Is A Quantum Stabilizer Code Degenerate or Nondegenerate for Pauli Channel?

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1 I A uu Szer Code Degeere or Nodegeere for Pu Che? Fgyg o wu Che Abrc g error ydroe o he error oeror he core of quu decodg ework d o he key e of recoery he defo of he b-f error ydroe rx d he he-f error ydroe rx were reeed d he he error ydroe of quu error were exreed er of he cou of he b-f error ydroe rx d he he-f error ydroe rx I o howed h he error ydroe rce of zer code re deered by check rx whch r o he cc ce So he error-deeco d recoery echque of cc er code c be ed o quu zer code fer oe odfco Soe ecery d/or uffce codo for he zer code oer GF( degeere or odegeere for Pu che bed o he reoh bewee he error ydroe rce d he check rx w reeed A ew wy o fd he u dce of he quu zer code bed o her check rce w reeed d foowed fro whch we roed h he erforce of degeere quu code ouerfor ( e he he e erforce odegeere quu code for Pu che Idex er zer code CSS code degeere code Pu che I INRODUCION uu foro c be roeced by ecodg o quu error-correcg code uu error-correcg code re que r o cc code y reec for exe error defed by eurg he error ydroe d he correced rore u he cc ce oweer here ereg c of quu code kow degeere code [][][3][4] oeg rg roery ukow cc code For cc error-correcg code error o dffere b ecery ed o dffere corrued codeword Bu for degeere quu code he error ydroe o uque d error ydroe re oy reeed whe F beog o he zer S yg h d F c he e wy o he codeword So hey c oee be ued o correc ore error h hey c defy he heoeo of degeere quu code or of good ew-bd ew uo for quu code he bd ew h oe of he roof echque ued ccy o roe boud o error-correco f dow becue hey c be ed o degeere quu code for exe he quu g boud he good ew h degeere quu code ee o be og he o ereg quu code hey re kow o ouerfor odegeere quu code for ery oy quu che (for exe Pu che[5][6][7] d he or co urfyg quu e[8] d rog he ecury of quu couco rooco[9]]0] I oe h hey re e o ore quu foro ore effcey h y odegeere code becue dc error do o ecery o Fgyg d Che wu re wh he Schoo of Couer Scece d geerg Souhe Uery Ng Jgu 0096 Ch he o ke he code ce o orhogo ce he decodg ework of quu code co hree r ey error deeco error correco decodg g error ydroe o error oeror he core of quu decodg ework d o he key e o reze quu error correco I he hrd eco we reeed he defo of he b-f error ydroe rx d he he-f error ydroe rx d he exreed he error ydroe of quu error er of he cou of he b-f error ydroe rx d he he-f error ydroe rx I he fourh eco howed h he error ydroe rce of zer code re deered by check rx whch r o he cc ce So he error deeco d correco echque of cc er code c be ed o quu zer code fer oe odfco U ow o echque h bee deeoed o deere wheher or o zer code degeere or odegeere oher h exhue erch I he ffh eco bed o he reoh bewee he error ydroe rce d he check rx of quu zer code oe ecery d/or uffce codo for he zer code oer GF( degeere or odegeere were reeed for Pu che A ew wy o fd he u dce of quu zer code bed o her check rce w o reeed d foowed fro whch we roed h he erforce of degeere quu code re ouerfor ( e he he e erforce odegeere quu code for Pu che We hoe h hee reu w hefu for deg quu degeere code [][] wh good erforce II PRLIINARY he couor d -couor[] of he oeror A d he oeror B re [AB] AB-BA d {A B} AB+BA reecey If [A B] 0 he he oeror A d he oeror B re d o coue If {A B} 0 he A d B re d o coue Four exreey uefu rce ced Pu rce wh her correodg oo re decrbed foowg: I Y fro whch we ee h [ Y] [Y ] [ ] Y he Pu grou G o qub he rx grou cog of he Pu rce I Y ogeher wh uce fcor ± ± e G { ± I ± I ± ± ± Y ± Y ± ± } he Pu grou G o -qub he grou geered by he oeror decrbed boe ed o ech of qub he eor roduc ber ce C

2 c g { I Y} c { 03} g g g g G g where I foow fro he defo of he Pu grou h f AB G he [ AB ] 0or{ AB } 0 Sce Y o y eee of c G c be deoed ( b g A hooorh fro G oo : G F where b F F defed ϕ ( ϕ g α β ϕ g ϕ g where ( h ( ( ( ( g c G ( b { 0 3} c α ( d β ( b b b F I foow fro he boe defo h he Pu rce I Y c be ed o he foowg bry ecor: ϕ( I (0 0 ϕ( ( 0 ϕ( (0 ϕ( Y ( A oorh fro F oo G defe : ϕ : F G G G ( h ϕ ( α β b ( ( { ± ± } For exe ϕ ( IYY ( d ϕ ( IYY Le g G uch h ( ϕ( g ϕ( g ϕ( g he he quu wegh of g wre w ( g he uber of cooe h o I h w ( g w( ϕ( g + w( ϕ( g w( ϕ( g ϕ( g where w(u he g wegh of u d he er roduc he yecc wegh of ecor ( where F wre w ( w ( w ( + w ( w ( A reu w( g w( ϕ ( g Le { } re he geeror of he zer S k where G ( k ( k rx oer GF( whoe row co he ecor ϕ( ϕ ( ϕ ( k h ϕ ( [ ] (3 ϕ ( k ced he check rx[] of S III RROR SYNDRO Suoe S k he zer of quu zer code CS ( Le he ecoded e φ w uffered fro oe ferece deoed G durg rg o oy che whch red he ecoded e o φ φ (y he error deeco c be doe by y eure he geeror of he zer how Fgure [] I w ge u of egeue he error ydroe whch w e u wheher he error coue or coue wh ech of he geeror ; f Fro fgure we ee h f [ ] { } 0 he he ecor ( k F 0 he 0 k reree he φ φ Fg Sydroe euree error ydroe of he error G where f 0 he 0 ; f{ } 0 he where k ( ( he oeror I I deoe he b-f error o ( ( he -h qub wre e I I he he error ydroe of ( k k F Se he row of o be he he ( k rx k k k k F ( ced he b-f error ydroe rx(bs ore geery f he b-f error he o ue qub c wre where c { } 0 3 F Suoe [ ] F f he b-f error he o he -h qub; f 0 he o error he o he -h qub Sce } k

3 3 Le c ( c ( ( u u u c ( ( u u u ( F herefore he error ydroe of ( u he u u c k (4 I foow fro q (4 h he error ydroe of y c error c be exreed er cobo of he error ydroe of ( ( ( he oeror I I ( deoe he he-f error o he -h qub Sry he rx k k ( k F k where f 0 he 0 0 he k ced he d f { } he-f error ydroe rx (PS If he he-f error he o ue qub h c where b { 03} F c Le b [ b b b ] F Sry he error ydroe of c k b b b b Fro whch we ee h he error ydroe of y error c c be exreed er cobo of of ( he error ydroe ( ( Suoe Y I Y I ( whch he ce of h wo kd of error (b-f d he-f heed coeorry o he -h qub Sce Y o Y (5 G I foow fro q (5 h rbrry error P c be exreed cp P where F cp { 0 3} Le [ ] d b [ b b b ] F uch h cp P b cp ( b + cp ( b + ( P he + b P cp P k k ( P k k b b b hereforehe error ydroe of b b b ( ( + (6 + b P I foow fro q (6 h he error ydroe of y error P G c be exreed er cobo of he error ydroe of of ( d he error ydroe ( Rerky ue error ydroe euree woud coe rbrry error (cudg cohere ueroo of b-f d he-f error o he dcree e of oy b-f d/or he-f error becue c be exreed ueroo of b oero he error b (whch here ge by he Pu rce Ad hee dcree Pu error c be ey reered o recoer he org e So oy h o fd ou he error ydroe of he b-f error ( d he he-f error ( o -qub d he he error ydroe of rbrry error c be exreed er of he cou of he b-f error ydroe rx d he he-f error ydroe rx ccordg o q (6 Fro reou y we kew h he error he

4 4 e { } ± ± c be ed o he e error k F ydroe ce fro obero o of ew he quu e corrued by he error wh dffere gob he fcor re dec For h reo we y gore he gob he fcor beg rree o he obered roere of he hyc ye d hg o ffec o he recoery I he re of he rce we w oy coder he error G IV ANALY RROR SYNDRO OF SABILIR CODS Suoe CS ( zer code wh S k whoe check rx zer ( k Wheher b-f error coue wh or coue wh deede o wheher or o he -h cooe of or Y If 0 or Y he{ } oherwe 0 I foow fro q (3 h he ydroe of he b-f error he -h qub ( correod o he +-h cou of whch o he -h cou of he ub-rx Sry he ydroe of he he-f error he -h qub ( o he -h cou of he ub-rx correod o he -h cou of whch h ϕ ( + + ( ( ϕ ( ( ( (7 (8 where ϕ ( he -h cooe of he -deo bry ecor ϕ ( d he eee row cou of rx k I foow fro (7 d (8 h for zer code CS ( he error ydroe of d ( ( + d where Le k k re ( ( where ( he -h cou of ( he -h cou of he ( ( d he foowg heore ow foow ey fer he dcuo boe heore If he check rx of zer code CS ( ( he d herefore he error ydroe of rbrry error G c be exreed er of he cou of he check rx e where ( ϕ ( ϕ( Λ ϕ( ( ϕ( + ϕ( 0 I Λ I 0 d ϕ( ϕ( ϕ( I h coe o gh h he error-deeco d recoery of he cc code re oy deede o her ry check rce heore ke coeco bewee he error ydroe rce (BS d PS d he check rx of he zer code hereby exreg he error ydroe ery cobo of he cou of BS d PS rfored o exreg ery cobo of he cou of he check rx So he error-deeco d recoery of he quu zer code re o oy deede o her check rce whch r o he cc ce herefore he error-deeco d recoery chee d echque of he cc code c be ed o quu zer code ey V IS A SABILIR COD DGNRA OR NONDGNRA? I h eco we decrbe oe ecery d/or uffce codo for zer code oer GF( degeere or odegeere bed o check rx for Pu che whch c o be ed o deorzo che Srcy ekg degeercy o roery of quu code oe bu roery of code ogeher wh fy of error deged o correc [5] ere we w how defo of degeere d odegeere zer code for Pu che Le G g G w ( g coder g fro G o F : h for g G [ ] { } k f : G F f ( g F ( k f { } 0 g 0 he 0 k If g he If f ogur we y h he zer code wh reer k + d k + re odegeere oherwe hey re degeere I he re of h eco we oy dcu he zer code wh reer k + d he cocuo c o be ed o zer code wh reer k + ery Le If he e of ecor { } deede he he u of y e of ecor of re dffere fro ech oher Proof: uoe he ecor d he ecor where

5 5 { } { } < fy c c wh k k k c d c c { 0} he we he c k + cb 0 k k ery deede herefore we roduce cordco hu f o be be check rx uch h y heore Le [ ] ( k 4 cou ery deede he he zer code C(S[[k+]] wh check rx[ ] odegeere Proof: Sce C(S odegeere f (d oy f for rbrry G d hey fy We w erfor wo e order o roe he heore ( Le u ue before h ( ( ue furher h cou of he u of he h h h h h h cou of d he b k + b Sce y 4 F h cou of [ ] re ery deede foow fro e h f d fy dffere wh reec o dffere e{ } d{ } ( Le k + he we w ge ( ({ } { } u F { } d { } u u u F where u d re uor of u d u reecey herefore ( k ( u u F ( ( b + b u + I foow fro q ( h he oeror correodg o u k ϕ ( u So he error ydroe of u u k b k b ( If he e { } d { } foowg codo: 0 + d { } { } fy he 0 wu ( wu ( d wu ( u w ( ( ( ( wu + wu wuu he herefore I foow fro q ( h f we ck dffere cou fro [ ] whch correod o dffere e { } d { } error G he hey w correod o dffere Fy cobg ( d ( we ge h he zer code C(S odegeere Noe h he reere of heore geery o rue h here woud ex e of 4 ery deede cou of [ ] ( ee f he zer code wh reer k [[k+]] odegeere Becue f he 4 ery deede cou fy he foowg codo: d { } { } < { } { } < he e oe of he wo correodg error h he he e error ydroe h quu wegh ger h whch doe o oe he defo of odegeere code We he he foowg corory Corory f he zer code C(S[[k+]] odegeere he y cou of check rx [ ] deede Proof: Suoe he re ery h h h cou of d he h h h cou of re ery deede where d { } d hey fy he codo 0 d { } { } + he b b Pck ube{ k } k k (9 + 0 fro { } d ube { } fro { } b eee where +b Le b b c kd c d wh eee wh b c { } { } d { } { k k k } b c d he w( b ( b b w I foow fro q (9 h b d b oe he e error ydroe herefore we roduce cordco he heore d corory re he uffce codo d ecery codo for zer code o be odegeere

6 6 reecey We he he foowg heore h decrbe ecery d uffce codo for zer code o be A odegeere 0 he for y eee A heore 3 A zer code C[[k+]] odegeere f d oy f he uber of eee he e we c fd uque r of d uch 0 0 h b ( b + So he error ed o he b { } + b error ydroe ϕ (({ 0 < { } { } } { } Becue he e A co he ecor h correodg o r 3 or 0 of d whch fy w( ({ } ({ } hereby he Proof: ( For Pu che zer code C odegeere f (d oy f he dffere error correce error e coeco of he error h correodg o r of G { I} re ed o dffere error ydroe Sce he uber of eee G { I} G {} I Le rbrry oeror of G { I} ( ( u ( uu u 3 d uoe ϕ wh u u u d where { } he d fy he foowg codo 0 0 < { } { } hu {} ϕ ( G I u ( he error ydroe of ydroe of he error G { I} 0 0 { } ( u ({ }{ } w (( u u + he he error re 0 0 b A ( + { } b 0 < { } { } he uber of eee A fe A (0 0 If he dffere error G { I} ydroe he G I A re ed o dffere error {} 0 (b Coerey foow fro q (0 h f he uber of eee G { I} d Fy C odegeere code By heore we he he foowg corory Corory Le be check rx uch h y 4 cou re ery deede bu here ex e of 4 ery deede cou he he zer code C(S defed by h he u d + If C(S odegeere he u dce dce fe + d 4+ Proof: he u dce of C(S he quu wegh of he eee N(S-S wh he u quu wegh where N(S he orzer of S Whou o of geery uoe he h h h cou of d he h h h cou of re ery deede e b b he we he d + 4+ Le u ue before h ϕ (({ } { } he NS ( d w w + (( ( { } { } Le u coder he ce wh + 4+: he Ce (: If { } { } ( w + ; < he f { } { } ( w > + Ce (: If { } { } ( 4 w + ; 0 he > 0 he f { } { } ( 4 w < +

7 7 I foow fro ce ( h w ( he d + If C(S odegeere he > + he we { w e e N S S} { w e e S} ( ( ( Fro whch we ee h d 4 we ge + d 4+ + Cobg ce ( d ce ( I ey o ee h oe equece of eeery row rforo d cou eruo w rfor he check rx o he foowg drd for [3]: r -k- r k r -k- r k r{ I A A B 0 C R R R r k { D I Fro whch we ee h he zer codec wh check rx R oorhc o he zer code C k rx herefore R + wh check C odegeere f (d oy f C R odegeere hu o deore h C R odegeere oy h o how h C odegeere heore 4 If r - k d here ex cou of B wh g wegh e h or equ o - he CR degeere Proof: Foow fro heore CSS code re ec kd of zer code wh he check rx of he foowg for: A 0 0 B I ey o ee h oe of he cou of c be exreed er cobo of he cou of herefore by heore we he he foowg corory Corory 3 he CSS code C [[k+]] odegeere f d oy f y cou of d re ery deede reecey Proof: he codo cery uffce he we w roe h ecery Whou o of geery uoe he re h h h h h h cou of b + 0 ery deede he we he b Le he error d be defe d reecey k k hu ( ( k w w I ey o ee h d b b k hu d he he e error ydroe herefore we roduce cordco Corory 4 Le be check rx uch h y cou re ery deede bu here ex e of ery deede cou he he CSS code C defed by h he u dce d + If C odegeere he u dce d + Proof: Our corory foow fro heore d corory 3 he boe ero r o he cc ce h he u dce of cc er code deered by ry check rx ce h he CSS code deec d correc quu error by kg ue of he error-correcg roere of he cc code We kow fro corory d corory 4 h he degeere quu zer code ouerfor he odegeere quu zer code whch hd roed referece [5] VI CONCLUSION We he reeed oe ecery d/or uffce codo for zer code oer GF( degeere or odegeere for Pu che I woud be ereg o ege he ecery d/or uffce codo for zer code oer GF(q degeere or odegeere I woud be o ereg o ege he ecery d/or uffce codo for zer code oer GF(q degeere or odegeere for oher quu che I redy roed h CSS code wh hbe ze q>4 where q re ower u obey he g boud [3] Soe ec ce of ere re zer code d CSS code oer ore ower hbe d hbe q <5 We hoe our work w hefu for og h roe d for deg quu degeere code wh good erforce RFRNCS [] ANeedLChug uu Couo d uu Iforo Cbrdge Br: Cbrdge Uery Pre 000 [] D Goe C of uu rror-correcg Code Surg he uu g Boud Phy Re A o 54 o3 86~868 Se 996 [3] D Goe Szer Code d uu rror Correco PhD dero Cfor Iue of echoogy Pde CA 997 [4] Cderbk A R R Shor P W e uu error correco code oer GF(4 I r If heory o 44 o Ju 998 [5] G Sh J A So Degeere uu Code for Pu Che Phy Re Le o 98 o J 007 [6] N J Cerf Pu Cog of uu B Phy Re Le o 84 o y 000 [7] D P DVcezo P W Shor J A So uu-che ccy of ery oy che Phy Re A o 57 o Feb 998 [8] K o F Chu Purfyg Greeberger-ore-eger e ug degeere quu code Phy Re A o 78 o Oc 008 [9] o-kwog Lo Proof of ucodo ecury of x-e quu key drbuo chee uu Iforo d Couo o o 8-94 Ar 00 [0] P W Shor J Prek Se Proof of Secury of he BB84 uu Key Drbuo Prooco Phy Re Le o85 o Ju 000 [] R Dko P ro New u-wed Degeere erry Ler Code I r If heory o 49 o Se 003 [] S A Ay A Keecker P K Sre Rerke Degeere uu Szer Code Dered fro Dudc Code quh/o607 J 006 [3] P Sre A Keecker Degeere quu code d he quu g boud Phy Re A o 8 o r 00