Content-Type: multipart/mixed; boundary="-------------0203041341850" This is a multi-part message in MIME format. ---------------0203041341850 Content-Type: text/plain; name="02-99.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-99.comments" Postscript fiel, 18 pages, 1 fig ---------------0203041341850 Content-Type: text/plain; name="02-99.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-99.keywords" Thermodynamics, Statistical mechanics, space of thermo-equlibrium, quantum information, Claude Shannon ---------------0203041341850 Content-Type: application/postscript; name="qph2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="qph2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: qph2.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o qph2.ps qph2.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.03.04:1430 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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b(Ho)m(w)m(ev)m(er,)47 b(for)41 b Fo(T)57 b(>)44 b Fx(0,)g(the)e(space) h(of)f(thermo)s(dynamic)e(equilibrium)0 3114 y(has)e(an)f(exp)s(onen)m (tially)f(large)g(dimension)g(\(see)j(the)e(line)f(preceding)i (equation)f(\(17\)\).)57 b(Our)37 b(attitude)0 3289 y(is)31 b(to)g(in)m(terpret)h(the)g(en)m(tire)g(space)h(of)e (thermo-equilibrium)c(as)32 b(a)g(kind)f(of)g('high)g(dimensional)e (ground)0 3463 y(state'.)42 b(It)25 b(turns)h(out)g(that)f(an)m(y)h (unit)f(v)m(ector)h(c)m(hosen)h(from)d(the)i(thermo-equilibrium)c (space)k(will)d(serv)m(e,)0 3637 y(equally)37 b(w)m(ell,)i(to)e (represen)m(t)j(the)e(ph)m(ysical)g(prop)s(erties)g(of)f(the)h(en)m (tire)g(equilibrium.)57 b(Here,)39 b(ph)m(ysical)0 3812 y(prop)s(erties)34 b(are)g(things)f(lik)m(e)h(energy)-8 b(,)35 b(momen)m(tum,)e(scattering)h(matrix,)e(and,)j(most)e(imp)s (ortan)m(tly)f(for)0 3986 y(our)f(purp)s(oses,)h(correlation)d (functions.)43 b(A)31 b(correlation)e(function)h(is)h(a)f(ph)m(ysical)h (quan)m(tit)m(y)h(asso)s(ciated)0 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b(emphasize)h(that)f(b)m(y)h(c)m (ho)s(osing)f Fo(C)35 b Fx(to)28 b(b)s(e)g(the)h(space)h(of)d(thermo)s (dynamic)g(equilibrium)f(and)i(the)0 5525 y(op)s(erator)c Fn(O)31 b Fx(=)c Fo(E)676 5489 y Fw(y)670 5550 y Fm(a)712 5525 y Fo(E)784 5540 y Fm(b)818 5525 y Fx(,)g(the)d(thermo)s(dynamic)f (correlation)g(function)h(condition)f(sa)m(ys)j(that)e Fn(h)p Fo( )t Fn(j)p Fo(E)3610 5489 y Fw(y)3604 5550 y Fm(a)3645 5525 y Fo(E)3717 5540 y Fm(b)3752 5525 y Fn(j)p Fo( )t Fn(i)1870 5774 y Fx(4)p eop %%Page: 5 5 5 4 bop 0 555 a Fx(has)30 b(the)g(same)f(v)-5 b(alue)29 b(for)g(all)e(unit)i(v)m(ectors)i Fo( )g Fn(2)d Fo(C)7 b Fx(.)43 b(Th)m(us,)31 b(for)e(the)h(subspace)h(of)e (thermo-equilibrium)0 730 y Fo(C)7 b Fx(,)33 b(the)g(quan)m(tum)f (error)h(correcting)f(condition)f(is)h(ac)m(hiev)m(ed.)125 904 y(The)j(sub)5 b(ject)37 b(of)d(thermo)s(dynamics)g(and)h(quan)m (tum)h(statistical)c(mec)m(hanics)k(is)e(treated)i(in)e(sev)m(eral)0 1079 y(textb)s(o)s(oks)e([11,)g(14)o(].)43 b(There)33 b(are)f(v)-5 b(arious)30 b(mo)s(dels)h(determined)g(b)m(y)h(di\013eren) m(t)g(Hamiltonians)c(describ-)0 1253 y(ing)40 b(di\013eren)m(t)h(in)m (teractions,)i(and)e(man)m(y)g(of)g(the)g(spin)g(c)m(hain)g(mo)s(dels)f (ha)m(v)m(e)i(relev)-5 b(ance)41 b(to)g(quan)m(tum)0 1427 y(information)e(theory)-8 b(.)75 b(W)-8 b(e)43 b(b)s(egin)e(our)i (w)m(ork)g(with)g(XX0)f(spin)h(c)m(hains,)i(whic)m(h)e(ha)m(v)m(e)h (already)f(b)s(een)0 1602 y(considered)37 b(in)e(the)i(con)m(text)g(of) f(quan)m(tum)g(computing,)g(though)g(for)g(purp)s(oses)h(di\013eren)m (t)g(than)f(ours.)0 1776 y(W.)c(W)-8 b(o)s(otters)31 b(suggested)i(that)e(quan)m(tum)g(spin)g(c)m(hains)h(b)s(e)f(used)i (for)e(the)g(study)i(of)e(en)m(tangled)g(qubits)0 1950 y([15],)h(and)h(this)f(in)m(v)m(estigation)g(con)m(tin)m(ues)h(\(see)h (for)e(example)g([16]\).)0 2320 y Fp(I)s(I.)104 b(AN)34 b(O)m(VER)-12 b(VIEW)35 b(OF)f(QUANTUM)h(ERR)m(OR)h(CORRECTION)125 2612 y Fx(In)30 b(this)h(section,)g(w)m(e)g(brie\015y)g(detail)e(the)i 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b(a)g(\014nite)g(dimensional)e(Hilb)s(ert)g (space)k(and)e(let)f Fo(C)43 b Fx(b)s(e)37 b(a)f(subspace)i(of)d Fo(W)14 b Fx(.)54 b(Consider)37 b(t)m(w)m(o)0 4032 y(collections)g(of)h (linear)e(transformations)h(of)h Fo(W)14 b Fx(,)39 b Fn(E)46 b Fx(=)37 b Fn(f)p Fo(E)2203 4047 y Fm(a)2282 4032 y Fx(:)h Fo(W)51 b Fn(!)37 b Fo(W)14 b Fn(g)2783 3996 y Fm(r)2783 4057 y(a)p Fj(=1)2952 4032 y Fx(and)38 b Fn(R)g Fx(=)g Fn(f)p Fo(R)3507 4047 y Fm(b)3578 4032 y Fx(:)g Fo(W)51 b Fn(!)0 4206 y Fo(W)46 b Fx(with)377 4132 y Fi(P)482 4158 y Fm(s)482 4235 y(b)p Fj(=1)623 4206 y Fo(R)698 4159 y Fw(y)697 4234 y Fm(b)734 4206 y Fo(R)808 4221 y Fm(b)888 4206 y Fx(=)g(Id)q Fn(g)1150 4170 y Fm(s)1150 4232 y(b)p Fj(=1)1274 4206 y Fx(.)76 b(The)44 b(co)s(de)g(space)h(is)d Fo(C)7 b Fx(,)47 b(the)c(quan)m(tum)h (medium)e(\(or)h(enco)s(ding)0 4381 y(space\))34 b(is)e Fo(W)46 b Fx(and)33 b(the)g(collections)e Fn(E)41 b Fx(and)32 b Fn(R)h Fx(represen)m(t)i(error)d(and)h(reco)m(v)m(ery)i(op)s (erations.)125 4555 y(Let)d(us)i(in)m(tro)s(duce)e(a)h(formal)d (mathematical)f(function)k(Err,)g(dep)s(ending)f(on)h Fo(C)7 b Fx(,)33 b Fn(R)p Fx(,)g(and)g Fn(E)9 b Fx(,)32 b(whic)m(h)0 4729 y(measures)h(the)g(exten)m(t)h(to)f(whic)m(h)g Fn(R)g Fx(can)g(correct)g(the)g(errors)g Fn(E)41 b Fx(on)32 b(the)h(co)s(de)g(space)h Fo(C)7 b Fx(:)701 5020 y(Err\()p Fo(C)r(;)17 b Fn(RE)9 b Fx(\))27 b(=)108 b(max)1311 5086 y Fm( )r Fw(2)p Fm(C)q(;)p Fw(j)p Fm( )r Fw(j)p Fj(=1)1744 4895 y Fm(r)1688 4925 y Fi(X)1703 5135 y Fm(i)p Fj(=1)1905 4895 y Fm(s)1849 4925 y Fi(X)1860 5135 y Fm(j)t Fj(=1)2009 5020 y Fn(j)p Fx(\()p Fo(R)2149 5035 y Fm(j)2186 5020 y Fo(E)2258 5035 y Fm(i)2308 5020 y Fn(\000)23 b(h)p Fo( )t Fn(j)p Fo(R)2616 5035 y Fm(j)2652 5020 y Fo(E)2724 5035 y Fm(i)2753 5020 y Fn(j)p Fo( )t Fn(i)p Fx(\))16 b Fn(j)p Fo( )t Fn(ij)3102 4971 y Fj(2)3158 5020 y Fo(:)576 b Fx(\(3\))0 5341 y(The)42 b(pair)f(\()p Fo(C)r(;)17 b Fn(R)p Fx(\))40 b(is)h(de\014ned)i(to)e(b)s(e)h(an)f Fn(E)9 b Fx(-correcting)40 b(co)s(de)i(if)e(and)h(only)g(if)f(Err\()p Fo(C)r(;)17 b Fn(RE)9 b Fx(\))42 b(=)h(0.)69 b(In)0 5515 y(this)33 b(case,)h(w)m(e)h(sa)m(y)f(that)f(the)h(errors)f Fn(E)42 b Fx(are)33 b(correctable.)46 b(This)33 b(is)g(the)g(precise)h (meaning)e(of)h(the)h(w)m(ord)1870 5774 y(5)p eop %%Page: 6 6 6 5 bop 0 555 a Fx(\\correctable")43 b(in)f(the)h(quan)m(tum)h(error)f (correcting)f(condition.)74 b(Notice)42 b(that)h(Err\()p Fo(C)r(;)17 b Fn(RE)9 b Fx(\))45 b(=)g(0)e(is)0 730 y(equiv)-5 b(alen)m(t)32 b(to)h(the)g(condition)e(that)h(for)g(eac)m(h)i Fo(a)27 b Fx(=)h(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(r)35 b Fx(and)d Fo(b)c Fx(=)g(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(s)p Fx(,)33 b(w)m(e)g(ha)m(v)m(e)1374 1004 y Fo(R)1448 1019 y Fm(b)1483 1004 y Fo(E)1555 1019 y Fm(a)1597 1004 y Fn(j)p Fo( )t Fn(i)27 b Fx(=)g Fn(h)p Fo( )t Fn(j)p Fo(R)2069 1019 y Fm(b)2103 1004 y Fo(E)2175 1019 y Fm(a)2217 1004 y Fn(j)p Fo( )t Fn(ij)p Fo( )t Fn(i)p Fx(;)1249 b(\(4\))0 1278 y(that)32 b(is,)h Fo(R)411 1293 y Fm(b)445 1278 y Fo(E)517 1293 y Fm(a)591 1278 y Fx(acts)h(as)e(a)h(m)m(ultiple)d(of)i (the)h(iden)m(tit)m(y)f(on)h Fo(C)7 b Fx(.)125 1452 y(F)-8 b(or)21 b(an)m(y)j(t)m(w)m(o)f(v)m(ector)h(spaces)g Fo(V)44 b Fx(and)23 b Fo(V)1570 1416 y Fw(0)1593 1452 y Fx(,)i(the)e (expression)h(Hom)o(\()p Fo(V)5 b(;)17 b(V)2688 1416 y Fw(0)2712 1452 y Fx(\))22 b(denotes)i(the)f(v)m(ector)h(space)g(of)0 1626 y(all)d(linear)h(transformations)g(from)g Fo(V)45 b Fx(to)23 b Fo(V)1593 1590 y Fw(0)1640 1626 y Fx(.)40 b(If)24 b(bases)g(for)f Fo(V)45 b Fx(and)23 b Fo(V)2543 1590 y Fw(0)2590 1626 y Fx(are)g(c)m(hosen,)k(Hom)o(\()p Fo(V)5 b(;)17 b(V)3501 1590 y Fw(0)3524 1626 y Fx(\))23 b(is)g(iden-)0 1801 y(ti\014ed)33 b(with)g(a)g(set)h(of)f(matrices.)46 b(One)33 b(can)h(c)m(hec)m(k)i(that)d(the)h(condition)e(Err\()p Fo(C)r(;)17 b Fn(RE)9 b Fx(\))28 b(=)h(0)k(means)h(that)0 1975 y(the)39 b(sup)s(erop)s(erator)g Fo(S)864 1990 y Fw(R)966 1975 y Fx(:)f(Hom)o(\()p Fo(W)m(;)17 b(W)d Fx(\))37 b Fn(!)h Fx(Hom)o(\()p Fo(W)m(;)17 b(W)d Fx(\))38 b(de\014ned)i(b)m(y)f Fo(S)2823 1990 y Fw(R)2888 1975 y Fx(\()p Fo(P)14 b Fx(\))37 b(=)3191 1900 y Fi(P)3296 1927 y Fm(s)3296 2004 y(b)p Fj(=1)3438 1975 y Fo(R)3512 1990 y Fm(b)3546 1975 y Fo(P)14 b(R)3698 1928 y Fw(y)3697 2003 y Fm(b)3772 1975 y Fx(in-)0 2149 y(v)m(erts)28 b(the)g(sup)s(erop)s(erator)f Fo(S)1074 2164 y Fw(E)1149 2149 y Fx(:)h(Hom)o(\()p Fo(W)m(;)17 b(W)d Fx(\))27 b Fn(!)h Fx(Hom)o(\()p Fo(W)m(;)17 b(W)d Fx(\),)28 b(de\014ned)g(b)m(y)g Fo(S)2970 2164 y Fw(E)3017 2149 y Fx(\()p Fo(P)14 b Fx(\))27 b(=)3301 2075 y Fi(P)3406 2101 y Fm(r)3406 2179 y(a)p Fj(=1)3554 2149 y Fo(E)3626 2164 y Fm(a)3668 2149 y Fo(P)14 b(E)3823 2113 y Fw(y)3817 2174 y Fm(a)3858 2149 y Fx(,)0 2324 y(up)33 b(to)f(a)h(constan)m(t)g(m) m(ultiple)e(when)j(restricted)f(to)f(the)h(subspace)i(Hom)o(\()p Fo(C)r(;)17 b(C)7 b Fx(\))27 b Fn(\022)i Fx(Hom)o(\()p Fo(W)m(;)17 b(W)d Fx(\).)43 b(This)0 2498 y(fact)33 b(can)h(b)s(e)g (regarded)g(as)g(motiv)-5 b(ation)31 b(for)i(the)h(de\014nition)e(of)h (Err)q(\()p Fo(C)r(;)17 b Fn(RE)9 b Fx(\).)45 b(Sometimes)33 b Fo(S)3490 2513 y Fw(E)3570 2498 y Fx(and)h Fo(S)3821 2513 y Fw(R)0 2673 y Fx(are)h(called)g(quan)m(tum)g(op)s(erations.)51 b(Matrix)35 b Fo(P)49 b Fx(is)35 b(a)g(densit)m(y)h(matrix.)50 b(Let)36 b(us)g(commen)m(t)f(for)f(mathe-)0 2847 y(maticians.)55 b(A)38 b(densit)m(y)g(matrix)e(arises)h(as)g(follo)m(ws:)52 b(giv)m(en)37 b(a)g(co)s(dew)m(ord)h Fo(c)e Fn(2)g Fo(C)43 b Fn(\032)36 b Fo(W)14 b Fx(,)38 b(one)f(has)h(the)0 3021 y(linear)g(functional)g Fo(c)785 2985 y Fw(\003)865 3021 y Fx(in)h(the)h(dual)f(space)i Fo(C)1730 2985 y Fw(\003)1809 3021 y Fx(de\014ned)h(b)m(y)e Fo(c)2337 2985 y Fw(\003)2417 3021 y Fx(:=)g Fn(h)p Fo(c)p Fn(j)17 b(i)p Fx(.)64 b(A)40 b(matrix)e Fo(P)54 b Fx(can)40 b(then)g(b)s(e)0 3196 y(iden)m(ti\014ed)32 b(with)h Fo(c)690 3159 y Fw(\003)751 3196 y Fn(\012)23 b Fo(c)32 b Fx(via)g(the)h(isomorphism)d Fo(W)1929 3159 y Fw(\003)1991 3196 y Fn(\012)22 b Fo(W)41 b Fn(')29 b Fx(Hom)o(\()p Fo(W)m(;)17 b(W)d Fx(\).)125 3370 y(The)37 b(follo)m(wing)c(lemma)h(indicates)h(that)h(capabilit)m (y)e(of)i(\()p Fo(C)r(;)17 b Fn(R)p Fx(\))35 b(to)h(correct)h(errors)f (applies)f(to)h(the)0 3544 y(en)m(tire)k(linear)e(space)i(spanned)h(b)m (y)f Fn(E)9 b Fx(,)41 b(not)f(just)f(the)h(collection)e Fn(E)48 b Fx(itself.)63 b(It)39 b(sa)m(ys)i(that)f(if)e(the)i(co)s(de)0 3719 y(corrects)34 b(the)f(mistak)m(es)f Fn(f)p Fo(E)1050 3734 y Fm(a)1092 3719 y Fn(g)p Fx(,)g(then)i(it)d(will)f(also)i (correct)h(an)m(y)h(linear)c(com)m(bination)h(of)h(the)h Fn(f)p Fo(E)3638 3734 y Fm(a)3680 3719 y Fn(g)p Fo(:)0 3976 y Fk(Lemma)k(I)s(I.1.)42 b Fv(Supp)-5 b(ose)34 b(that)i Fn(E)43 b Fv(and)35 b Fn(E)1563 3916 y Fh(0)1624 3976 y Fv(ar)-5 b(e)34 b(two)h(c)-5 b(ol)5 b(le)-5 b(ctions)34 b(of)h(op)-5 b(er)g(ators)35 b(on)f(a)h(Hilb)-5 b(ert)36 b(sp)-5 b(ac)g(e)34 b Fo(W)0 4150 y Fv(that)c(sp)-5 b(an)29 b(the)h(same)e(subsp)-5 b(ac)g(e)29 b(of)g Fx(Hom\()p Fo(W)m(;)17 b(W)d Fx(\))p Fv(.)42 b(Then)29 b Fx(\()p Fo(C)r(;)17 b Fn(R)p Fx(\))29 b Fv(is)h(an)f Fn(E)9 b Fv(-c)-5 b(orr)g(e)g(cting)28 b(c)-5 b(o)g(de)29 b(if)h(and)f(only)0 4325 y(if)35 b Fx(\()p Fo(C)r(;)17 b Fn(R)p Fx(\))34 b Fv(is)h(an)f Fn(E)711 4265 y Fh(0)737 4325 y Fv(-c)-5 b(orr)g(e)g(cting)34 b(c)-5 b(o)g(de.)125 4582 y Fx(W)d(e)36 b(complete)f(this)h(section)g(b)m(y)h(presen)m(ting)f(a)g(form)m (ulation)d(of)i(error)h(correctabilit)m(y)f(in)g(terms)h(of)0 4756 y(a)j(reco)m(v)m(ery)-8 b(.)66 b(It)39 b(is)g(the)g(basis)g(for)g (the)h(error)f(correcting)g(conditions)f(in)h(section)g(\(I)16 b(A\).)64 b(Giv)m(en)39 b(only)0 4931 y(the)i(co)s(de)h(space)g Fo(C)48 b Fx(and)41 b(the)g(set)h(of)e(errors)i Fn(E)9 b Fx(,)42 b(it)e(is)h(p)s(ossible)f(to)h(determine)f(whether)j(or)d (not)h(one)0 5105 y(can)30 b(construct)h(a)f(collection)d(of)j(reco)m (v)m(ery)i(op)s(erators)e Fn(R)g Fx(making)e(\()p Fo(C)r(;)17 b Fn(R)p Fx(\))30 b(in)m(to)f(an)g Fn(E)9 b Fx(-correcting)29 b(co)s(de.)0 5279 y(This)k(is)f(the)h(con)m(ten)m(t)h(of)e(the)h(next)g (theorem.)1870 5774 y(6)p eop %%Page: 7 7 7 6 bop 0 555 a Fk(Theorem)41 b(I)s(I.2.)i Fv(L)-5 b(et)38 b Fo(W)52 b Fv(b)-5 b(e)37 b(a)h(Hilb)-5 b(ert)38 b(sp)-5 b(ac)g(e,)38 b(let)g Fo(C)44 b Fv(b)-5 b(e)38 b(a)g(subsp)-5 b(ac)g(e)37 b(of)g Fo(W)14 b Fv(,)38 b(and)g(let)g Fn(E)j Fx(=)33 b Fn(f)p Fo(E)3671 570 y Fm(a)3713 555 y Fn(g)38 b Fv(b)-5 b(e)0 730 y(a)34 b(c)-5 b(ol)5 b(le)-5 b(ction)34 b(of)g(line)-5 b(ar)34 b(op)-5 b(er)g(ators)35 b(on)f Fo(W)14 b Fv(.)44 b(Then)34 b(ther)-5 b(e)35 b(exists)f(a)g(c)-5 b(ol)5 b(le)-5 b(ction)34 b(of)g(op)-5 b(er)g(ators)34 b Fn(R)i Fv(making)0 904 y Fx(\()p Fo(C)r(;)17 b Fn(R)p Fx(\))34 b Fv(into)h(an)g Fn(E)9 b Fv(-c)-5 b(orr)g(e)g(cting)33 b(c)-5 b(o)g(de)35 b(if)f(and)g(only)h(if)1399 1178 y Fn(h)p Fo( )t Fn(j)p Fo(E)1611 1137 y Fw(y)1605 1203 y Fm(a)1646 1178 y Fo(E)1718 1193 y Fm(b)1753 1178 y Fn(j)p Fo( )t Fn(i)27 b Fx(=)h Fn(h)p Fo(\036)p Fn(j)p Fo(E)2221 1137 y Fw(y)2215 1203 y Fm(a)2256 1178 y Fo(E)2328 1193 y Fm(b)2362 1178 y Fn(j)p Fo(\036)p Fn(i)1274 b Fx(\(5\))0 1452 y Fv(for)35 b(al)5 b(l)34 b(unit)h(ve)-5 b(ctors)35 b Fo( )t(;)17 b(\036)27 b Fn(2)h Fo(C)42 b Fv(and)34 b(for)h(al)5 b(l)34 b Fo(a;)17 b(b:)0 1822 y Fp(I)s(I)s(I.)103 b(QUBITS)125 2114 y Fx(Let)40 b(us)h(remind)e(the)i (reader)g(that)f(an)h(ordinary)e(classical)g(bit)h Fo(b)h Fx(tak)m(es)g(only)f(t)m(w)m(o)h(v)-5 b(alues)40 b Fn(f)p Fx(0)p Fo(;)17 b Fx(1)p Fn(g)p Fx(.)0 2288 y(So)35 b(the)g(classical)e (bit)h(is)g(an)h(elemen)m(t)f Fo(b)e Fn(2)g(f)p Fx(0)p Fo(;)17 b Fx(1)p Fn(g)p Fx(.)49 b(A)34 b(quan)m(tum)h(bit,)g(or)f(a)h (qubit,)g(is)f(an)h(elemen)m(t)f(of)h(a)0 2463 y(t)m(w)m(o)c (dimensional)c(complex)i(v)m(ector)i(space)g Fg(C)1709 2427 y Fj(2)1755 2463 y Fx(.)42 b(Usually)-8 b(,)30 b(one)g(denotes)h (the)f(basis)g(v)m(ectors)h(b)m(y)g Fn(j)p Fx(0)p Fn(i)e Fx(and)0 2637 y Fn(j)p Fx(1)p Fn(i)p Fx(.)65 b(Information)39 b(con)m(tained)h(in)f Fo(k)k Fx(qubits)e(is)f(carried)f(b)m(y)i(elemen) m(t)f(of)g(the)h Fo(k)s Fx(-fold)d(tensor)j(pro)s(duct)0 2811 y Fg(C)66 2775 y Fj(2)136 2811 y Fn(\012)24 b(\001)17 b(\001)g(\001)22 b(\012)j Fg(C)545 2775 y Fj(2)590 2811 y Fx(.)53 b(Ordinarily)-8 b(,)33 b(a)j(quan)m(tum)f(co)s(de)h(em)m(b)s (eds)g(a)g(co)s(dew)m(ord)g(consisting)f(of)g Fo(k)j Fx(qubits)e(in)m(to)0 2986 y(a)41 b(quan)m(tum)h(medium)e(consisting)h (of)g Fo(n)g Fx(qubits.)71 b(The)43 b(co)s(de)e(space)i Fo(C)48 b Fx(has)42 b(dimension)e(2)3467 2950 y Fm(k)3551 2986 y Fx(and)i(the)0 3160 y(enco)s(ding)35 b(space)h Fo(W)49 b Fx(has)36 b(dimension)e(2)1506 3124 y Fm(n)1553 3160 y Fx(,)i(with)f(a)g(condition)f Fo(n)e(>)g(k)s(:)k Fx(An)f(elemen)m(t)h(of)e Fo(W)49 b Fx(is)35 b(though)m(t)0 3335 y(of)e(as)h(a)f(message)i Fo(n)e Fx(qubits)h(long)f(whic)m(h)h (carries)g Fo(k)i Fx(qubits)e(of)f(information.)44 b(No)m(w)34 b(imagine)d(that)j(the)0 3509 y(quan)m(tum)28 b(medium)e(is)g(exp)s (osed)j(to)e(errors,)i(whic)m(h)f(tak)m(e)g(the)g(form)e(of)h(linear)f (op)s(erators)h(on)h Fo(W)14 b Fx(.)41 b(Often,)0 3683 y(one)33 b(considers)g(errors)g(whic)m(h)g(a\013ect)g Fo(t)g Fx(or)f(few)m(er)i(of)e(the)h Fo(n)g Fx(qubits)g(in)e(the)i (medium.)125 3858 y(Let)40 b(us)h(clarify)e(the)i(situation.)65 b(Let)41 b Fo(n)g Fn(2)h Fg(N)55 b Fx(and)40 b(de\014ne)i Fo(W)55 b Fx(=)41 b Fn(\012)2720 3821 y Fm(n)2720 3882 y(j)t Fj(=1)2847 3858 y Fo(W)2939 3873 y Fm(j)2975 3858 y Fx(,)i(where)e(eac)m(h)h Fo(W)3654 3873 y Fm(j)3731 3858 y Fx(is)e(a)0 4032 y(t)m(w)m(o)35 b(dimensional)e(v)m(ector)j (space)f(with)g(basis)g Fn(fj)p Fx(0)p Fn(i)1916 4047 y Fm(j)1951 4032 y Fo(;)17 b Fn(j)p Fx(1)p Fn(i)2111 4047 y Fm(j)2147 4032 y Fn(g)p Fx(.)49 b(Eac)m(h)36 b Fo(W)2610 4047 y Fm(j)2678 4032 y Fx(=)31 b Fg(C)20 b Fn(j)p Fx(0)p Fn(i)2967 4047 y Fm(j)3033 4032 y Fn(\010)k Fg(C)c Fn(j)p Fx(1)p Fn(i)3316 4047 y Fm(j)3392 4032 y Fx(is)35 b(iden)m(ti\014ed)0 4206 y(with)k(a)h(space)g(of)f(qubits)h (in)f(the)h(eviden)m(t)h(w)m(a)m(y)-8 b(.)66 b(Let)39 b Fo(\033)2140 4170 y Fm(x)2184 4206 y Fo(;)17 b(\033)2287 4170 y Fm(y)2328 4206 y Fo(;)40 b Fx(and)g Fo(\033)2651 4170 y Fm(z)2730 4206 y Fx(b)s(e)g(the)g(P)m(auli)f(matrices.)63 b(On)0 4381 y Fg(C)66 4344 y Fj(2)139 4381 y Fx(=)28 b Fg(C)20 b Fn(j)p Fx(0)p Fn(i)27 b(\010)c Fg(C)c Fn(j)p Fx(1)p Fn(i)38 b Fx(w)m(e)c(can)f(write)f(them)g(as)863 4718 y Fo(\033)922 4677 y Fm(x)993 4718 y Fx(=)1097 4517 y Fi(0)1097 4697 y(@)1184 4640 y Fx(0)50 b(1)1184 4815 y(1)g(0)1331 4517 y Fi(1)1331 4697 y(A)1435 4718 y Fo(;)114 b(\033)1635 4677 y Fm(y)1705 4718 y Fx(=)1808 4517 y Fi(0)1808 4697 y(@)1895 4640 y Fx(0)50 b Fn(\000)p Fo(i)1903 4815 y(i)89 b Fx(0)2105 4517 y Fi(1)2105 4697 y(A)2208 4718 y Fo(;)115 b(\033)2409 4677 y Fm(z)2476 4718 y Fx(=)2580 4517 y Fi(0)2580 4697 y(@)2667 4640 y Fx(1)88 b(0)2667 4815 y(0)49 b Fn(\000)p Fx(1)2892 4517 y Fi(1)2892 4697 y(A)2996 4718 y Fo(:)738 b Fx(\(6\))0 5049 y(It)38 b(is)f(common)g(in)g (quan)m(tum)h(information)d(theory)j(to)g(use)h(the)f(notation)f Fo(X)8 b Fx(,)39 b Fo(Y)21 b Fx(,)39 b(and)f Fo(Z)45 b Fx(instead)38 b(of)0 5224 y Fo(\033)55 5239 y Fm(x)99 5224 y Fx(,)k Fo(\033)223 5239 y Fm(y)265 5224 y Fx(,)g(and)e Fo(\033)586 5239 y Fm(z)626 5224 y Fx(.)66 b(F)-8 b(or)39 b(an)m(y)i Fo(j)46 b Fx(=)41 b(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(n)40 b Fx(and)g(sup)s(erscripts)h Fo(\013)g Fx(=)g Fo(x;)17 b(y)t(;)g(z)43 b Fx(de\014ne)f(a)d(lo)s(cal)f(op)s(erator)1870 5774 y(7)p eop %%Page: 8 8 8 7 bop 0 555 a Fo(\033)59 519 y Fm(\013)55 580 y(j)136 555 y Fx(:)28 b Fo(W)41 b Fn(!)28 b Fo(W)46 b Fx(b)m(y)1386 835 y Fo(\033)1445 794 y Fm(\013)1441 860 y(j)1495 835 y Fn(j)1523 850 y Fm(W)1590 858 y Fl(m)1679 835 y Fx(=)1783 601 y Fi(8)1783 691 y(>)1783 721 y(<)1783 900 y(>)1783 930 y(:)1872 748 y Fx(Id)149 b(if)32 b Fo(j)h Fn(6)p Fx(=)28 b Fo(m;)1872 957 y(\033)1931 921 y Fm(\013)2110 957 y Fx(if)k Fo(j)h Fx(=)28 b Fo(m:)3761 835 y Fx(\(7\))0 1151 y(In)39 b(other)f(w)m(ords,)k Fo(\033)762 1115 y Fm(\013)758 1176 y(j)849 1151 y Fx(acts)d(non)g(trivially)c(on)k(the)g Fo(j)6 b Fx(-th)38 b(qubit)g(as)h(the)g(P)m(auli)e(matrix)g Fo(\033)3355 1115 y Fm(\013)3443 1151 y Fx(and)h(lea)m(v)m(es)0 1326 y(all)f(other)j(qubits)g(unc)m(hanged.)66 b(The)41 b(follo)m(wing)36 b(con)m(v)m(enien)m(t)42 b(notation)c(is)h(suggested) i(for)e(a)g(basis)h(of)0 1500 y Fo(W)14 b Fx(.)50 b(Ev)m(ery)36 b(basis)f(v)m(ector)h Fn(j)p Fo(b)1071 1515 y Fj(1)1111 1500 y Fn(i)1150 1515 y Fj(1)1212 1500 y Fn(\012)25 b(j)p Fo(b)1383 1515 y Fj(2)1422 1500 y Fn(i)1461 1515 y Fj(2)1524 1500 y Fn(\012)f(\001)17 b(\001)g(\001)22 b(\012)i(j)p Fo(b)1935 1515 y Fm(n)1982 1500 y Fn(i)2021 1515 y Fm(n)2103 1500 y Fx(can)35 b(b)s(e)g(referred)g(to)g(b)m(y)g(the)h(shortened)g (name)0 1674 y Fn(j)p Fo(b)69 1689 y Fj(1)109 1674 y Fo(b)150 1689 y Fj(2)206 1674 y Fn(\001)17 b(\001)g(\001)e Fo(b)380 1689 y Fm(n)427 1674 y Fn(i)p Fx(.)75 b(In)44 b(this)e(w)m(a)m(y)-8 b(,)47 b(bit)c(strings)f(of)h(length)g Fo(n)g 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Fo(:)1172 b Fx(\(8\))0 3028 y(as)37 b(a)f(creation)f(op)s(erator.)55 b(By)36 b(setting)h Fo(\033)1569 2987 y Fw(\000)1565 3054 y Fm(j)1662 3028 y Fx(=)1782 2989 y Fj(1)p 1782 3005 36 4 v 1782 3063 a(2)1827 3028 y Fx(\()p Fo(\033)1924 2992 y Fm(x)1920 3053 y(j)1992 3028 y Fn(\000)26 b Fo(i\033)2187 2981 y Fm(y)2183 3054 y(j)2228 3028 y Fx(\),)38 b(one)e(can)h(obtain)e(ev)m (ery)j(basis)f(v)m(ector)g(in)0 3203 y Fo(W)46 b Fx(b)m(y)34 b(acting)d(on)i Fn(j)p Fx(0)p Fn(i)e Fx(b)m(y)j(pro)s(ducts)f(of)f(the) h Fo(\033)1727 3161 y Fw(\000)1723 3228 y Fm(j)1786 3203 y Fx(.)125 3377 y(The)38 b(errors)h(w)m(e)f(wish)g(to)g(consider)g(are) g(lo)s(cal)e(errors;)41 b(i.e.,)d(linear)f(com)m(binations)f(of)h(pro)s (ducts)i(of)0 3551 y(the)33 b(op)s(erators)f Fo(\033)658 3515 y Fm(\013)654 3576 y(j)708 3551 y Fx(.)43 b(If)33 b Fo(E)38 b Fx(is)33 b(a)f(pro)s(duct)h(of)f Fo(t)g Fx(suc)m(h)i(op)s (erators)1532 3825 y Fo(E)g Fx(=)27 b Fo(\033)1800 3784 y Fm(\013)1845 3793 y Ff(1)1796 3850 y Fm(m)1858 3859 y Ff(1)1898 3825 y Fo(\033)1957 3784 y Fm(\013)2002 3793 y Ff(2)1953 3850 y Fm(m)2015 3859 y Ff(2)2071 3825 y Fn(\001)17 b(\001)g(\001)e Fo(\033)2263 3784 y Fm(\013)2308 3792 y Fl(t)2259 3850 y Fm(m)2321 3858 y Fl(t)3761 3825 y Fx(\(9\))0 4099 y(with)34 b(distinct)g Fo(m)664 4114 y Fm(j)701 4099 y Fx(,)h(w)m(e)h(sa)m(y)g(that)e Fo(E)41 b Fx(is)34 b(an)h(error)f(of)h(w)m(eigh)m(t)g Fo(t)p Fx(.)50 b(Fix)34 b Fo(t)h Fx(and)f(let)g Fn(E)3055 4114 y Fm(t)3119 4099 y Fx(b)s(e)h(the)g(sp)s(eci\014c)h(set)0 4273 y(of)c(errors)h(consisting)f(of)g(pro)s(ducts)h(of)f Fo(t)h Fx(or)f(few)m(er)i(P)m(auli)d(matrices)1006 4547 y Fn(E)1059 4562 y Fm(t)1116 4547 y Fx(=)c Fn(f)p Fo(E)39 b Fx(acting)32 b(on)g(W,)h(with)f(the)h(w)m(eigh)m(t)28 b Fn(\024)g Fo(t)p Fn(g)p Fo(:)833 b Fx(\(10\))0 4821 y(An)26 b(imp)s(ortan)m(t)f(class)h(of)g(co)s(des)h(are)f(able)f(to)h (correct)h(the)g(errors)f Fn(E)2464 4836 y Fm(t)2493 4821 y Fo(:)h Fx(The)g(signi\014cance)f(of)f Fn(E)3407 4836 y Fm(t)3463 4821 y Fx(correcting)0 4996 y(co)s(des)g(is)g(that)f (they)i(are)e(able)g(to)h(correct)g(arbitrary)f(errors)h(on)f Fo(t)h Fx(or)g(few)m(er)g(bits.)41 b(This)25 b(is)f(a)g(consequence)0 5170 y(of)38 b(lemma)e(I)s(I.1)i(since)g(the)h(errors)g(of)e(w)m(eigh)m (t)i Fn(\024)f Fo(t)g Fx(form)f(a)h(basis)g(for)g(the)g(linear)f(space) i(of)f(all)e(errors)0 5344 y(a\013ecting)c Fo(t)h Fx(or)f(few)m(er)i (qubits.)1870 5774 y(8)p eop %%Page: 9 9 9 8 bop 0 555 a Fp(IV.)104 b(AN)35 b(O)m(VER)-12 b(VIEW)34 b(OF)h(XX0)g(SPIN)g(CHAINS)125 854 y(A.)104 b(The)35 b(partition)f(and)h(correlation)h(functions)125 1147 y Fx(W)-8 b(e)44 b(study)g(a)g(mo)s(del)e(of)h(in)m(teracting)f(spins)i (on)g(a)f(one)h(dimensional)d(lattice.)75 b(Eac)m(h)45 b(spin)e(is)g(an)0 1321 y(elemen)m(t)32 b(of)g Fg(C)537 1285 y Fj(2)583 1321 y Fo(:)g Fx(Let)h Fo(n)28 b Fn(2)g Fg(N)48 b Fx(and)32 b(de\014ne)i Fo(W)46 b Fx(as)33 b(b)s(efore)1159 1595 y Fo(W)42 b Fx(=)27 b Fn(\012)1473 1554 y Fm(n)1473 1620 y(j)t Fj(=1)1600 1595 y Fo(W)1692 1610 y Fm(j)1729 1595 y Fo(;)114 b(W)1962 1610 y Fm(j)2026 1595 y Fx(=)28 b Fg(C)20 b Fn(j)p Fx(0)p Fn(i)2312 1610 y Fm(j)2376 1595 y Fn(\010)i Fg(C)e Fn(j)p Fx(1)p Fn(i)2657 1610 y Fm(j)2699 1595 y Fo(:)986 b Fx(\(11\))0 1869 y(W)-8 b(e)38 b(call)d(the)j(in)m(teger)f Fo(n)g Fx(the)h(length)e(of)h(the)h (lattice.)56 b(It)37 b(is)g(the)g(n)m(um)m(b)s(er)h(of)f(in)m (teracting)f(spins.)57 b(The)0 2043 y(space)42 b Fo(W)55 b Fn(')43 b Fx(\()p Fg(C)640 2007 y Fj(2)685 2043 y Fx(\))723 2007 y Fw(\012)p Fm(n)866 2043 y Fx(is)d(the)i(Hilb)s(ert)d(space)j(of) e(the)i(mo)s(del.)66 b(The)42 b(Hamiltonian)c Fn(H)k Fx(:)g Fo(W)56 b Fn(!)41 b Fo(W)55 b Fx(is)0 2218 y(de\014ned)34 b(b)m(y)1180 2408 y Fn(H)29 b Fx(=)e Fn(\000)1541 2284 y Fm(n)1491 2314 y Fi(X)1501 2524 y Fm(j)t Fj(=1)1651 2328 y Fi(\000)1697 2408 y Fo(\033)1756 2367 y Fm(x)1752 2433 y(j)1800 2408 y Fo(\033)1859 2367 y Fm(x)1855 2433 y(j)t Fj(+1)2004 2408 y Fx(+)22 b Fo(\033)2161 2361 y Fm(y)2157 2434 y(j)2202 2408 y Fo(\033)2261 2361 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b(the)h(partition)0 3280 y(function)e(de\014ned)i(in)e(equation)g(\(13\).)43 b(One)33 b(has)844 3554 y Fn(Z)i Fx(=)28 b(tr)16 b(2)1206 3485 y Fh(\000H)p 1206 3498 101 3 v 1234 3539 a Fl(T)1349 3554 y Fx(=)1596 3460 y Fi(X)1452 3670 y Fj(eigen)n(v)n(ectors)26 b Fm(v)1900 3554 y Fx(2)1949 3513 y Fw(\000)2014 3480 y Fl(E)s Ff(\()p Fl(v)r Ff(\))p 2014 3498 129 3 v 2056 3539 a Fl(T)2184 3554 y Fx(=)2426 3460 y Fi(X)2287 3672 y Fj(eigen)n(v)l(alues)f Fm(E)2724 3554 y Fx(2)2773 3513 y Fm(nS)2867 3554 y Fx(2)2916 3513 y Fw(\000)2981 3486 y Fl(E)p 2980 3498 48 3 v 2981 3539 a(T)3688 3554 y Fx(\(A1\))0 3884 y(where)31 b(the)f(factor)f(2)769 3848 y Fm(nS)892 3884 y Fx(is)h(the)g(degeneracy)h(of)f(the)g(energy)h(lev)m(el)e Fo(E)36 b Fx(and)29 b Fo(S)36 b Fx(is)29 b(en)m(trop)m(y)-8 b(.)44 b(In)30 b(thermo)s(dy-)0 4059 y(namics)35 b(limit)d Fo(n)h Fn(!)g(1)i Fx(b)s(oth)g(energy)i(and)f(en)m(trop)m(y)h(increase) f(linearly)e(with)h Fo(n)p Fx(.)53 b(So)36 b(w)m(e)g(can)g(use)h(the)0 4233 y(metho)s(d)32 b(of)g(steep)s(est)i(descen)m(t)h(to)d(ev)-5 b(aluate)32 b(the)h(partition)e(function)1180 4507 y(lim)1156 4567 y Fm(n)p Fw(!1)1357 4507 y Fn(Z)k Fx(=)52 b(lim)1568 4567 y Fm(n)p Fw(!1)1768 4412 y Fi(X)1813 4624 y Fm(E)1929 4507 y Fx(2)1978 4466 y Fm(nS)t Fw(\000)2133 4439 y Fl(E)p 2132 4451 V 2133 4492 a(T)2221 4507 y Fn(!)2349 4371 y Fi(Z)2465 4507 y Fx(2)2514 4466 y Fm(nS)t Fw(\000)2669 4439 y Fl(E)p 2668 4451 V 2669 4492 a(T)3688 4507 y Fx(\(A2\))0 4811 y(The)34 b(metho)s(d)d(of)i(steep)s(est)h(descen)m(t)h(leads)d(to) g(the)h(v)-5 b(ariational)29 b(equation)1591 5085 y Fo(\016)1655 4944 y Fi(\022)1728 5085 y Fo(S)f Fn(\000)1926 5017 y Fo(E)p 1926 5062 79 4 v 1930 5153 a(T)2014 4944 y Fi(\023)2115 5085 y Fx(=)f(0)p Fo(:)1394 b Fx(\(A3\))0 5359 y(This)33 b(brings)f(us)h(to)f(the)h(k)m(ey)h(de\014nition:)1845 5774 y(16)p eop %%Page: 17 17 17 16 bop 0 555 a Fk(De\014nition)38 b(A.1.)k Fx(The)36 b(subspace)g(of)e(thermo)s(dynamic)e(equilibrium)g(is)h(de\014ned)j(to) e(b)s(e)g(the)h(span)g(of)0 730 y(the)e(set)g(of)f(eigen)m(v)m(ectors)j (that)d(solv)m(e)h(equation)f(\(A3\).)125 979 y(F)-8 b(or)33 b(the)j(mo)s(del)d(at)h(hand,)i(w)m(e)f(can)g(b)s(e)g(more)f (sp)s(eci\014c.)51 b(Using)34 b(v)-5 b(ariational)31 b(calculus,)k(one)g(has)g(for)0 1153 y(XX0)740 1417 y Fo(\016)t(S)f Fx(=)984 1282 y Fi(Z)1084 1308 y Fm(\031)1039 1507 y Fw(\000)p Fm(\031)1158 1417 y Fo(dp)1275 1277 y Fi(\024)1327 1417 y Fn(\000)p Fo(\016)t(\032)p Fx(\()p Fo(p)p Fx(\))17 b(log\()p Fo(\032)p Fx(\()p Fo(p)p Fx(\)\))22 b(+)g Fo(\016)t(\032)p Fx(\()p Fo(p)p Fx(\))17 b(log)2521 1277 y Fi(\022)2634 1350 y Fx(1)p 2605 1394 108 4 v 2605 1486 a(2)p Fo(\031)2745 1417 y Fn(\000)22 b Fo(\032)p Fx(\()p Fo(p)p Fx(\))3019 1277 y Fi(\023)q(\025)3688 1417 y Fx(\(A4\))0 1706 y(and)1389 1995 y Fo(\016)t(E)33 b Fx(=)1645 1860 y Fi(Z)1745 1886 y Fm(\031)1700 2085 y Fw(\000)p Fm(\031)1819 1995 y Fo(dp)17 b Fx([)o Fn(\000)p Fo(\016)t(\032)p Fx(\()p Fo(p)p Fx(\))p Fo(\017)p Fx(\()p Fo(p)p Fx(\)])h Fo(:)1191 b Fx(\(A5\))0 2275 y(Setting)33 b Fo(\016)401 2194 y Fi(\000)446 2275 y Fo(S)28 b Fn(\000)644 2236 y Fm(E)p 644 2252 56 4 v 646 2309 a(T)709 2194 y Fi(\001)784 2275 y Fx(=)i(0)j(giv)m(es)h(the)h(solution)d(giv)m(en)i (in)e(equation)i(\(38\).)46 b(Therefore,)35 b(the)f(span)h(of)e(the)0 2449 y(eigen)m(v)m(ectors)24 b(that)f(corresp)s(ond)g(to)f(the)h (function)f Fo(\032)28 b Fx(=)2075 2410 y Fj(1)p 2054 2426 79 4 v 2054 2484 a(2)p Fm(\031)2142 2449 y Fo(\022)e Fx(comprise)c(the)h(space)g Fo(C)30 b Fx(of)22 b(thermo)s(dynamic)0 2623 y(equilibrium.)40 b(W)-8 b(e)33 b(sa)m(w)h(that)e(the)h(equation)g Fo(\016)1752 2543 y Fi(\000)1798 2623 y Fo(S)28 b Fn(\000)1995 2584 y Fm(E)p 1995 2601 56 4 v 1997 2658 a(T)2061 2543 y Fi(\001)2134 2623 y Fx(=)g(0)k(de\014nes)i Fo(\032)p Fx(\()p Fo(p)p Fx(\).)p 924 2841 2039 5 v 1178 2842 1529 7 v 1433 2843 1020 9 v 1688 2844 510 11 v 45 3178 a Fu([1])51 b(C.)30 b(E.)h(Shannon,)e(Bell)g(Syst.)i(T)-8 b(ec)m(h.)31 b(J.)f(p.)g(379)i(\(1948\).)45 3358 y([2])51 b(E.)31 b(Knill)c(and)j(R.)g(La\015amme,)h(Ph)m(ys.)f(Rev.)h(A)f Fp(55)p Fu(,)i(900)f(\(1997\).)45 3539 y([3])51 b(C.)30 b(Benett,)i(D.)f(DiVincenzo,)g(J.)f(Smolin,)e(and)i(W.)h(W)-8 b(o)s(otters,)33 b(Ph)m(ys.)d(Rev.)h(A)f Fp(54)p Fu(,)h(3824)h (\(1996\).)45 3719 y([4])51 b(J.)58 b(Preskill,)k Fb(L)-5 b(e)g(ctur)g(es)59 b(notes)p Fu(,)65 b Fa(www.theory.caltech.edu)42 b(/people/preskill/ph219/)51 b Fu(\(1997-)191 3900 y(1999\).)45 4081 y([5])g(I.)38 b(L.)f(Ch)m(uang)g(and)g(M.)h(A.)g(Nielsen,)h Fb(Quantum)g(Computation)j(and)e(Quantum)f(Information)47 b Fu(\(Cam-)191 4261 y(bridge)29 b(Univ.)h(Press,)g(2000\).)45 4442 y([6])51 b(A.)31 b(Kitaev,)g(A.)f(Shen,)g(and)g(M.)h(Vy)m(alyj,)f Fb(Classic)-5 b(al)35 b(and)e(Quantum)g(Computations)40 b Fu(\(AMS,)31 b(2002\).)45 4622 y([7])51 b(A.)31 b(R.)f(Calderbank)f (and)h(P)-8 b(.)31 b(W.)g(Shor,)e(Ph)m(ys.)i(Rev.)f(A)h Fp(54)p Fu(,)g(1098)h(\(1996\).)45 4803 y([8])51 b(A.)31 b(M.)g(Steane,)g(Ph)m(ys.)f(Rev.)h(Lett.)h Fp(77)p Fu(,)f(793)g (\(1996a\).)45 4983 y([9])51 b(D.)31 b(Gottesman,)h(Ph.D.)f(thesis,)f (Caltec)m(h)g(\(1997\),)k Fa(quant-ph/9705052)p Fu(.)0 5164 y([10])51 b(M.)31 b(F)-8 b(reedman,)31 b(A.)g(Kitaev,)g(M.)f (Larsen,)h(and)e(Z.)h(W)-8 b(ang,)32 b Fa(quant-ph/0101025)p Fu(.)0 5345 y([11])51 b(V.)27 b(E.)e(Korepin,)h(N.)g(M.)h(Boguliub)s(o) m(v,)e(and)h(A.)g(G.)g(Izergin,)h Fb(Quantum)i(Inverse)f(Sc)-5 b(attering)30 b(Metho)-5 b(d)29 b(and)191 5525 y(Corr)-5 b(elation)35 b(F)-7 b(unctions)39 b Fu(\(Cam)m(bridge)30 b(Univ.)f(Press,)h(1993\).)1845 5774 y Fx(17)p eop %%Page: 18 18 18 17 bop 0 555 a Fu([12])51 b(F.)31 b(Colomo,)g(A.)f(Izergin,)g(V.)h (Korepin,)e(and)h(V.)h(T)-8 b(ognetti,)31 b(Ph)m(ys.)f(Lett.)i(A)e Fp(169)p Fu(,)i(243)f(\(1992\).)0 736 y([13])51 b(A.)31 b(Its,)g(A.)f(Izergin,)g(V.)h(Korepin,)e(and)h(N.)h(Sla)m(vno)m(v,)f (Ph)m(ys.)g(Rev.)h(Lett.)h Fp(70)p Fu(,)f(1704)h(\(1993\).)0 917 y([14])51 b(M.)22 b(T)-8 b(ak)j(ahashi,)23 b Fb(Thermo)-5 b(dynamics)27 b(of)d(One-Dimensional)g(Solvable)h(Mo)-5 b(dels)30 b Fu(\(Cam)m(bridge)20 b(Univ.)h(Press,)191 1097 y(1999\).)0 1278 y([15])51 b(W.)31 b(K.)f(W)-8 b(o)s(otters)33 b(\(2000\),)g Fa(quant-ph/0001114)p Fu(.)0 1458 y([16])51 b(H.)31 b(F)-8 b(u,)31 b(X.)g(W)-8 b(ang,)31 b(and)f(A.)h(I.)f (Solomon,)g Fa(quant-ph/0105075)p Fu(.)0 1639 y([17])51 b(E.)31 b(Lieb,)e(T.)h(Sc)m(h)m(ultz,)h(and)f(D.)h(Mattis,)g(Annals)e (of)h(Ph)m(ysics)g Fp(16)p Fu(,)h(461)h(\(1961\).)0 1819 y([18])51 b(A.)26 b(G.)f(Izergin,)h(A.)g(G.)f(Pron'k)m(o,)i(and)e(M.)h (B.)f(Zv)m(onarev,)i(Zap.)e(Nauc)m(h)g(Semin.)f(POMI)h Fp(268)p Fu(,)i(207)g(\(2000\).)0 2000 y([19])51 b(R.)31 b(F)-8 b(eynman,)31 b Fb(F)-7 b(eynman)34 b(L)-5 b(e)g(ctur)g(es)33 b(on)g(Computation)40 b Fu(\(P)m(erseus)31 b(Publ.,)d(1996\).)0 2181 y([20])51 b(D.)31 b(Gottesman,)h(Ph)m(ys.)f(Rev.)f Fp(A57)p Fu(,)i(127)f(\(1998\).)0 2361 y([21])51 b(R.)31 b(Landauer,)f(IBM)g(J.)h(Res.)f(Dev.)i(\(1961\).)0 2542 y([22])51 b(R.)31 b(Landauer,)f(Ph)m(ys.)g(T)-8 b(o)s(da)m(y)31 b(pp.)e(23{29)k(\(1991\).)0 2722 y([23])51 b(J.)31 b(C.)f(Maxw)m(ell,)g Fb(The)-5 b(ory)34 b(of)f(He)-5 b(at)39 b Fu(\(Longmans,)31 b(Green)g(and)e(Co.,)i(1871\).)0 2903 y([24])51 b(J.)31 b(Preskill,)d(Pro)s(c.)i(Ro)m(y)-8 b(.)32 b(So)s(c.)e(Lond.)g Fp(A454)p Fu(,)h(385)h(\(1998\).)0 3083 y([25])51 b(J.)37 b(Preskill,)g(in)f Fb(Intr)-5 b(o)g(duction)42 b(to)d(Quantum)h (Computation)p Fu(,)h(edited)c(b)m(y)g(H.-K.)h(Lo,)i(S.)d(P)m(op)s (escu,)i(and)191 3264 y(T.)30 b(Spiller)e(\(W)-8 b(orld)30 b(Scien)m(ti\014c,)f(1998\).)0 3445 y([26])51 b(B.)31 b(Sc)m(h)m(umac)m(her,)g(Ph)m(ys.)f(Rev.)h(pp.)f(2614{1628)k(\(1996\).) 0 3625 y([27])51 b(B.)31 b(Sc)m(h)m(umac)m(her)g(and)f(M.)h(A.)f (Nielsen,)g(Ph)m(ys.)g(Rev.)h(pp.)e(2629{1635)35 b(\(1996\).)0 3806 y([28])51 b(A.)31 b(M.)g(Steane,)g(Rept.)g(Prog.)g(Ph)m(ys.)f(pp.) g(117{173)j(\(1998\).)1845 5774 y Fx(18)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0203041341850--