quantum capacty of th -th us of th quantum channl [,39,5]. In th supactvaton poblm, w hav to us dffnt quantum channl modls and [9,5]. Fo two quantum channls and, th Q ( Ä asymptotc quantum channl capacty of th jont stuctu can b xpssd by supball adus sup ( Ä = Q( Ä ( ( Än = lm Q ( Ä. n n (5 As w hav dpctd n Fg. 9, th quantum nfomaton supball dffs fom th classcal Eucldan ball: n compason, t has a dstotd gomtcal stuctu [,44,45]. Th adus of th quantum nfomatonal supball dscbs th Q ( Ä asymptotc quantum capacty of th Ä jont channl stuctu.
Fg. 9. Th quantum supball dfnd fo th analyss of supactvaton of zo-capacty quantum channls. In Fg. 0, w summaz ou gomtcal taton pocss as follows. Th nputs of th gomtc constucton of th quantum supball a th channl output stats of th two Än Än spaat quantum channls and. Ths pocss ylds th supball adus as dscbd abov. Th tatd jont channl constucton s dnotd by ( Ä Än, whl th output of th oundd box s th adus supball. sup ( Ä of th quantum nfomatonal Channl modl Input stats n n Ou gomtcal appoach Supball adus Channl pobablts Fg. 0. Th cusv algothm tats on th nput, channl modls, and o pobablts of th channls to fnd a combnaton fo whch supactvaton holds. Th cusv tatons a mad on th paamts: quantum channl modls, channl pobablts, and st of nput stats. Accodng to th lngth of th supball adus, th taton stops f th condtons fo supactvaton hold.
5. Effcnt Cost Constucton of Channl Output Stats Th cost tchnqu has dp lvanc to classcal computatonal gomty. A cost of a st of output quantum stats has th sam bhavo as th lag nput st, so clustng and oth appoxmatons can b mad wth small costs. Th cost can b vwd as a small nput st of channl output stats; hnc, t can b usd as th nput to an appoxmaton algothm. Th wghtd sum of os of th small cost s a ( - appoxmaton of th lag nput st. Th bound on ths o can b dcasd only f th cnt ponts that fom a fnt st a usd n th appoxmaton. Ths costs a calld wa costs [5], and ths mthod can b appld n quantum spac btwn quantum stats. Usng wa costs, th un tm of ( + cost algothms [5,3] wth spct to th quantum nfomatonal dstanc can b mpovd. To constuct th cost mthod analyzng th supactvaton of zo-capacty quantum channls [0], w hav to ntoduc th dfnton of smla quantum nfomatonal dstancs and wa costs of quantum stats. 5. Smla Quantum Infomatonal Dstanc Th quantum nfomatonal dstanc s asymmtc and contans sngulats snc th a s ( dnsty matcs and fo whch D s, =. Th smla quantum nfomatonal dvgnc functon dos not contan ths sngulats, and th dstancs a appoxmatly symmtc. To us a smla quantum nfomatonal dvgnc functon, w fst dfn t as follows. Th quantum nfomatonal dstanc functon D ( s btwn dnsty matcs and s s m -smla fo a postv al constant m f th s a postv dfnt matx A such that: A( ( (. md s D s D A s (6 d Fo quantum nfomatonal dstancs, f th doman s gvn as = élg, ù ë û Í R+, thn m = l g and I l calculatd as. If w hav 0 < l < g, thn th quantum nfomatonal dstanc functon can b D ( ë ( û å ( s, T é log logs ù d = - - - s on th doman = élg, ù ë û Í R
l []. Th quantum nfomatonal dstanc s m -smla f m = and g A = I. In ths cass, l th quantum nfomatonal dstanc functon s m -smla bcaus t s stctd to a subdoman, whch avods th sngulats [], [0]. It can b asly povn that th quantum nfomatonal dstanc functon s stctly convx and that all scond-od patal dvats xst and a contnuous on th doman d = élg, ù ë û Í R wth paamts l m = and g A = I l []. Th appld cost algothm was ognally psntd by Chn [6,7]. W show that ths mthod can b usd to gnat a cost basd on a smla quantum nfomatonal dstanc functon. To obtan an nhancd vson of pvously nown cost appoxmaton algothms, w must dfn wa costs. 5. Wa Cost of Quantum Stats Wa costs nclud all th lvant nfomaton qud to analyz th ognal xtmly lag nput st [5]. Th cost appoach has sgnfcantly low computatonal complxty; hnc, t can b appld vy ffcntly [3]. In ou mthod, wa costs a appld to m - smla quantum nfomatonal dstancs snc, n ths substs, th dstancs btwn quantum stats a symmtc; hnc, sngulats can b avodd and fast Eucldan mthods can b appld [30,34,54]. Th supactv quantum channls can b dscovd wth appoxmaton o ( by usng th small m -smla subst of nput quantum stats. Usng th sults fom Chn s wo [6,7] w show that, by usng ths algothm, th supactvaton of quantum channls can b appoxmatd wth o ( usng th small m -smla subst of nput quantum stats. Th goal of th algothm s to fnd a st of sz such that th sum of os of quantum nfomatonal dstancs s mnmzd; hnc, o n, = å mn ( s ( s = D s. Th algothm solvs th -mdan poblm wth spct to th quantum nfomatonal dstanc D n quantum spac [0]. Th output of th algothm s a st of quantum stats fo whch th functon o (, s s mnmzd. W gnalz th -mdan poblm fo quantum s
nfomatonal dstancs. Lt us assum that w hav two quantum stats and s n doman. W would l to constuct a subst of D OUT ( = ( of quantum stats, fo whch:, OUT mn s. (7 Î D s Th -mdan poblm fo quantum stats can b statd as follows. W would l to us only a fnt st of quantum stats fom th ognal lag spac. Fo a st, w OU T would l to constuct a st of -quantum stats, fo whch (, OUT = å ( OUT o D s Î s mnmzd; hnc [0]: ( ( o, = å mn D. s Î Th o of th optmal soluton fo nput stats s dnotd by opt (, and OUT th lmnts of th output st a th mdan-quantum stats of st. To constuct a mo ffcnt algothm, w us only th m -smla quantum nfomatonal dstancs; hnc, (8 th st of nput quantum stats can b avodd []. s stctd to quantum stats fo whch th sngulats Th supactvaton popts of quantum channls can b dscovd by usng m - smla quantum nfomatonal dstancs and th cost constucton mthod. Fo any st of sz n quantum stats and fo any fnt æ log n log log n ç ö è constuctd n tm ( ( ø Í, th s a wa cost of sz. Ths -wa cost of quantum stats can b ( n æ ö log( n log log + dn ç, wh s th numb of è ø quantum stats n st OUT, n s th numb of nput stats, and d s th dmnson of th ponts. Th pvous sult can b ntgatd nto ou analyss as follows. Usng m -smla quantum nfomatonal dstancs and th -wa cost of quantum stats, th supactvaton of quantum channls can b analyzd by an ( + æ ö n a un tm d og + l n + dn ç. çè ø -appoxmaton algothm
Usng th sult of Banj t al. [3], th optmal -mdan of any gvn nput st n quantum spac can b unquly dfnd by th cntod c = å. Usng th fact that an Î optmal soluton of th -mdan clustng poblm can b appoachd by ( - lnaly spaabl substs, t can b shown that, fo any st, most n stats hav to b consdd optmal -mdan quantum stats of [,5]. W us a small st fom, whch s a d small wghtd st that has th sam clustng bhavo as th lag nput st. Th cost mthod usd n ou appoach can b dfnd by th o of th appoxmaton n tms of th quantum nfomatonal dstanc btwn quantum stats as follows [0]: o (, = å w ( D (, (9 w OUT OUT Î o and ths o s a ( -appoxmaton of (, fo any st of quantum stats of sz =. Fo th wa cost constucton, lt us assum that w hav a st OUT OUT OUT of quantum stats and a st. If th wght functon s dfnd by: å w = (, (30 thn th wghtd st s a -wa cost of, ff fo all OUT Î of sz OUT = ; w hav: (, - (, ( ( o o o,. OUT w OUT OUT (3 Ths -wa cost s calld th (, wa-cost of. To gt ths constucton wth ths o bound, w us th sults of Chn [6]. 5.. Cost Mthod fo Quantum Infomatonal Dstancs To apply th modfd cost mthod, w hav to constuct a éab, ù ë û bcta appoxmaton algothm to gt th st of mdan quantum stats M = { s s s } (,, of a -mdan clustng of, fo whch o, M a opt and M = b. Usng th ( sults of [,3] th bcta algothm can b summazd as follows [0]:
Bcta algothm to channl analyss. Choos an ntal quantum stat s unfomly at andom fom. Lt M b th st of chosn quantum stats fom. Stat Î s chosn wth ( D M pobablty as th nxt stat of M. o (, M 3. Rpat stp untl M contans quantum stats. At th nd of th bcta algothm, w hav a st of mdan quantum stats { } ( ( M = s, s, s, fo whch o, M aopt and M = b. Aft applcaton of th bcta algothm, w us th modfd cost constucton mthod psntd n [6] fo th quantum stats as follows [0]: Cost algothm to quantum channl analyss. Patton nto,, by assumng ach quantum stat Î to th closst s Î M. (. Lt Î ff s = ag mn D s. sîm 3. Lt R= o (, M. an 4. Dfn quantum nfomatonal ball ( s = D( x s. { j } of j, = Ç ( s fo =,,,. = Ç( j ( s j- ( s j ( s wth adus and cnt s as follows: 5. Dfn th patton of by j 6. Lt \ fo =,,, and j = =,, g, wh g = élog ( an ù ê ú. 7. Fo j, lt ba unfom st fom of sz = m. j j 8. Lt w ( = j b th wght assocatd wth Î j. m 9. Dfn th wa cost of nput quantum stats as follows: = j of sz = mg = mb élog ( an ù ê ú. j, Th algothm has appoxmaton o ( ; howv, th un tm of th poposd mthod s mo ffcnt snc t uss th -wa cost of st gnatd by Chn s algothm nstad of th ognal nput st. + 5.3 Dtmnaton of Mdan-Quantum Stats Th clustng mthod of [] fo a wa st of quantum stats and m -smla quantum nfomatonal dstancs can b summazd as follows [0]:
CLUSTER : Clustng of channl output stats. Lt b th st of manng nput stats, wth w (. Lt w b th wght functon on nput quantum stats = n 3. Lt m b th numb of mdan-quantum stats yt to b found 4. Lt C b th st of mdans alady found 5. f m = 0 thn tun C 6. ls 7. f m ³ thn tun C È 8. ls 9. Sampl a multst of quantum stats 96 of sz fom md 0. T Lt c th wghtd cntod of Í, 3 wth = md. fo all c Î T do. C ( c CLUSTER, wm, -, CÈ c 3. nd fo ( { } 4. Patton of th st of nput quantum stats nto st N and \ N such that: ( ( 5. " Î N, s Î \ N : D C D s C and 6. n w( N = w( \ N = 7. Lt w th nw wght functon on \ N ( N q m C 8. Lt C CLUSTER \,,, 9. C ( c tun o C wth mnmum o 0. nd f. nd f In Fg., w llustat th clustng of channl output stats. In th clustng pocss, ou algothm computs th mdan-quantum stats dnotd by s usng a fast wa cost and clustng algothm. In th nxt stp, w comput th convx hull of th mdan quantum stats and, fom th convx hull, th adus of th smallst quantum nfomatonal ball can b obtand. Th smallst supball masus th channl capacty; hnc, th adus of th supball s qual to th sum of th ad of th quantum balls of ndpndnt channl outputs. Th output stats a masud by a jont masumnt sttng.
Fg.. Clustng of quantum stats wth th computd mdan quantum stats. To summaz, ou quantum channl supactvaton mthod combns th wa cost mthod of Chn and th clustng algothm psntd by Acmann t al. []. To us Chn s mthod [6,7] to constuct th wa cost, w apply a bcta algothm [5] to fnd th qud paamts, and thn w apply th sult of Acmann t al. [] n quantum spac. In both mthods, w us quantum nfomatonal dstanc functons as dstanc masus. 6. Illustatv Exampl In ths scton, w show that ou mthod can b usd to confm th sults of [5,53] and that th mthod can b xtndd to a lag st of possbl channls. Smth and Yad showd that any pvat oodc channl can b combnd wth oth symmtc channls; hnc, two zo-capacty quantum channls can b combnd to alz a capacty gat than 0.0 [5]. Th tm supactvaton dscbs ths ffct, whch cannot b magnd fo classcal systms,.., t has no analogu n classcal systms. Two zo-capacty quantum communcaton channls can b usd to tansmt nfomaton, and t s possbl to actvat on channl wth th oth channl so that th zo-capacty of th channls can b ncasd. At psnt, w hav no thotcal bacgound to dscb all possbl combnatons; hnc, th should b many oth possbl combnatons of supactv zo-capacty quantum channls. Ou mthod povds a fundamntally nw and ffcnt algothmc soluton to dscov all possbl supactvatd channls, and t can b appld to analyz th capacty of ths channls. Smth and Yad had lft opn th quston as to whth how many dffnt
channl modls can b usd fo supactvaton [5]. W wll also us th convxty of quantum channl capacty. Th fact that quantum capacty s not a convx functon of th channl can b usd n ou computatons. Th a zo-capacty quantum channls that can b combnd to achv hgh capacty than th avag capacty of th ndvdual quantum channls; hnc, fo ths channls, convxty dos not hold. W dv th jont channl capacts fom th ndpndnt channl capacts, and w ntoduc a nw psntaton, th supball psntaton, whch was usd fo channl addtvty analyss by Gyongyos and Im n []. In ths pap, w dscb only th clustng pocss of channl output quantum stats to constuct mdan-quantum stats. Th mthod of convx hull calculaton fom ths mdan-quantum stats s basd on quantum Dlaunay tssllaton, a mthod poposd by Gyongyos and Im n []. In th ntoducd supball psntaton, th lngth of th supball adus s qual to th sum of th ad of th quantum nfomatonal balls of th ndpndnt channl capacts, masud n a jont masumnt sttng. 6. Confmaton of Rsult fo Supactvaton In [5] th authos usd a latonshp btwn classcal pvat and quantum channl capacts that could hold fo any quantum channl. Thy consdd a quantum channl ( ( fo whch I ( X : B - I ( X : E P, wh ( ( P s th pvat capacty of channl. Accodng to ou mthod, th channl capacty s masud by th adus of th smallst quantum nfomatonal ball; w us th followng quaton to dscb th pvat capacty of supactvatd zo-capacty quantum channls: wh pvat ( ( ( ( = max I X : B - I X : E = P, (3 X pvat masus th sngl us classcal pvat capacty of channl. Evy oodc channl satsfs th laton P ( > 0 ; hnc, th s an pvat nput fo whch > 0. Th combnaton of a oodc channl and a 50%-asu channl -channl can sult n th followng supball adus: ( = Q ( ( ( ( : ( : Ä = I X B I X E P ( - = = pvat, (33
: whl fo th supactvatd asymptotc jont quantum capacty wth adus ( Ä ( ( ( pvat Q Ä = P (. Ä ³ = = (34 W show that ou mthod fnds that supactvatd channl combnaton, wth appoxmaton o. Th authos of [5] dfnd a pvat oodc channl such that pvat = 0.0, whos combnaton wth has a channl capacty ( ( 0.0 0.0 Ä > =. Fo th dfnd fou-dmnsonal oodc channl, th followng quaton gvs th pvat channl capacty: pvat ( ( ³ -qlogq - -q log - q > 0.0 (35 wh q =, a paamt usd n th Kaus psntaton of th channl [5]. +, w hav usd th fact that th map of any quantum channl can b wttn n ( Kaus fom as = å N N, wh N dnots th Kaus matcs, wth å NN = I. In th cas of a oodc channl, th channl can b spcfd by sx Kaus matcs [39,5]. W show that th sults of [5] fo th supactvaton of a oodc channl and an asu channl, can b found and confmd by ou fundamntally nw gomtc appoach, wth appoxmaton o. As w conclud, ou mthod can b xtndd to oth possbl channl modls and combnatons of oth channls and channl pobablts. To dscb gomtcally th supactvaton of zo-capacty quantum channls, w ntoduc th channl modl and channl paamt p as follows: ( p = p Ä 0 0 + - Ä, (36 wh 0 p. Th dfnd channl modl s th convx combnaton of two zocapacty channls Ä 0 0 and Ä. Lt us assum that w us ths two quantum channls and th poduct channl psntaton Ä. Th man goal of ou
gomtc analyss s to fnd a channl pobablty paamt p fo whch th jont capacty of th tnso poduct channl Ä s gat than zo. Th channl constucton tchnqu fo th supactvaton of zo-capacty channls s llustatd n Fg.. To supactvat zo-capacty quantum channls, w must us th convx combnaton of dffnt channl modls and th pobablstc mxtus of ths channls to alz supactvaton. p Zo-capacty channl -p Q p Zo-capacty channl -p Fg.. Channl constucton fo th supactvaton of zo channls. In ou taton pocss, w sach fo th optmal channl pobablty paamt. In ths cas, w us th channl modl of [5]; hnc, w hav fxd channl modls, and w hav to tat on paamt p only. Th jont channl quantum capacty Q ( Ä as adus sup ( Ä n functon of channl pobablty p can b dscbd as follows: ( ( ( ( ( ( + - p p ( Ä + - p ( Ä sup Ä = Ä + p - Ä p p. (37 - p Ä As shown n [5], th tm ( ( can b nglctd snc th quantum capacty of 0 ths combnaton s zo,.., ( Ä =. As follows, th channl modl s ducd to ( sup ( ( Ä ( ( Ä Q Ä = Ä = p + p - p, (38 and th adus of th smallst supballs can b dscbd as ( (, p ( Ä = pq Ä (39
o ( ( ( ( ( p - p Ä = p - p Q Ä, (40 wh 0 < p <. w not, th notaton Ä mans usng th jont channl constucton Ä two-tms, whch sults n dffnt supactvatd asymptotc jont quantum capacts at th channl outputs. nc, fo ths channl constucton, w obtan adus lngth ( Ä wth wght p and w obtan supball adus ( Ä wth wght p( - p n ( Ä. But: usng channl combnaton and, th sup tm p ( Ä can nv b gat than zo, bcaus th quantum capacty of th oodc channl s zo [5], ( ( Q = = 0,.., adus ( Ä wll always hav zo lngth. Now, w focus on supball adus ( Ä. Th adus ( Ä can b xpssd as follows ( ( Ä ³ +, (4 wh th ad and of th smallst nclosng quantum nfomatonal balls masu th sngl-us pvat classcal capacts of th channls. As follows, n (4, th and psnt th pvat classcal capacty of th channls and, wh P > and ( 0 ( 0 P = (, nstad of th quantum capacts Q and Q of th ndvdual ( channls and. Th adus ( Ä s qual to zo fo channl paamts outsd th doman p Ï [0, 0.004]. In Fg. 3, w show th smallst quantum nfomatonal balls n th ang 0 < p < 0.004. In ths cas, th channls hav postv supactvatd quantum capacty,.., ( ( ( pvat 0 < ( Ä ³ + = P =. (4
wh pvat s th sngl-us pvat classcal capacty of th oodc channl. Th channls and wth zo quantum capacts ndvdually can b supactvatd and a postv capacty can b alzd on th output of th channls. Fg. 3. Th smallst nclosng balls and ad fo oodc channl (a and fo asu channl (b n th sngl channl vw. If th zo-capacty channls a supactv, th jont capacty wll b postv n th gvn channl paamt doman 0 < p < 0.004. (Th ad psnt th supactvatd quantum capacty of th jont stuctu, usng sngl channl vw psntaton. In Fg. 4, w show th smallst nclosng quantum nfomatonal balls and th ad of two zo-capacty channls, and fo channl pobablts p = 0 and p ³ 0.004. Th ad ( and ( a qual to zo fo channl paamts outsd th doman 0 < p < 0.004. Th ad ( and ( xpss th quantum capacts of th ndvdual ( channls and, Q = Q =,..: ( 0 sup ( Ä = ( 0 ( Ä = + = ¹ +. (43 Fg. 4. Outputs of two zo-capacty quantum channls (n sngl channl vw. Th ad of th smallst quantum nfomatonal balls a qual to zo. (Th ad psnt th quantum capacty of th jont stuctu, usng sngl channl vw psntaton.
s Th sults of th supactvaton as a functon of dffnt p pobablts, wh ( Ä th adus of th supball, whch dscbs th jont capacty of th jont stuctu ( Ä a shown n Fg. 5. Fg. 5. Th output of th optmzaton algothm dscbs th adus of th supball, whch wll b postv only fo a gvn doman of th channl paamt. As can b obsvd, th lngth of th adus of th quantum nfomatonal supball s ( Ä = 0.0 fo channl paamts n th doman 0 < p < 0.004. In Fg. 6, w show w show th lngth of supball adus sup ( Ä, whch dscbs Q ( Ä, s (38. As w hav convx combnatons of channls, th supball adus sup ( Ä wll b masud as ( 0 Ä = wth zo wght, and ( Ä = 0.0 ( wth wght p - p, whch lad to sup ( Ä = p( - p ( Ä = p( - p ( 0.0. W hav compad th wghts of th ad as a functon of channl paamt. Fg. 6. Th lngth of supball adus ( Ä as functon of th channl paamt. sup
It can b concludd fom ou sults fo a channl paamt n th doman 0 < p < 0. that th output of th algothm wll sult n a channl capacty ( Ä = 0. 0. W hav usd th sub-doman 0 < p < 0. of paamt p snc th ctcal valu s 0 < p < 0.004, as found by ou algothm. 0 Th lngth of th fst supball adus s ( Ä = ; howv, ths output has zo wght. Th scond supball adus ( Ä has a lngth of 0.0; ths output has a much hgh wght, 0.008, fo fxd channl paamt p = 0.004, whch s th upp bound of th possbl ang 0 < p < 0.004. Th maxmum wght of th ( Ä can b obtand fo channl pobablty p = 0.004. W can conclud fom ou numcal analyss that, f w hav two zo-capacty channls Ä, thn th convx combnaton of ths channls can sult n gat than zo capacty fo a small subst of possbl paamts p. As ou gomtcal analyss vald, f w hav two fxd channl modls and w tat on possbl valus of paamt p, thn, fom th adus of th smallst supball, w can dtmn th possbl valus of th supactvaton paamt. W post that stong combnatons fo supactvaton can b constuctd fom th lag st of quantum channl modls and possbl paamts. 7. Conclusons Ths pap xhbtd a fundamntally nw algothmc soluton fo th supactvaton of th asymptotc quantum capacty of zo-capacty quantum channls. Usng ou mthod, a lag st of supactv zo-capacty channls can b dscovd vy ffcntly, and ou mthod can bdg th gap btwn thotcal and xpmntal sults. To analyz channl supactvaton, w ntoducd th supball psntaton, wh th adus of th smallst quantum supball s qual to th sum of th ad of th smallst quantum balls of th analyzd quantum channls. Th tatons a basd on th computd adus of th supball; th tatons a xcutd on channl nput stats, channl modls, and o pobablts. W hav shown that th poposd mthod s an ffcnt xpmntal algothmc alzaton of
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