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1 11 −1
) g=
1 0 0 00 1 0 00 0 0 10 0 1 0
π/8 =
(
1 0
0 eiπ/4
)
H( <*9':B 1$!1*$*7*L$45Ygp(N6*BF,B;h+/<"5
π/8( _*I+.*WSBq$ & *BC
π/8C
f`#1+& N)6N H,g
, π/8 <*XJ;B'<& =+'= C4 NQ& )N H,
g, π/8 (N N( $*BOC?N)7/J. Y')"N+*
- .*/ ' U LBN<>\)N.*B$9%= h6' Yh<*\"+*/dnM"%*B<`'+"<GnWB
δ')*B X3 < h$9%.*/`':/dX)0.<)0
1/δA,A8 m<*[\*41N.*$*B+$q0B_*4WSBZ$& *('#'L!/ 8*WSBk(N2C?= h$#"N B* <3N#N;F$9O_*#BN#%7*1h(u3V 2AC
P=,C+%798[? 8 (UV"-,CC Qx=(5 ,C+'5 8,C&'2<& 3BN"N#N;#"Na"N+"EN$ 9'':/dXBB>W*
6? ,C.NM,<&'D8 D)+'A5=8+'F= &' +<"-,C ,<&'`798,Vs)+' ; !&L 9&<+'xs+' 9/1AB,`8 $='5 +<,C)&<,AB
=85 ,C+'5 8,C&Qx
h [? 8&<,CC, !&%7!8-=)AB+<&'%=I3LJ;B'-<& =+'=;: &:UuAB 5 &<,AB # k&' "-))ABC AB"-,3s |x| Qe[>B5 &<,AB : !&<+' )+<,'>);+(>=,O +' !&':);+<&<,(,C !&' +0,' 9&G= *8 )+'AB&'A!5ABI=E)+' 8/1,C 9&' +B5 &<,C/147!8; 9&<,7!8I' +'AB 9&=5 +<,C&':);+:=>5 ,C+'5 8,C&'7!8; 9&<,7!86QOe[B5 &<,AB G=8k/1 +<,_; 8+6[=AB !&G3L)8,' 5-<&C,C"-,C&'t' +'AB 9&G' 9&<, C "4 9&I=5 +<,C&');+G=Uu"-,CC%=45 ,C+'5 8,C&'G8 ,UuAB+<"4 "4 9&01 +'% k&' "-)%)ABC AB"-,3* AB8% +'AB )C8G)+'5 ,= G3F'5 &<,AB K5AB+<+'<)OAB = 9&'|NYAB+'79844)+'AB8/1 8+I<&E8&'AB+<,'LzD8&<,CC,' +=%5 ,C+'5 8,C&'Uu,i 9&Y,C 9&' +</1 ,C+T=YAB) +&' 8+'Y8 ,C&,C+'T+<J,C&<+,C+'61 &Y [?q<&s);BYC,C"-,C&'.z*3>J;B'f<& =+'=@Q
,VG f(3;!(3*#87g(313()/ "$:2;I/JN7n!;kP#$7!(3P#$:2;
e[f &&'7!8(V?qAB D5AB <,=H +'' +'AB !&=AB f);+f=2"#&<+<,5.= ',C&'')5 ,_;/15:8 5 +<&,C :)+'5 ,<,AB [Q* *"#&<+<,5%= <,C&'5AB+<+'<)OAB =Xz8 :=,<&<+<,CJ8&<,AB `=*)+'ABJ;J,CC,C&'()AB8+5'Z;B5 8 *=Y"4<8+'7!8fV?qAB 4)OAB8+<+,C&@5 &<8 +='<86QRS<&=AB 5. ;&<8+' ;=.=,C+'.7!8= 8M"#&<+<,5*= ',C&'E'AB !&()+'A5'Z>AB+''7!8)OAB8+*5Z;B798E"4<8+'798V?qAB ^)OAB8+<+,C&*@5 &<8 +><8+5:"#&<+<,5%= <,C&'O=,'&<+<,CJ8&<,AB %=I)+'ABJ;J,CC,C&'%'8+=,O +' !&'G+'<8C&&')OA'<,CJ=:5 &<&'*"4'8+'!AB+'798[?q C<&))C,7!8Gz%5Z;B5 8 G=f= 8MP"#&<+<,56'AB !&f)+'A!5Z6Q*AB85AB"-"4 AB (=AB 5);+( AB8.,C 9&' +'' +*8MX"4<8+'(<8+(=,<&<+<,CJ8&<,AB (=)+'ABJ;J,CC,C&'|Q
lAB,C&w
8 :=,<&<+<,CJ8&<,AB D=*)+'ABJ;J,CC,C&'=,'5 +'H &':<8+2+'<8C&&'ABJ&' !8f);+T8 X5 +<&,C ));+' ,CVQl!8))OA'AB >798V?r));+' ,Cs'AB,C&(=NUu5 &<8 8M@;5?q<&Sz=,C+'E798[? ,Cg=AB GJOAB $+'<8C&&/15)+'ABJ;J,CC,C&'
1−ε &t"#8/,b/15T)+'ABJ;J,CC,C&'
εQl!,
w′ <&t3.=,<&<+<,CJ8&<,AB =)+'ABJ;J,CC,C&'<8++'<8C&&'*=:V?r));+' ,Cb=NUu5 &<8 8MOAB+'
∑
j|w′
j − wj| ≤ 2 ε
(5 ,C)+'A798 "4 !&6t<,5 &<&'#,C 09C,C&'#<&%/1 +<,_;bAB+'w′ ) 8&%j &<+'PNM)+<,C"4#5AB"-"443D<85N
5'<,AB =`= 8M\)+'A5'<8 `)+' "-, +DO5 &<8`8 &<,C+01^6&'AB,C+'' AB 3kAB,w2 &P
'5AB =-)O +<&<8+<J(.+'<8C&&f/15()+'ABJ;J,CC,C&'εQ9etG=,<& 5( ;&<8+' C*'8+=,<&<+<,CJ8&<,AB f=
)+'ABJ;J,CC,C&'(<&*=AB 5:3I AB+<"4`1
‖w − w′‖1 =∑
j|w′
j − wj| +C,'AB *5 &<&'G=N_ ,C&<,AB `z-="#&<+<,5*= <,C&'%+<J,C&<+,C+'6Q
!
N *L AB+<"4&<+B5R$& (')"N+* Od a"+*/ < A(
‖A‖ = Tr(√
AA∗)
et>)+'AB)OA<,C&<,AB #<8,C/ 9&'2UV,C&YfC, #/15.=,<&<+<,CJ8&<,AB =f)+'ABJ;J,CC,C&'2<8+sY+'<8C&&'=I=,O +' !&'I"4<8+'I798LV?qAB p) 8&IO5 &<8 +4'8+E8 <<&'H "4P7!8; 9&<,7!8X=5 +<,C&I);+8 "#&<+<,5G= <,C&'Q
4 a/u ;ρ, γ
$9 !L8*hhi(Q$9N) B"CK_B(‖ρ − γ‖ = maxU |Tr U(ρ − γ)|
/JG8*M! (NI'nE %. (1(8*hh h.*/ <GCl!, Π0,Π1
'AB !&%= 8M^)+'A<5 &' 8+'%AB+<&<ZAB01AB ;8MdB'A!5 ,6O5AB+<+'<)OAB = 9&IzL8 -"4<8+')+'A5 &<,C/1IJ,C ;,C+'[AB+':V?qAB)O +&' 8+
U = Π0 − Π1<&:8 ,C&,C+'Q@etP)+'AB)A',C&<,AB k)+'5= 9&'
':+'5 +<,C&
|Tr (Π0ρ) − Tr (Π0γ)| + |Tr (Π1ρ) − Tr (Π1γ)| ≤ ‖ρ − γ‖ (,C ',V[3F"4<8+'#)+'A<5 &<,C/1 Π0,Π1
)O +<"4 &=L=,<&<,C 08 +ρ
=γ
,t=AB L8 K+'<8C&&=,O +' !&.<8+
ρ &<8+
γ2/15:)+'ABJ;J,CC,C&'%8X)C8 ‖ρ− γ‖ Qet AB+<"4:&<+B5:"4'8+'>=AB 5:3
)+'ABJ;J,CC,C&'."#M,C"#(/15.3B798 C8 f"4'8+') +<"4 &T==,<&<,C 08 += 8MI"#&<+<,5= <,C&'Q
>AB8P,C !&<+'A!=8,AB D8 k= 8M,H "4"4'8+'^=^3K)+'AM,C"-,C&'== 8M &&'D7!8; 9&<,7!86);+UhAB,.)C8)+&<,798EzI8&<,CC,' +(7!8:3 AB+<"4G&<+B5 3%^$#"d B" Q uN
ρ, γ ∈ D(N )$9 !$8*hhi(7$!N)(hB"MC6NI*m`$#"d B"
F (ρ, γ)$!
ρ9$9
γ(N^$#"-u;':*BF (ρ, γ) = max|〈ξ|η〉|2 : Tr F (|ξ〉〈ξ|) = ρ, Tr F (|η〉〈η|) = γ
/J C-8*M!/ (N6'(O -. (>C9' T,*WW) |ξ〉 5 |η〉 $9 ρ
γ I. >(4.':*1i(
* !/dS*/ '( F $!O$/ #N)WBdimF = dimN C
et_;= C,C&'%<&+' C,%zI3 AB+<"4&<+B5G);+.)+'AB)+<, &'('8,C/B !&'6Q
4 CF (ρ, γ) = ‖√ρ
√γ‖2
C (1 − ‖ρ − γ‖
2
)2
≤ F (ρ, γ) ≤ 1 −(‖ρ − γ‖
2
)2
, + I[/J!7n:2#80; "8: 46:2;N#8!:
>AB8+)) AB (+),= "4 9&>,5 ,[:UuAB+<"#C,<"48&<,CC,'G)AB8+(=5 +<,C+'%.AB)O +&<,AB *="4N<8+' -"456 ,7!8.798; !&<,7989 =AB +== &,C6Q9Y 456B=fJO'AB,C [AB -)AB8+<+G'f+'NUh +' +zX~q v BQ
lAB,C& F 8 <);B5XZ +<"-,C&<, [ &ρ ∈ D(F)
8 $"#&<+<,5$= <,C&'D=5 +<,C/ 9&#V?q &&P=@? 8 <<&'H "47!8; 9&<,7!8Q|e[&)="4<8+')C8t01 +798YV?qAB E)8,''@5 &<8 +b'8+
ρ|?r)) C
8 $#( 'O':4 W gAB8 : n':4 W ('NL*1.BXp*/ +$k4\*4 <NQs( P&' CP"4<8+'#<&
=5 +<,C&'X);+I8 D5ABC5 &<,AB Xa, a ∈ A =L"#&<+<,5-)OA<,C&<,C/1-'X'AB"-"# 9&#z`V? ,= !&<,C&'
ΣaXa = IdQe[AB+''7!8:5 &<&':"4<8+':<&f))C,7!8G8D<<&'H "4
ρAB XABJ&<, !&>+'<8C&&
a ∈ A/15Y)+'ABJ;J,CC,C&'Tr (Xaρ)
Q6 @56B[Aρ = |Ψ〉〈Ψ| 5AB+<+'')AB =%z28 E &&[)8+632)+'ABJ;J,CC,C&'
=@?qABJ&' ,C+a
= /!, !& 〈Ψ|Xa|Ψ〉 Qe[(4 <(`'+Ll6<W ('AB 9&T8 (5 3B''*)+<,AB+<,;)C8T+'<&<+' ,C !&'*=@?qAB)O +&' 8+'T=("4<8+' C'AB !&T=AB );+8 .5ABC5 &<,AB Πa, a ∈ A =)+'A5 &' 8+'2B''A5 ,6Q v )O = 9&6!,C<&5AB 98^798[? 8 : ) 8&*&'AB8 AB8+'*j &<+'%<,C"E8G);+.8 %"4<8+'G)+'A<5 &<,C/1%<8+(8 F &&()C80+ =@@ d AB8& !&G=798J,C&'E8M,CC,3,C+'6Q@*,C <,VO= G5+))OAB+<& AB8 AB8:+'<&<+' ,C =+'AB 'AB8/1 9&4z`5AB <,= +' +4="4<8+')+'A5 &<,C/1|g5 )O = 9&45 3` [?q !&<+ X);B=P)O +<&'L=01 +C,C&'Q
Y n( nX 2:b
l AB W3^=N_ ,C&<,AB \= v AA9e[ /,C [s8 <<&'H "4X=L)+' 8/14 <&-5AB <&<,C&<8D=D= 8M"#B5'Z,C 2=.a[8+<,C 0
A &
B1+'<)5 &<,C/1 "4 !&TF'L % &4"N f Q9et)+'AB8/1 8+2=,')A'
=@? 8 D)8,' 5F W&' "-)4,CCC,C"-,C&'TAB+'#798DX/1 +<,_; 8+L<&-)OABC! AB"-,3VQet#= 8M 'AB 9&= &' +<"-,C ,<&'6)OA''H= 9&:8 I 9&<+'-5AB"-"E8
xO &*,C !&' +0,'' !&%=34"# ,H +'-<8,C/ 9&'
A565 8F8 ^5'Z;
y=`AB 08 8+L)OABC AB"-,3^z);+<&<,C+X=
x2 &#V?q5 +<,C&X<8+P8 +<8J;
<)O5 ,3t798B
)O 8&.C,C+'QB
565 8EAB+'.8 GUhAB 5 &<,AB fL(x, y)
&*B55 )&'E<,t &>' 8 "4 9&><,fL(x, y) = 1
Qetd5 3B''F 56)&<8+',C ',(P3 0901
L&' P7!8[? 8 `)+' 8/1`=@?r));+<&' ; 5=@? 8
5 +<&,C X 9&<+'x ∈ Σ∗ z
L)8,''Lj &<+'L5 +<,C&'#8 #UhAB,)OAB8+%&'AB8&'Dh5 &<&'L)+' 8/1X= ) =
=x
&=LNO &*/1 +<,_;-=E"# ,H +'-N #56B5);+:3#<8,C&'QOY o m] AB=fB'' +6OF,56C,Y &
*B51A ~ *]|)+'AB)OA' 9&G=I01 +C, +%5 &<&'-=N_ ,C&<,AB [[ &G=-5AB <,= +' +G=G'<&'H "4=)+' 8/1K ; L*16W (NQ1e? ,=*<&T798V?qAB P/B%8&'AB+<,' +3G"#B5'Z,C
BzG)A' +T=7!8<&<,AB Tz
A5'Z;B7!8:798'&<,AB D)AB8/ 9&= )O =+'=T+' )OAB 'f+' 8.8MX798<&<,AB f)+'5= !&'6Q v 5 ,
wT+' =-3>/,fJ685AB8)4)C8<,C"-)Ty5AB"-"4,CY=, 9& 8MV"4j "4B
)AB8/ 9&61 IUuAB 5 &<,AB =+' )OAB '.=
A= "# = +*=.5 3,C+'5 ,' "4 9&'><8+&' CGAB8$&' C:);+<&<,E=:3I)+' 8/1Q
=N_ ,C&#,C <,8 8*1N] $9 ( ; L*16W `^aRiQs* X&' CX"#B5'Z,C `=,<)A=@? 8 #+<8J; X=.&<+6/B,CV=@? 8 #+<8J; X=@?r6B69=@? 8 L+<8J; P=@?q !&<+' &T=(= 8M4+<8J; 2<)O5 ,38M=%5AB"-"E8 ,56&<,AB [V? 8 `+'' +</1IzI3-5 &<8+' &(V?r8&<+'z-V?q5 +<,C&<8+'Q> 8MF^aRTUhAB+<"4 !&*8 );,C+'4=I"#B5'Z,C =-a[8+<,C 0`,C 9&' +B5 &<,C/1<,Y CG);+<&01 !&%I"4j "4-+<8J; d=@?q !&<+'t &G<,V? 8 G5 +<,C&(<8+:+<8J; ^zI5 &<8+'G' 8G=:V?r8&<+'% &./!,5NV/1 +'iQ
uNc, s : N → [0, 1]
5m : N → N
$9( 3WSB)`( = s < c
m ∈ polyC f
IPc,s(m)<*$3<*4(8$!(?<*B#* 9(
LP>= & +!/N OK':*/ ' (A,B)
$98*1N# (O$9 8 ;N+*1W (%/"N( `*B
r; -. x ∈ L"< F > +* $ & N;h\"'P\BI j
A9j
B5F*('(8= A
B*/VN;;"<N:*B#"
m(|x|) 4,3* 9(,5B
*1'B'O7*B<9'L#\*#PhB"7* 8/ )c(|x|) C
r; I. x /∈ L. I7*9N# >$9 R NL*9W B A′ 5`<*Q':*/ < (A′, B)
*1B'\& ;h+"
x*B<'L#\*#PhB";* ' s(|x|) 5s*('(^= m(|x|) 4,3* 9(^*/VN;s" B";"< *B#"GC
m
f $/ +*[*/<B(Q= (A,B)( <'&'H "4=)+' 8/1,C 9&' +B5 &<,Uh': L
Cet#)8,' 5I=5*)+'AB&'A5AB=E)+' 8/1:,C 9&' +B5 &<,C/1:Uh8&> &<8=,I= ::=,M^
798,<8,C/,C+' 9&g 8+b,C !&<+'A!=85 &<,AB [5 8C"-,C ; !&s 8 T AB8/1 CT=N_ ,C&<,AB -=T3.5 3B''2 .|5AB"-"4V?q ' "%JE=3 0901*B="4 &<& !&*=()+' 8/1/1 +<,_J*=:"# ,H +'X\\*1i Q
>AB8bCAB g=@?rJAB+'=%+)) +b[)+<,C 5 ,C);8M%+'<8C&&'b5AB 5 +< ; 9&st'<&'H "4b=Y)+' 8/1,C 9&' +B5 &<,C/16)8,4 AB8P /1 +<+'AB 48 F))C,56&<,AB z^3 $/ \ B"Z$ & *(''LM! 8*WSB;Q v &<&'))C,56&<,AB D AB8E &'>8&<,C:= 2:56B=+'>=@? 8 X)+'AB&'A5AB=()+' 8/1:,C !&' +B5 &<,C/1G7!8; 9&<,7!8:z= 8MP)+'AB8/1 8+')OAB8+f>)+'ABJH "4 w >R ^:lR >*eO^Ya v2 RS &(zE&<+/1 +'C8,O)OAB8+235 3B'' *=AB !& AB8: [?r/1AB );B 8&' "-)=&' +<"-,C +V?q &<8=4/ 9&3L'AB8"-,'',AB =5:+))AB+<&6"#,(7!8,[<&.5 ) = !&(J+<,H /1 "4 !&*=,'5 8&'G= (3);+<&<, Q
G3 :2;I8/J 246:2;L-Lu^-
3%_ D=. (f +6 AB=fB'' +6:,CC,3 $ &f ,C01= +'AB ~ :@01 +C,' 9&Y3:5 3B'fR -,C !&<+'A!=8,i 9&s)+' 8/1,C 9&' +B5 &<,C/1fz>)C8<, 8+'Y)+'AB8/1 8+' Bf/1 +<,_; 8+) 8&g"#,C 9&' ; !&)OA' +b=s7!8<&<,AB 4z.8 0+ = AB"%J+'=T)+'AB8/1 8+'6B798, T5AB"-"E8 ,798 !&);Bf 9&<+' 8ML &f [?qAB !&);B.B55H.8ML5'Z; 01(=*"401f798,@AB 9&C, 8D !&<+':*/1 +<,_; 8+. &(8&<+')+'AB8/1 8+'6Q
lAB, 9&P1, . . . , Pk
=b"#B5'Z,C s=Ta[8+<,C 0:,C 9&' +B5 &<,C/1s=)8, 5T=T565 89 AB JAB+< &
V8 4"#B5'Z,C L=4at8+<,C 0F,C 9&' +B5 &<,C/1L&'AB8+< ; !& &' "-)%)ABC AB"-,3VQ v Z;B798
PiX8
+<8J; =^5AB"-"%8 ,56&<,AB n 5 +<,C&<8+'' 8^798,>C8,*' +<&DzK 9/1A61 +D=L"4'01DzV
QV
k+<8J; 4 5 +<,C&<8+'D 8D798,fC8,' +</1 9&Lz` !/1A61 +#=4"4'0148M
PiQ \)) C
(P1, . . . , Pk, V )8 'L1.6\/Q NL*9W ?j
k'+ B (NQ
/VN;c, s : N → [0, 1]
5m : N → N
$!(3WSB)$C( = s < c
m ∈ poly5.
k p ;Wu GC fMIPc,s(k,m)
<*$3<*4,68$9?<*B* 9(L
P7= & +!/N '+.6\C? ;N+*1W Oj k
'+ , (P1, . . . , Pk, V )4"N ^*B;
r; D. x ∈ L"< E L +* $ & N;h\"'`\BI j
V 1* ! Pi
51*('(= V*/V-N4/X#"7* ' m(|x|) #(,G* 9>jDN *#= Pi5 2' C ( k · m(|x|) \"B':B)6(,5
V*1B'*B<9'+#+*]dhB">* 8/ )c(|x|) C
r; %. x /∈ LT. k
i 'C F$9G8*1N# (O$9 7 ;N+*1W ( (P1, . . . , Pk)5V*1B'R\& N;h\"'
x*B'm'L#\*#PhB"R* ' s(|x|) 59*(' ([*p/ 4 p/X#"R* ' m(|x|)#((3* 9?jP *= Y'+ B (5( ' C (G+"2':B)6(GC
~ : *)+'AB8/1 !&X7!8$V?qAB )O 8&#&'AB8 <AB8+'LD+"4 +$zk8 )+'AB&'A!5AB^=$)+' 8/1z$= 8M)+'AB8/1 8+'6Qg 8 )+'HV? ,C !&<+'A!=85 &<,AB W=#5I<<&'H "4E=#)+' 8/1E,C 9&' +B5 &<,C/16s 8+)8,' 5NM5 )&<,AB C7!8,@ [?q &,C&.);B.=8F&'AB8&('AB8)OAB 8798:3z<&. &JC,
# ; * r
=r` >g
! #" N$ 5 " N% &(') r='* r
(2, 1) =+$,%r
*,C <,V|&'AB8&s3 0901f= s* )O 8&gj &<+'2/1 +<,_;2);+s8 2"#B5'Z,C =2a[8+<,C 0)+'ABJ;J,CC,<&' `&' "-)>)OABC! AB"-,3VQe? ,C !&<+'A!=85 &<,AB d=EV? ,C 9&' +B5 &<,C/,C&'4))OAB+<&',C <,g8 ^09,C =E)8,' 5
o
5AB <,= +JQ9 .J8&=("-, 8M-5AB"-)+' =+'*'<&'H "4T=)+' 8/1T,C 9&' +B5 &<,C/16(+'AB+ &TlU +#h &=@?r8&<+'T6/B !&T 8M;b)+'AB)A' !&T=fUV,C+'.));+ &<+'.(<,3 <(>8&<,CC,');+8 <<&'H "4E=)+' 8/1,C 9&' +B5 &<,C/1#NM)C,5 ,C&' "4 !&G= :3#=N_ ,C&<,AB =3L5 3B''=I5AB"-)NM,C&'Qv +'''AB8+'5*'AB 9&*)+<,C 5 ,C); "4 9&>3-7!8; 9&<,C&'%=@?r6B*8&<,CC,');+(/1 +<,_; 8+6,C <,b798 AB"%J+'D=L"4'01I5Z; 016YAB8 )C8I)+'5 ,' "4 9&-L AB"%J+'D=LJ,C&'I5AB"-"%8 ,7!84);+V?q ' "%J-=G)+'AB8/1 8+'%8/1 +<,_; 8+6Q v 5 ,=AB I3L=N_ ,C&<,AB d=3P5 3B'-=I5AB"-)NM,C&'PCP
)OAB8+.)+'ABJ;J,CC,<&<,56CC$5'Z5BJ%)+'A!AUui
:T E DbqXp N#PCP
(N`_* $"'>$ & %7*9N# ?$9 ( 1'+#+*#3iPN FNK'-':PXBB>S*/0BC:4"N 2 2A>*/X*B6*9'(Oj $ +*B .')"<GW*/π0.<*K'' 2ACI- r, q3B9$9( 3WSB)8$9
N$#*B)
N59B $/ +*q= & p4"N 2 LN
(r, q)i'( h< ;K(5F 4N *= N;h\"'m$99<B n
5 Wdn6E* ' r(n)hK$ & *<"+*/,5s d 6* ' q(n)
hK$9K<*;'< MC/VN;c, s : N → [0, 1]
C%= s < cC-N *I3_*/,9rmgIr
c,s[r, q]\B)n N\& NNm$9(
_** 9L': QC(G= oQb+!n ^4"N 2 V (r, q)
i'(Nh' ; = r; I. x ∈ L
5hg+!n OY'< πC>= V (x)
*1'2'π*B<E'+#+*#d B"* 8/ )
c(|x|) C r; %. x /∈ L
52T. ;'' B π5V (x)
*1'2'π*<K'+#+*]dhB">* ' s(|x|) C
Y UV, !&D/B+<, +$$);+"4H &<+'r
&qAB ABJ&<, 9&F8 3+<01 /1 9&,C=k5 3B''$=
5AB"-)NM,C&' );+NM "-)1&<+<,C/,3 "4 9&f = v
1,1/2[0, poly(n)]Q9>Y
=^R +'5 +<,C&
09 "4 9&G>Y = v
1,1/2[poly(n), poly(n)]QOY k='5 = 9&E5%+'<8C&&%zP ( &)+'H
)C8<, 8+'("4 C,AB+&<,AB ><855'<,C/1|AB `JAB8&<,C&>8$&<ZAB+'H "4 v
# : % 5 9N ' + & 9r =r1gIr
1,1/2[O(log n), O(1)]C
etp5AB <& !&''AB8 !&' =8);+DO(1)
K);+X3p<8,C&' &'J;,''kz3257!8,<&
AB)&<,C"#VQ* X"#B5'Z,C $=Dat8+<,C 0d)+'ABJ;J,CC,<&'D)O 8&4=AB 5X/1 +<,_; +4&'AB8&43 0901F=D &' "-))OABC! AB"-,3V@ k8&<,CC, !&%' 8 "4 !&G8 AB"%J+'-AB09+<,C&<Z"-,7!8#=J,C&'G=@?r6B6@ &G 8&<,CC, 9&*5 &*6B)OAB8+.+' 09+'= +(' 8 "4 !&(&<+'AB,.J,C&'(=3)+' 8/1Q
GV, [$[g13(3%W7n!() 137 "$( %M#&1K/ " 7[$[&! 8(54 7n!()
-5 &<&'$'5 &<,AB [s AB8I"4AB !&<+'AB #5AB"-"4 9&PX&<ZAB+'H "4 v )O +<"4 &4=@?qABJ&' ,C+#=+'<8C&&'X=@? ,C ;))+'AM,C"#J,CC,C&'d=^5 +<&,C X)+'ABJH "4D5AB"%J,C ;&'AB,C+'6( AB8X5AB 5 9&<+ !&<8+GV?qNM "-)#=
3:R `*aR *>eO^Ya v2 R
3:NQb*AB8E5AB"-"4 AB );+E=5 +<,C+'
5:)+'ABJH "4)8, AB8*=AB AB .8 D&<ZAB+'H "4E=@? ,C ;))+'AM,C"#J,CC,C&'E5AB 5 +< ; !&6Q
Dt1 N.*Bi3 ' $!%.*C
n( h<*R$B"Q$!%h+/EN) 83C(
U, V,W$9
\*B+$/ */d B"n52 F$ & 3 (i<N)6NC M ⊂ U × V × W
C f $/ +*R= & I)& * /h^$ & O ) .*B':4 W 1)& t+!n >$9 ! 3 l6<WSB) π : U → V
σ : U → WCQ=
∀u ∈ U (u, π(u), σ(u)) ∈ M
*)mC\*4E,B;h+*/ '(5^ $/ +*R= & I)& * /h`$ & Om )N.*%" *1W C # Nt;'+# N#
3 ' (N 9r^i\BK' BC
oo
" 7)Nh+*WSBCTetk)+' 8/1F)+'A!5H=`);+#+'=85 &<,AB z);+<&<,C+X=3 lYaQT>AB843d=5 +<,C/1AB
J+<,H /1 "4 9&256+ C' /!,C+:z: &JC,C+V? ,C ;))+'A6M,C"#J,CC,C&'QlAB,C&ϕ
8 f,C <& 53 lYaQ - AB&'
(ϕj)j=1,...,m5 38f));+, 9&f=
ϕ! &
(xi)i=1,...,n/+<,3J6QetG5AB '&<+<85 &<,AB X=
V? ,C 9& 53: 5AB+<+'<)OAB = 9&''*Uu,C&( D&<+'AB,* &)|Q
d5AB"-"4 5P);+G,C 9&<+'A=8,C+'#=G/B+<,3Jxi[j],¬xi[j] ∈ U, ai[j] ∈ V
&bi[j] ∈ W)OAB8+%&'AB8
(i, j)&' %7!84'AB,C&
xib'AB,C& ¬xi
));+,'#= ϕj
Q ,C ' +'#G&<+<,C) &'(xi[j], ai[j], bi[j])
&(¬xi[j], ai[j
′], bi[j])=
MgA
j′<&P)+' "-, +,C =,5
j′ > j "4A=8Am
t&' 7!8xi
));+,''= ϕj′
Q v s&<+<,C) &''AB 9&=s&<+<,C) &'=@?@*/,( B*1WWB$9%4"NhB" Q
Y <8,C&')OAB8+>5Z;B7985 38'ϕj
AB `,C !&<+'A!=8,C&=Ep*BW*#3C(8$9QG*W G*1WSBc1[j] ∈ V &
c2[j] ∈ W &:V?qAB ,C 'H +'&'AB8:>&<+<,C) &'
(xi[j], c1[j], c2[j])AB8
(¬xi[j], c1[j], c2[j])&' (7!8:+'<)O5 &<,C/1 "4 9&xi
AB8 ¬xi));+,''%=
ϕjQ
g,C ; "4 9&6@AB `,C 9&<+'A=8,C&>=Ep*BW*]CY': ' pi,k[j] ∈ V &
qi,k[j] ∈ W)AB8+>&'AB8&
(i, j)&' 7!8I'AB,C&
xi@'AB,C& ¬xi
));+,''-= ϕj
&&'AB8&k ∈ 1, . . . , 4 Q k,C 5 C8&
AB+'&'AB8:>&<+<,C) &'(xl[j], pi,k[j], qi,k[j])
&(¬xl[j], pi,k[j], qi,k[j])
@)OAB8+k
C3 !&G=1
z4&' .7!8G'AB,C&
xl'AB,C& ¬xl
));+,'G= ϕj
QR: [?q<&FAB+'$);B$=, #5 ,C=/1AB,C+F7!8[? 8 ,C <& 5
3 lYa 5AB =8,C&`zp8 ,C '& 5
3:
)A',C&<,C/1(<,O &T' 8 "4 !&<,3UhAB+<"E83 lYa &,C&f&<,_JQe? ,=:<&T7!8*&'AB82&<+<,C) &'
5AB+<+'<)OAB = 9&L8Mai[j]
&bi[j]
5AB <,<&' 9&Lz= &' +<"-,C +#8 FB'',C0 ;&<,AB \5AB <,<& !&'$=/B+<,3Jt4&<+<,C) &'L5AB+<+'<)OAB = 9&D8M
c1[j] &
c2[j]09+ 9&<,' 9&X798F3k5 38'
ϕj<&
&<,_J &2_ ; "4 9&.2/+<,3Jf)AB8JO C:' +</1 9&.' 9&<, C "4 !&*z%5:7!8U
V
&W, 9&."4j "4%56+'=,C ;C,C&'Q
^AB 9&<+'AB Yz.)+' 9&s79825)+'ABJH "4fB="4 &s8 DwYUuA''gy5?q<&Sz=,C+'2798[? ,C<&g=8+g=Uu,C+'3D=,@ +' 5P !&<+'P=E,C <& 5E)A<,C&<,C/1|[ &E=E,C <& 5798,AB 9&%AB,C =@?qj &<+'4)OA<,C&<,C/16Qv AB"-"4 AB ();+"4AB !&<+' +(V?qNM,<&' 5G=@? 8 DUuA'':)OAB8+
3 lYaQ
2Wεi #*('i
3iB 1'+# N#R p*B;K" .*B;m$B;"' GBN 3
i 5Y$#" NN> 7(. (74G_* 73B;;( .*B"N#N;Y3*W^*#(,5K 3uNc(9* ' O N+*1WSB 1 − ε
$916(%3_* 6(13; .*Ba"N#N;G*W`*#3C(MC4 b+!/N ε1 > 0
= ε1i *'i
3i. G/h 9rFi \BK' BC
" 7)Nh+*WSBC v :)+'ABJH "4<&5 3,C+' "4 !&(= . bQ gAB8+."4AB 9&<+' +(7!8>&'AB8&f)+'ABJH "4G6? +'=8,C&61V? ,=*<&T=@? 8&<,CC,' +T8 v K)OAB8+T bQlAB,C&T=AB 5
L ∈ p &69 AB 4.&<ZAB+'H "4 o w 18 /1 +<,_; 8+
V8&<,CC, 9&
O(log n)J,C&:=@?r6B6Uu,i 9&:8 ` AB"EJ+'-5AB <& 9&
q=%+'7!8j &'z#8
)+' 8/1π &.&' g798',
x ∈ LAB+'.,CbNM,<&'
π&' b798
V (x)B55 )&'
π/15E5 +<&<,C&<8=AB+'
798%<,x /∈ L
)OAB8+(&'AB8&')+' 8/1πV (x)
B55 )&'E6/15G)+'ABJ;J,CC,C&'I8$)C81/2
5?q<&Sz=,C+'798[?r8D"4AB,C (3"4AB,C&<,%='(5'Z; (=@?r6B(5AB =8,' 9&:zI+' < &' +
πQ
gAB8+E5'Z;B7!845Z; r
=O(log n)
J,C&'E=@?r6B6@-/1 +<,_; 8+E+' 09+'=q
J,C&'i1, . . . , iq
=3L)+' 8/1QtRSY)+' =dAB+'%iX=5 ,<,AB d kUuAB 5 &<,AB d=45J,C&'|[5?q<&Sz=,C+'#7!8[? ,C565 848 UhAB 5 &<,AB `JAAB
fr,x : 0, 1q → 0, 1 &B55 )&'E',t &> 8 "4 9&:<,fr,x(i1, . . . , iq) = 1
Q ) 8&I&<+ UuAB+<"4 +
fr,x=L"# ,H +'D<& =+'=W 8 W ' "%JD=@?r8 )C8
2q 5 384z3/B+<,3J.);+<"-,
i1, . . . , iq&' C.7!8
fr,x(i1, . . . , iq) = 1<,[ &.' 8 "4 !&(<,@&'AB8&'.5(5 38'
'AB 9&&<,<_J6QY X)+'A!5= !&.=*5 &<&'>"# ,H +'AB DABJ&<, 9&2O(log n) · 2q = poly(n)
5 38'|
o$>
5AB+<+'<)OAB = 9&%z#&'AB8&':>5Z; G=@?r6B6Q +6OAB k,C&>7!8O<,x ∈ L
,CYNM,<&'E8 )+' 8/1798
V (x)B55 )&'G/15G)+'ABJ;J,CC,C&'
1Q v ?q<&Sz=,C+'G7!8[? ,C[NM,<&':8 GB<,C0 ;&<,AB $=/+<,3J
ik&' CD7!8X&'AB8&'-45 38-'AB, 9&#i&<,Uu,C&'|Q s+#5AB !&<+'<,
x /∈ L8W)C8I3^"4AB,C&<,
=-5'Z; P=@?r6Br
5AB =8, 9&4X/1 +<,_; 8+PzkB55 )&' +6 &4=AB 5$8W"4AB,C 43^"4AB,C&<,D=fr,x
6?q /C8 9&4z0b &48p"4AB,C I8 L=I5 38'-5AB+<+'')AB = !&'- [?q<&);Bi&<,Uu,C&'L);+
V?rB'<,C0 ;&<,AB ^5AB+<+'<)AB = !&:zπQ;(,C <,V; $)Ai 9&
ε1 = 1/2q+1 ;8 :Uh+B5 &<,AB ^8$)C81 − ε1=
poly(n)5 38'.7!8:V?qAB `-ABJ&' 98*'AB 9&*<,C"%8C& "4 !&>&<,<_J6Q
etX/ 8+EAB)&<,C"##)AB8+ε<&
1/8QgRY [?q'&%);B%=, #5 ,CP=-/1AB,C+E7!8[? 8 4<&<+&' 0,#0AB8
&'AB O5AB <,<& 9&%z#)+' =+'I:5 38':8 );+:8 &GzLB'<,C0 +:>/B+<,3J: AB k 5AB+'= &' +<"-,C I=4"# ,H +'Dz$&<,<Uu,C+'#3$5 38'P<,)OA'<,CJP5AB =8,C&Iz$8 LB'',C0 ;&<,AB K&<,Uu,i 9&.8 GU +B5 &<,AB
7/8=*5 38'( F"4A61 Q F)O 8&.)+'AB8/1 +*7!8G55 ,t<&*AB)&<,C"#b
8&<,CC, 9&( v \= B<&B=k~ o Qe[:UuA'':7!8:V?qAB `IABJ&' 98`)AB8+
ε1V09)
3 lYa 6?q &' =Fz
3>pQ
2I1εi *('i
3 ' Cs'+# N#` p*B; " .*;s$B" h )N.* (U, V,W,M)$93 ' 5$#"Ni> N9)& -)& * /h6$& K )N.*Bh':4 W m (5') :. (m3 l6<WSB) π : U →
V
σ : U → W5^* ' NL*9WW 1 − ε
$9h~' (u, π(u), σ(u))
56': u ∈ U5h3;
$*B)MC
4 b+!/N ε2 > 0 = ε2
i *'i3 ' 3/h Kr^i \9'C BC
" 7)Nh+*WSB +'=8,C&DzK);+<&<,C+D=ε1V09)
3 lYaQbAAB
ε2 = ε1/6Q.l!8))OA'AB X798
V?qAB ,E= 8M^J, 5 &<,AB π
&σ
&' C7!8[? 8 GUh+B5 &<,AB k<8) +<, 8+'-z1 − ε2 = (6 − ε1)/6
=&<+<,C) &'
(u, π(u), σ(u))'AB, !&.=
MQ>AB8f,C0 AB+'AB f&'AB82&<+<,C) &'.798,O :'AB !&);B=
MQ v AB"-"4PAB V?rD/8kAB+'%=#3X+'=85 &<,AB K=
3 lYa z
3:p@8 d5'ZAB,Md=4&<+<,C) &'%/C,=
5AB+<+'<)OAB =^z#8 B<,C0 ;&<,AB k=(/+<,3J6QORS5 ,VAB ^ABJ&<, !&G=AB 5%8 IB'<,C0 ;&<,AB );+<&<, C7982V?qAB 4 &' =#z*&'AB8&'Ys/+<,3J
xi=f"# ,H +'.+<J,C&<+,C+'Q +' "#+'7!8.7!8f5'Z;B7!8
xi[j],C 9&' +</, 9&s= b8E)C86&<+<,C) &'6Q v AB"-"4TAB E.<8))A798[? 8 U +B5 &<,AB
(6−ε1)/6=b&<+<,C) &'
&, 9&E/BC,=6b5?q<&%7!8[?r8"4AB,C G8 4U +B5 &<,AB 1 − ε1
=#5%&<+<,C) &'%5AB+<+'<)OAB = 9&-zD=&<+<,C) &'T=*&<,UVB5 &<,AB [Q9e[25 38'T798,O));+,'' 9&f= 25&<+<,C) &'2'AB 9&2&'AB8&'Ti&<,Uu,C&');+V?rB'',C0 ;&<,AB P7!8.V?qAB LG5AB <&<+<8,C&'h56+
π &
σAB !&2=J, 5 &<,AB NNQ Pi&<,Uu,C&2=AB 5.8
U +B5 &<,AB 1− ε1
=25 386 &2=AB 5>V? ,C <& 5ε1V09)
3 lYa &,C&f)OA<,C&<,C/1!5:7!8,5AB 5 C8&f3
+'=85 &<,AB [Q
o w
Y n( :b : Y 2YbO. Y
>AB8t)+'' !&'AB st)+<,C 5 ,C);8MI<<&'H "4t=T)+' 8/1g,C 9&' +B5 &<,C/17!8; 9&<,7!8Y/15T8 ' 8)+'AB8/1 8+27!8,AB !&2 &'* &<8=, 8'7!8[?rz)+'' !&6! &T AB8T L=AB AB 2)+<,C 5 ,C);2)+'AB)+<, &'6Qv 5 ,O AB8) +<"4 &=>+))O +.5 +<&,C +'<8C&&'2UuAB ="4 9&8M$5AB 5 +< ; 9&>52'<&'H "4=)+' 8/1697!8V?qAB P+' 8&<,CC,' +% '8,C&'.= *56B=+'(=T<'&'H "4Tz:)C8<, 8+'T)+'AB8/1 8+'6Q!*AB8=AB AB * 09 "4 !&=.)+'ABJH "4:5AB"-) &'()OAB8+*5Z;B5 8 I=G5*5 3B'6 `*NM)+<,C"# 9&=D"# ,H +'^5ABZ +' 9&'25$798,() +<"4 &L=$/1AB,C+4 8+'L=,O +' 5L=@?qNM)+''<,C/,C&'$5AB"-"4F=5'Z; 01 "4 !&'*=G7!8; 9&<,_;56&' 8+'*,CtNM,<&'7!8 [798'AB,C&*= )+'ABJH "4>5AB <,= +'6Q
+ 3 76%M1)79;N;:I -
e[g<<&'H "4[=)+' 8/1g,C !&' +B5 &<,C/1g7!8; 9&<,7!8gAB 9&s &',C 9&<+'A=8,C&'g o mmm);+gEQ:,C&B / &Qsd&<+'AB8$~q: s \ ;AB0,F/15X4'<&'H "4I=X)+' 8/1-,C !&' +B5 &<,C/1#5 3B'',7986Q:X"# ,H +'$,C UuAB+<"4 CY8 \<<&'H "4D=X)+' 8/1$,C !&' +B5 &<,UG798; !&<,798`<&45AB"-)AD=$= 8M);+<&<, 8 $/1 +<,_; 8+(=,')A !&(=@? 8 FAB+'=,C ;&' 8+(7!8; 9&<,7!8; &.7!8,[=AB,C&(6?qNM5 8&' +. $&' "-))ABC AB"-,3V &f8 D)+'AB8/1 8+(7!8, [?q'&2C,C"-,C&'7!8:);+ffAB,.=>3"456 ,7987!8; 9&<,7!8Qe[)+'AB8/1 8+G &:%/1 +<,_; 8+G5AB"-"E8 ,798 !&zP&<+/1 +':8 k56 ;7!8; 9&<,7!8Q@e[J8&:=8)+'AB8/1 8+<&:=5AB !/B,C 5 +'4E/1 +<,_; 8+G=V?r));+<&' ; 54=@? 8 - !&<+'
x5AB"-"%8 4zP8 3 0901
LQ
e[-/1 +<,_; 8+=AB,C&%/1 ,CC +z$5AB 9&<+'AB +G,C UuAB+<"#&<,AB E7!8-C8,YUuAB8+< ,C&%-)+'AB8/1 8+6b56+%,C ) 8&fUV,C+'G5AB _ 5%z45= +< , +6Q
AB+<"4 C "4 !&698 D4" ( 2 %= *B;WS= *<&Y8 fUuAB 5 &<,AB V : Σ∗ → Σ∗ 565 83J( -&' "-)
)ABC AB"-,3! E3AB 08 8+s='AB 9&<+'Q bAB8+s5'Z;B7!8x ∈ Σ∗ V (x)
<&[,C !&' +<)+' &'25AB"-"4T8 k(|x|) V&<8) (V1(x), . . . , Vk(|x|)(x))
)OAB8+(8 5 +<&,C EUhAB 5 &<,AB k
JOAB+< %);+>8 ^)ABC B"4QbAB8+T5'Z;B7!8
iVi(x)
<&Y3='5 +<,C)&<,AB P=@? 8 #5 ,C+'5 8,C&T7!8; 9&<,7!8(0,' !&<8+qV (|x|)+qM (|x|)798J,C&'|A
qV &
qM'AB !&*= 8MDUuAB 5 &<,AB (JOAB+< *);+(=()OABC B"46Q;e[
qV (|x|) )+' "-, +'798J,C&'#'AB 9&-,C !&' +<)+' &'P5AB"-"4FV?q<);B5X)+'AB)+'F=8W/1 +<,_; 8+62AB+'4798X
qM (|x|) 8&<+'5AB <&<,C&<8 9&bV?q<);B5=t"4'016Q|b,C ; "4 !&6|)+' "-, +b7!8J,C&t=YV?q<);B5)+<,C/1=8G/1 +<,_; 8+<&.=<,C0 G5AB"-"4% & 9&(7!8J,C&(='AB+<&<,Q
* ?'L m= *B;WS= >'&=N_ ,;=f"# ,H +'* ;AB08 95?q<&Y8 fUuAB 5 &<,AB P : Σ∗ → Σ∗
<8+3B7!8 CAB $ [? ,C"-)AG85 8 +'<&<+<,5 &<,AB F=G565 83J,CC,C&'Q bAB8+*5'Z;B7!8x ∈ Σ∗ P (x)
<&,C 9&' +<)+' &'*5AB"-"48
l(|x|) V&<8) (P1(x), . . . , Pl(|x|)(x)))AB8+Y8 (5 +<&,C fUhAB 5 &<,AB
lJOAB+<
);+b8 )OABC! B"4QgAB8+s5Z;B798iPi(x)
<&g3(=5 +<,C)&<,AB I=@? 8 -5 ,C+'5 8,C&Y7!8; 9&<,7!8f0, 9&s<8+qP (|x|)+ qM (|x|) 798J,C&'|9A
qM'&8 UuAB 5 &<,AB PJOAB+< .);+T8 P)OABC B"41"#,T,C; [? 4);B
=+'<&<+<,5 &<,AB <8+qP
QetqP (|x|) )+' "-, +'%798J,C&''AB !&,C 9&' +<)+' &'G5AB"-"4IV?q');B5I)+'AB)+'
=8D/1 +<,_; 8+6AB+'.7!8qM(|x|) 8&<+'(5AB <&<,C&<8 9&:zI AB8/168$V?q<);B5G=("4'016Q
* K)+'AB8/1 8+# &I8 /1 +<,_; 8+47!8; 9&<,7!84' +'AB 9&-=,C&'$\9')*WS3C(F',Vb)OAB8+I&'AB8&'D 9&<+'xY5'Z;B7!8
Vi(x) &
Pi(x)6?rB55AB+'= 9&L<8+-X AB"%J+'`=D798J,C&'P=XV?q<);B5$=-"4016
&G<,k(|x|) = bm(|x|)/2 + 1c &
l(|x|) = bm(|x|)/2 + 1/2c )OAB8+8 -5 +<&,C -UuAB 5 &<,AB m
+' )+'' 9& !&G AB"EJ+'-=G"4'01:5Z; 016Q ^=,C+P= *556B:798
V<&(8 ^/1 +<,_; 8+
zm
"4'016O &P
8 ^)+'AB8/1 8+Gzm
"4'016Q >&'AB8&'E3P<8,C&'OAB <8))A%&'AB8 <AB8+'798*25AB8)2UhAB+<"4f=@? 8 L)+'AB8/1 8+. &2=@? 8 X/1 +<,_; 8+798*V?qAB X5AB <,=H +':'AB 9&5AB"-);&<,CJ6Q
o o
& !&P=AB P8 /1 +<,_; 8+V
Y8 )+'AB8/1 8+P
 F 9&<+'xAB \=N_ ,C&P8 \5 ,C+'5 8,C&
(V, P )(x)0,' !&<8+
q(|x|) = qV (|x|) + qM(|x|) + qP (|x|) 798J,C&':=E3#UuBAB <8,C/ 9&'Q[l!,m(|x|) <&,C"-);,C+6.5 ,C+'5 8,C&'
P1(x), V1(x), . . . , P(m(|x|)+1)/2(x), V(m(|x|)+1)/2(x)'AB 9&>))C,798(.8 *z34<8,C&'G=*8&<+'|'AB,C&*8M
qV (|x|) + qM(|x|) 7!8J,C&'(=8F/1 +<,_; 8+6'AB,C&.8M
qP (|x|) + qM (|x|) =8D)+'AB8/1 8+6Ql!,m(|x|) <&f);,C+65?q<&f:/1 +<,_; 8+(7!8,[))C,798:
)+' "-, +(5 ,C+'5 8,C&AB $NM5 8&'3-798 5
V1(x), P1(x), . . . , Pm(|x|)/2(x), Vm(|x|)/2+1)(x)
g,C ; "4 9&6)OAB8+f8 > !&<+'x3G)+'ABJ;J,CC,C&'%798*:5AB8)
(V, P )B55 )&'
x<&2=N_ ,
5AB"-"4% & !&*3I)+'ABJ;J,CC,C&'I=@?qABJ&' ,C+ |1〉 AB+'(=@? 8 %"4<8+'%=8^798J,C&>=G'AB+<&<,%=8`/1 +<,_; 8+= X3pJ;B'56 AB ,798
(|0〉, |1〉) :)+'H$NM5 8&<,AB =85 ,C+'5 8,C&(V, P )(x)
'8+LV?q &&D,C ,C&<,3|0〉⊗q(|x|) Q
>AB8)AB8/1AB *zI)+'' 9&*=N_ ,C+.3-5 3B''=5AB"-)NM,C&'x>R
UT /VN;m : N → N
c, s : N → [0, 1]
$93WSB)E(>= s(n) < c(n)': E. nC f r
c,s(m)<*D3<*4,6>$9(O<*B* 9(
Lo>= & s+!/N k4" ( 2 V
jm
#(,G* 9(%3= +!/N Z'+ B P
jm
#(,G* 9(X^= (5`': L. x ∈ L5
(V, P )(x)*1B'
*BB<K'+#+*#3d B">* 8/ c(|x|) C
r; >. '+ Pj
m#(,G* 9((5Y g. x /∈ L
5(V, P )(x)
*1'2'1*BB<h'+#+*#d B"* ' s(|x|) Cfq*(''
c ':*B+*B h'O$!:\BK'@" $!O$ '+.6\/(52
s(N GBQ':*B+*B h<Q$9:\i'<WSBC
f`N+*R$!F' r(m) =
r1, 1
2
(m)': %. m
C
V1(x) V2(x) V3(x)
P1(x) P2(x)
tGupv4wyx.z.w3Lz uk.w/Lu
tGupv4wyx.z ;kzSz kz
tGupv4wyx.z.w3Lz u.u3Lup
tGu4vpw~x zS(Sx.w~
^F > l5'Z "#4=@? 8 `<'&'H "4=:)+' 8/1,C 9&' +B5 &<,C/1798; !&<,798etP798'&<,AB ` ;&<8+' CI7!8%V?qAB 'E)A'z4)+'AB)OA:=()+'AB&'A5AB=E)+' 8/1*,C !&' +B5 &<,C/1
798; !&<,798-'& 8+4NM)+''<,C/,C&'QsY );+<&<,5 8C, +65AB"-"4 9&P'X5AB"-);+'NV& CFz^5 CD=P35 3B''=:)+' 8/1(,C 9&' +B5 &<,C/1>5 3B'<,7!8.R
o ]
>AB8G+' "#+'7!8AB 7!84V? ,C 5 C8',AB pR ⊂ x>R n [?q<&%);BE&<+<,C/!,3QbY K@ &6[3$&<+B=85 &<,AB ;&<8+' C*=@? 8 P)+'AB&'A!5AB>R K P8 #)+'AB&'A5AB:x>R K5AB ',<&'.)OAB8+(/1 +<,_; 8+fz% 9/1A61 +fV?q 5A=01=798'&<,AB .5 3B'<,7!8798[? ,Ct8+,C&)OA'= :)+'AB&'A5AB5 3B'<,7!8= =. &&'798; !&<,798=E3PJ;B'56 AB ,798@ &zL<'&' "#&<,798 "4 !&>"4'8+' +*+' )OAB ':7!8[? ,Cb+'AB,C&=8I)+'AB8/1 8+= s3*J;B'f56 AB ,798Q v )O = 9&6955 , [?q "-)Oj5'Z.);BsT)+'AB8/1 8+T=@?q 5'Z /1jN&<+' +'s=,O +' !&'+' )OAB 'Y 9&<+'( C6 &,C <,=@?q /1 9&<8 C "4 !&2), 01 +Yf/1 +<,_; 8+6Q 5' 6 *(>)+'AB8/1.8 `wT&<ZAB+'H "4 v K798; !&<,798TyL~ >|];798,"4AB !&<+'(' 9&<, C "4 9&27983I5 3B''G=5AB"-)NM,C&'%5AB+<+'')AB = !&>z-=.'<&'H "4(=)+' 8/1.,C !&' +B5 &<,C/1:zI= 8MD&'AB8+'h798;&<+'*"4'01NNAL*)+' "-, +&'AB8+2<&f7!8; 9&<,7!8> &f>= 8M,H "4:5 3B'',7985AB+<+'<)OAB =Xz3-5 3B''%*Y (; &(<&>,C ',[J685AB8)`)C8()8, 9&'zI3IUuAB,(7!83-5 3B'G5AB+<+'<)OAB = 9&z#= 8M^&'AB8+'5 3B'<,7!86)8,'7!8R
=Tl v ~ l!Z;m > O &7983L5 3B''I5AB+<+'<)OAB = 9&%z
= 8ML&'AB8+'*798; !&<,798()8,'7!85AB"-"4G AB8(CAB (:/1AB,C+6x>R ⊂ Y .Qv )O = 9&6d&<+'AB8L^)+'AB8/1 <8,C/ 9^)+' 8/1F5 3B'<,7!8Y7!8 Tl v B="4 &
&,C&*8 k<<&'H "4E=E)+' 8/1>,C !&' +B5 &<,C/1G798; !&<,798G6/158 ^ AB"EJ+'I5AB <& 9&G=E"4'01~ d&mmQY $&' +<"4*=5 3B''.=%5AB"-)NM,C&'.,C 5 C8<,AB (<8,C/ 9&'*'AB 9&*5AB 98
# ZTs 5 * r^:r >g ⊂ r1, 1
2
(3) ⊂ $,KretL)+' 8/14=43X= 8M,H "4-,C 5 C8',AB );B'');+GV? 8&<,CC,&<,AB K=I3['+B+*BI7*1WWBc ?i
$#"-u QO 8 k)+'AB&'A5AB4=)+' 8/1G,C !&' +B5 &<,C/1%7!8; 9&<,7!86@V?rB5 &<,AB =8k)+'AB8/1 8+6[J, 798D5AB"-)8&&<,AB C "4 !&D,CCC,C"-,C&'T'&-+' 0,$);+4#AB,#=D3"456 ,7!8F7!8; 9&<,7!8Y );+<&<,5 8C, +**5 ,C+'5 8,C&'*7!8[? ,Cb))C,7!8Ez-5Z;B798 &)O) 8/1 !&>j &<+'E=5 +<,C&'.);+>=."#&<+<,58 ,C&,C+'6 &V?q &&>=.'f+' 0,<&<+'*<&8 "#&<+<,5G= <,C&'Q
lAB,C&(V1, V2)
8 /1 +<,_; 8+)OAB8+:8 ^)+'AB&'A5AB4x>R z#&<+'AB,:"4'01|Q AB&'( &+' 0,<&<+'%+'<)5 &<,C/1 "4 !&=8d/1 +<,_; 8+6t=G"401G &%=8)+'AB8/1 8+6g & V M & P <);B5.= ,CCJ +<&.5AB+<+'')AB = !&'6Q $ AB&'>_ ; "4 !&
q q
&q
AB"%J+'*=798J,C&'=5'Z;B5 8 ^=&<+'AB,.+' 0,<&<+'|Q>N_ ,'AB ..AB)O +&' 8+'
T1 :
D(M) −→ D(V)
ρ 7−→ TrM (V1(|0 q 〉〈0 q | ⊗ ρ)V ∗1 )
&T2 :
D(B q −1+q ) −→ D(V)
ρ 7−→ TrM (V ∗2 (|1〉〈1| ⊗ ρ)V2)
R!<&sAB+'g5 3,C+s798V?qNM,<&' 52=@? 8 )+'AB8/1 8+g'Uu,i 9&YB55 )&' +s6/15T)+'ABJ;J,CC,C&'1,C"-)C,7!8
V?qNM,'&' 5=-= 8M^"#&<+<,5G= <,C&'ρ1
&ρ2
&' C7!8T1(ρ1) = T2(ρ2)
Q "4AB 9&<+'-+'5 ,)+'A!7!8 "4 9&.7!8<,O3E)+'ABJ;J,CC,C&'"#M,C"#:)OAB8+f8 X)+'AB8/1 8+.=@?qj &<+'GB55 )&'<&f,C Uh +<, 8+'z
1/2AB+')OAB8+.&'AB8&'"#&<+<,5(= ',C&'
ρ1 &
ρ2
‖T1(ρ1) − T2(ρ2)‖ ≥ 2−q −4
);+<&<,C+2=.3z,C; [?q<&);B=, #5 ,C*=('.+"4 +2z:3:+'ABC8&<,AB P=@? 8 #)+'ABJH "4*=.)+'AB0+"I"#&<,AB $' "-,=N_ ,G<8+.=f"#&<+<,5*=>&,CCGNM)OAB 9&<, C&' [798:3= &' +<"-,C ;&<,AB `=:3/B 8+g=3('ABC8&<,AB EAB)&<,C"#f/15T8 )+'5 ,',AB ENM)AB !&<, C2'8 #'z.= &' +<"-,C +Y6? ,C!NM,<&'8 -)+'AB8/1 8+)OAB8/B !&'2UV,C+'.B55 )&' +T/15)+'ABJ;J,CC,C&'
11AB84<,;845AB 9&<+,C+'>85 8 4)+'AB8/1 8+
o
I) 8&E'Uu,C+'4B55 )&' +I/15-)+'ABJ;J,CC,C&'P<8)O +<, 8+'#z1/2
Q v AB"-"44AB d5AB ; &E=%C01A+<,C&<Z"4:)ABC AB"-,38Mk=E+'ABC8&<,AB =:)+'AB0+"-"4%' "-,=N_ ,6@AB k k==8,C&V? ,C 5 C8<,AB x>R
(3) ⊂ Y *Qetf)+' 8/1=5 &<&'Y,C 5 C8<,AB %AB8C,C0 YsC, G ;&<8+' ! !&<+'Y)+' 8/1[,C !&' +B5 &<,C/1b7!8; 9&<,7!8
&*)+'AB0+"-"#&<,AB "-,=N_ ,C, 7!8,b ; &:=>AB,*=E34"456 ,7!8I798; !&<,798 8 ^ &&798; !&<,798I<&:=N_ ,Y);+>8 E"#&<+<,5-= ',C&'5?q'&Sz=,C+'I8 E"#&<+<,5)OA<,C&<,C/1I=&<+B5
1Q
*AB8-=AB +'AB Lz^3^);+<&<,F]Q o Q w 8 $))C,56&<,AB =D5DC, = -D56B=+'$=4'<&'H "4=F)+' 8/1Dzd)C8<, 8+'X)+'AB8/1 8+'6f #+' C,3 !&Fz8 C01AB+<,C&<Z"4=@?r))+'AM,C"#&<,AB )AB8+`* v >aQ
>AB8I&' +<"-,C AB 4 W=AB ; 9&48 L)+'AB)+<, &'$=L+'=85 &<,AB \=LV?q +<+' 8+# &-=8 AB"%J+'$="4'01)+'AB8/1();+(:,C&B /X &.d&<+'AB8.AB+'.=GV? ,C 9&<+'A=85 &<,AB `=3I5 3B'Ex>R (Q
4 T-T a/u ;k, p ∈ poly
5ε ∈ poly−1 5%
c, s : N → [0, 1](q=
∀n ∈ N, c(n) − s(n) ≥ ε(n)CK_B(
QIPc,s(k) ⊂ QIP1,2−p(3)C
+ V, -.! N %W *13:2; #8
:J015(3 PX#$79*!(5PX#$:2;
e[ 8M4=.&S!)O.*+<&<Z98+^ +<C,C [9,C !&<+'A!=8,C&'T);+ J;,g~fJ;]B= *56B5 3B'<,7!8)8,);+Y^+<+<,AB&<& &sd&<+'AB8.~q`oB= Yf56BY7!8; 9&<,7!85AB <&<,C&<8 9&8 25 3B'2+'<&<+' ,C 9&'.=)+' 8/1,C !&' +B5 &<,C/16= 3B7!8 C*.+'<8C&&f=&<,C+01f6&'AB,C+'fO5 &<82);+2(+<&<Z!8+2<&)8JC,5Q!*+<&<Z98+2=<,C0 *./1 +<,_; 8+6 &T^ +<C,C #.)+'AB8/1 8+6Q v AB"-"4*` +<C,C X<&&'AB8&V)8,'i 9&6 &5AB ; &5'ZAB,MP6&'AB,C+'2O5 &<8);+(+<&<Z!8+28-Uh8+ &Tz"4<8+'(7!85 C8,5 ,;Uu,C&6,C [?q<&.);B( 5',C+'Ez4(+<&<Z!8+*=@?q 9/1A|1 +="4'01>z4^ +<C,C G` +<C,C `) 8&((= &' +<"-,C +C8,V"4j "4# d-J;B 9&<8+GV?r6B%)8JC,5Qt>AB8G AB8G,C !&' +'''AB E)C8G);+<&<,5 8C,H +' "4 !&,5 ,8MX= 8MD5 3B''(=G5AB"-)NM,C&'%<8,C/B !&'6Q
UT 21 ' c,s
+& N)6N7$9(1<*B* 9(>*#$B# u.*B; X4N N#?$9Y'' = *B;WS= $9kWX(' *N i ' (d j :4,3* 9(5$9':*L*B h'(c$9\BK'@" $! c
\BN<'WWsC
f ' 'c,s
+& N)6NC8$9(O<*B* 9(>*#$B# u.*B; X4N N#>$9Y'' 8= *BWW= $9EWX(' *N i ' Nd jLhL/#(,G* 9((5^$9;':*B+*B h'(?$9\BK'@" $9 c f\BN'<WSB
sC
e[*)+'AB&'A!5AB=8`&)-x>^ 5AB <,<&' 9&<,C"-) "4 !& `3#=AB =@? 8 5 ,C+'5 8,C&V
)AB8+I/1 +<,_; 8+6tzD))C,7!8 +G<8+I"4'01-7!8,YC8,<&G 9/1A|1-);+%^ +<C,C [@5AB <,<&<8-=@? 8 d &&798; !&<,798G)8+6Qx>` <&(,C <,@8 ^ ;AB08%7!8; 9&<,7!8 ;&<8+' t=3I5 3B'G (Q
d<&T35 3B''.=T3 0901fB="4 &<& 9&28 .)+' 8/1>=AB 08 8+T)OABC AB"-,39/1 +<,_J -&' "-)Y)OABC! AB"-,3VQ1eb:798<&<,AB P=/1AB,C+',= 8MI)+' 8/1)OAB8+<+, 9& AB8j &<+')C8Y8&<,C798[? 8 (' 8 ')OA'f);B61)8,'798f,C<8 4+,C&= 8M&)OA' +YT= 8MI)+' 8/1T 48 .' 8=AB 9&*3-AB 08 8+(+''&' +,C&*)OABC1AB"-,3Q v )O = 9&6@/15E=.)+' 8/1>798; !&<,798|;*5'ZA' ('AB !&);BT<,<,C"-) 9= 8M# &&'2798; !&<,798T)O 8/1 9&fj &<+'.)C8T8&<,CT7!8[? 8 P' 8V!56+69= *56BAXV?qAB $= 8MP^ +<C,C 6V?qAB Xi,C&f7!8>5'Z;B5 8 F=f= 8MP AB8 !/1AB,:8 D &&2)8+[AB+'798= :I56B=@? 8 k' 8s)+'AB8/1 8+6O,Cg AB8'&*,C"-)OA'<,CJI=I/1AB,C+<,s3P)+' 8/1I798,s AB8<&=AB '=5AB"-)OA':J, $ X8 D)+'A!=8,C&&' 'AB+<, |Ψ1〉 ⊗ |Ψ2〉
Q f3);+<&<,<8,C/ 9&' AB8: AB8:,C 9&' +''AB %8M^)+'AB&'A5ABG=)+' 8/1,C !&' +B5 &<,C/1EzP)C8<, 8+')+'AB8/1 8+'G &6@
o !
9&<,5 ,C); !&8 I) 8[ AB8CAB s)+'' !&' +,5 ,8 -+<08"4 !&8<&<,_ 9&Y7!83)+'AB8/1 8+'(h &Y=AB 5
k)+'AB8/1 8+'.)OAB8+
k ≥ 32 G'AB !&.);B.)C8(8&<,C(7!8G= 8MO;' AB d~qFc w Qeb47!8<&<,AB F=
/1AB,C+T<,= 8M-)+'AB8/1 8+''AB !&)C88&<,CT798[? 8 4' 8;+'<&'f8 .7!8<&<,AB #AB8/1 +<&'Eh5NUO3'5 &<,AB NQ
nTW /n
>AB825AB"-"4 AB .);+f=5 +<,C+'>8 $C01AB+<,C&<Z"4:,C"-)OAB+<& 9&.= f:56B=+':=*3%/1 +<,_;56&<,AB =Y<<&'H "4s=T)+' 8/1798; !&<,7986B2&'<& v l! gQ>AB8s+' 8&<,CC,' +'AB z*)C8<, 8+'s+' )+<,'5 &(C01AB+<,C&<Z"4%= ()+'AB&'A!5AB>zI= 8MD)+'AB8/1 8+'(=:3I)+'A5'Z;,C E'5 &<,AB [Q /n #<8))OA'7!8V?qAB #=,<)OA'=&<+'AB,+' 0,<&<+'7!8; 9&<,7!8Y
1 &
2Q$5AB !&<, 9&28 48 ,798
798J,C&E= %V?q && |0〉 sAB+'%7!81 &
25AB !&<, 9&-+'<)O5 &<,C/1 "4 9&G"#&<+<,5I= <,C&'
<8+n
798J,C&'ρ
&σQe?rC01AB+<,C&<Z"4%)+'A!5H=G5AB"-"4E<8,C&
o Q.*))C,798 +(3&<+ UhAB+<"#&<,AB ^= B="#+'=H
zYQ> Q.*&<,CC,' +k5AB"-"4G7!8J,C&(=5AB 9&<+ B%)AB8+.5Z; 01 +
1 &
2',!k<&.= V?q && |1〉 AB+'.5Z; 01 +*(5AB 9&' 98>="
1 &(="
2 &( :+<, DUu,C+'',!<&.= V?q && |0〉 Q
w Q.*))C,798 +3L&<+ <UhAB+<"#&<,AB p= B="#+'=dz#YQ@^<8+' +$W= 3LJ;B-56 AB ,798 &(B55 )&' +><,@:+'<8C&&(<& |0〉 Q
e? 8&<,CC,C&'E=5 &*C01AB+<,C&<Z"4G)+'A|/!, !&*=8D "-"4%'8,C/B !&6Q
% T NI*Q'+#+*]dhB"= ><*Q':* '8$9I7*1h(u(8$9N) B"(ρ, σ)
3/h;*1'B'OB"'1':*B>\& */ /i(h 9#[g i. KrnN2\!*1N#N
1/2 + Tr (ρσ)/2C
" 7)Nh+*WSBCeb:)+' 8/1 [?q<&);B=, #5 ,CQ9>AB8Y3:=AB AB = Y(56BAρ &
σ'AB !&=
&&'Y)8+'656B87!8 AB 4'f+"4H .= s56B=f"#&<+<,5= <,C&'.7!8 5AB 7986Q9l!8))OA'AB =AB 5G7!8
ρ &
σ'AB, !&(=3EUuAB+<"4
ρ = |ϕ〉〈ϕ| &σ = |Ψ〉〈Ψ|
(8X= J8&=(V?rC01AB+<,C&<Z"4T+' 0,'&<+'%Y1 &
2'AB !&2= fV?q && |0〉|ϕ〉|Ψ〉 Q*)+'HfNM5 8
&<,AB $=>3)+' "-,H +':&<+ UhAB+<"#&<,AB `= B="#+'=D &=:V?q5'Z; 01%5AB 9&<+'AB5 &. &&.= /, 9&
1√2
(
|0〉|ϕ〉|Ψ〉 + |1〉|Ψ〉|ϕ〉)
Y8,GAB d))C,7!84zL AB8/1688 I&<+ UuAB+<"#&<,AB K= B="#+'=@Q[Y k+' 0+'AB8); !&%&' +<"4 XUhAB 5 &<,AB `=8$)+' "-, +(7!8J,C&6V?q &&.&'AB&t<&(AB+'
1
2
(
|0〉(
|ϕ〉|Ψ〉 + |Ψ〉|ϕ〉)
+ |1〉(
|ϕ〉|Ψ〉 − |Ψ〉|ϕ〉)
)
et)+'ABJ;J,CC,C&'=:"4<8+' +.)+' "-, +(7!8J,C&(= V?q && |0〉 '&.=AB G);+
P (0) =1
4‖|ϕ〉|Ψ〉 + |Ψ〉|ϕ〉‖2
=1
2+
1
2|〈ϕ|Ψ〉|2
o
+Tr (ρ σ) = |〈ϕ|Ψ〉|2 5798,@)+'AB8/1%: "-"4G= .56B(=( &&')8+'6Q
etL)+' 8/1-=-5I "-"4I"4AB !&<+'4798@= G56B%=@?q &&')8+'63L)+'ABJ;J,CC,C&'4=-<855H=%V?rC01AB+<,C&<Z"4 v l! '&(C,I8`)+'A=8,C&'563,C+' |〈ϕ|Ψ〉|2 <,b> &&' |ϕ〉 & |Ψ〉 'AB !&,= 9&<,7!86;AB+'G&'<&(B55 )&'%/15%)+'ABJ;J,CC,C&'
1AB+'.7!8G<,@5. &&'*'AB 9&(AB+<&<ZAB01AB ;8M
()+'ABJ;J,CC,C&'.=f<855Hs [?q<&Y798f=1/2
Qe[2&'<& v l!^ <&Y=AB 528 -&'<&Y7!8,! AB8Y)O +<"4 &== &' +<"-,C +(3I)+'A6M,C"-,C&'%=G= 8MX &&'|Q
v AB <,= +'AB 4z`)+'' !&8 3 0901L
=AB 9&-V?r));+<&' ; 5$)8,''Lj &<+'P)+'AB8/1X);+I8 )+'AB&'A!5ABL=P&)P(+<&<Z!8+` +<C,C [z$8 K"4'i01# &&<+'AB,E)+'AB8/1 8+'|QslAB,C&
x ∈ Σ∗ g & V
/1 +<,_; 8+(5AB+<+'<)OAB = 9&6Q F<&.= .V? 8 `=.= 8MX56B(<8,C/B !&'o Q*lAB,C&I,CNM,'&'P&<+'AB,- &&' |Ψ1〉
|Ψ2〉 & |Ψ3〉
&' -798V
B55 )&'$/15X)+'ABJ;J,CC,C&'F8"4AB,C
c<8+V?q !&<+' |Ψ1〉 ⊗ |Ψ2〉 ⊗ |Ψ3〉
5?q<&(56Bx ∈ L
Q> Q*lAB,C&6)OAB8+.&'AB8&.&<+<,C) &*=@?q &&' |Ψ1〉
|Ψ2〉 & |Ψ3〉
V
B55 )&'/15)+'ABJ;J,CC,C&'8$)C8s<8+V?q 9&<+' |Ψ1〉 ⊗ |Ψ2〉 ⊗ |Ψ3〉
;5?q<&G56Bx /∈ L
Q>AB8#/1AB8AB X)+'AB8/1 +D7!8
L) 8&Xj &<+'`+'5AB 98 );+L8 )+'AB&'A5ABkzd' 8 "4 !&D= 8M
)+'AB8/1 8+'6Q v AB <&<+<8,'AB 8 K AB8/168 /1 +<,_; 8+V ′ zD);+<&<,C+I=
VQV ′ F5AB"-"4L+' 0,<&<+'P=
&<+/,CfP"4j "4L+' 0,<&<+' 798V
b)C88 +' 0,<&<+'# 5AB <&<,C&<8D=@? 8 W' 8798J,C&,C ,C&<,3C,'= V?q && |0〉 Qo Q
V ′ +'AB,C&(= 8MX &&' |D1〉= .+' 0,<&<+'
1
1 & |D2〉
= .+' 0,<&<+' 2
2Q
> QV ′ O5 &<8G5Z;B5 8 F=(= 8MX&'<&'.'8,C/B !&'*/15G)+'ABJ;J,CC,C&'
1/2
u1Xab<&G=I' );+J,CC,C&'Q[*))C,798 +V?rC01AB+<,C&<Z"4 v l! 8M`&<+'AB,+' 0,'&<+'"Y 1 &
2Q
JXab<&G=-5AB <,<& 5Q[*))C,7!8 +V
8M^+' 0,'&<+' ( 1
2 &
1Q@55 )&' +%<,
V8+,C&*B55 )&'QetF5AB"-) &<8=P=P54)+'AB&'A5AB#<&<,C"-)PzX/1AB,C+ t<,E &&' |Ψ1〉
|Ψ2〉 & |Ψ3〉
5AB++'<)OAB =, 9&zL8 #<&<+&' 0,4AB)&<,C"##=%)+'AB8/1 8+'%= %I)+'AB&'A!5ABLzL&<+'AB,G)+'AB8/1 8+'6AB+'%,5 ,2<,#)+' "-, +E)+'AB8/1 8+- 9/1AB,PV?q && |Ψ1〉 ⊗ |Ψ3〉
&%P= 8M,H "4#)+'AB8/1 8+- 9/1AB,|Ψ2〉 ⊗ |Ψ3〉
8+P)+'ABJ;J,CC,C&'`=`<855H#'&c′ = (1 + c)/2
)8,'7!8[? ,CP);B'' 9&PD&'<&L=' );+J,CC,C&'%6/15G)+'ABJ;J,CC,C&'
1Q
etK5AB+<+'5 &<,AB <&L)C8L= C,56&'f &L AB8L `=AB AB X);BLV?r+<08"4 9&$5AB"-) &X,5 ,-h5NUQ~q$cG w NQge? ,=X<&7!8g<,T= 8Mp &&'I !/1A61I);+I%)+'AB8/1 8+'I);B' 9&IP&'<&=' );+J,CC,C&'15?q<&7!8[? ,C'AB !&Y)+'A!5Z*h I_;= C,C&'|s=@?q &&'=8-&) |Ψ1〉⊗|Ψ3〉
& |Ψ2〉⊗|Ψ3〉Q
l? ,C@);B'' !&tY&''&[=5AB <,<& 55?q<&t7!8Y@&<+'AB,t &&' |Ψ1〉 |Ψ2〉
& |Ψ3〉5AB+<+'<)AB =, !&
z*8 .<&<+&' 0,=Y)+'AB8/1 8+'7!8,;8+,C& &'(B55 )&');+Y2)+'AB&'A5AB(z3)+'AB8/1 8+'6Q +55 ,
[?q<&)OA'<,CJ7!8[?r/15G)+'ABJ;J,CC,C&'8D)C8s &(V?qAB F $==8,C&.3-5AB+<+'5 &<,AB ^=8D)+'AB&'A!5ABQ
nTnT ` v 3B'<,7!8 "4 9&6fC, ( 9&<+':f)+'AB&'A5AB.=>&):*+<&<Z98+^ +<C,C F &ff)+'AB&'A!5AB.=
)+' 8/1.,C !&' +B5 &<,C/1uz-6B()+<,C/1if'AB !&(<8,C/ 9&'6Q
o m
# ZT r; %. GBWSB m : N → NjX +/,3*>* ' F':/dXBB>W*C(5``1
'(m)
<*G3<*4,69$!(^<*B#* 9(9*$B# u.*; (X4N 49$9'< (Y ;N+*1W (Y$ WX(' *N i' NP $*B)m+= * ' NP "'N *B!N;m(|x|) #((3* 91 m+& N;h\"' x
CK<B( *] '
(m) = '
(2) r
(k) ⊂ '(m + 2) ⊂ r
(m + 2)C
>AB8b)+' 9&'AB Y,5 ,8 I+'<8C&& ;AB08= sT56Bs7!8; 9&<,7!8798,"4AB 9&<+'7!82V?qNM)+'<,C/,C&'2=)+'AB&'A!5AB*+<&<Z98+^ +<C,C Pz>&<+'AB,s"4'i01<&s3>"4j "47985 C=)+'AB&'A!5AB=)+' 8/1,C !&' +B5 &<,C/127!8; 9&<,7!8f/15.&<+'AB,T"4'01|Q1(,C ',V1<,8 #3 0901>B="4 &8 #'<&'H "4=-)+' 8/1x>R (tAB ) 8&E<8))A +%7984V?r6BG8&<,CC,'#);+G-/1 +<,_; 8+E= E5#<<&'H "4-<&)8JC,5Q
# ZT ' ' ' r
" 7)Nh+*WSBCtlAB,C& L ∈ x>R 1,2−n(3)
@ &GAB,C&V
/1 +<,_; 8+%5AB+<+'<)OAB = 9&6b=AB I);+= 8ML&<+ <UhAB+<"#&<,AB *8 ,C&,C+'
V1 &
V2Qe[G)+'AB&'A!5ABIx>`> )OAB8+
L<&'8,C/B !&6Q
o Q *5 /1AB,C+.8 D+' 0,<&<+' \=3);+<&*=G` +<C,C [Q> Q 1 &' +(8 ),H5%8$Z;B+'=@ &* 9/1A61 +*+'<8C&&*zI^ +<C,C [Qw Q *5 /1AB,C+8 k+' 0,<&<+' =I3L);+<&G=-^ +<C,C [Qtl!,Y3P),H54=IV?q &)OI)+'5= 9&'4 &,C&
&'AB"EJ:<8+2),C))C,798 +V2
z(,
) &B55 )&' +f', &' 8 "4 !&f',*+'<8C&&=@? 8
"4<8+'E=8`7!8J,C&*=E'AB+<&<,%=AB |1〉 Q@l!,t3I),H5 &,C&(&'AB"EJ<8+UVB5))C,7!8 +V ∗
1z(,
) &(B55 )&' +*<,[ &. 8 "4 9&*',@&'AB8..7!8J,C&'.= \'AB !&(= .V?q && |0〉 Q
RSf'&EUVB5 ,CX=L/1AB,C+4798X5L)+'AB&'A!5ABD'&I5AB+<+'5 & g*+<&<Z98+-/1 +<,_;D7!8X` +<C,C '&J, 56);J(=fUV,C+');B' +'+' 0,'&<+'=@? 8 P &&T5AB+<+'<)AB = !&fzV?q &&,C ,C&<,3z8 P &&TB55 )& 9&6Qe[E)AB,C !&>5 NU<&>798E(+<&<Z!8+*)OA''H=%%+' 0,'&<+' =H>E= );+<&6 &:=AB 5GV?q &&:=E5+' 0,<&<+'<&Y,C = )O = 9&2=8 &' +=f),H5(798,<&Y@5 &<8(z>3:= 8M,H "4. &)(=8-)+'AB&'A!5ABQ
+ VG -.!:# H*:2;M:J! $ 1):2" :
et-5 3B'=3 0901*B="4 &<& !&>=(<'&'H "4.=:)+' 8/1.,C !&' +&<,C/1*7!8; 9&<,7!8 6 +'A! A =01*5AB '&<,C&<8(8 P,C 9&' +<"4=,3,C+'>,C 9&' +''i 9&f 9&<+'*T5 3B'2x>^ &2x>R *Q9>AB8 AB8,C 9&' +''AB f,5 ,@z%35 3B''xl =23 0901.B="4 &<& 9&.8 >)+' 8/1 6 +'A ! A =01G<&&<,&<,798,C 9&<+'A=8,C&'I);+ Qd&<+'AB8-~ d& > QRS 9&<8,C&<,C/1 "4 !&6[AB =,C+P7!8[? 8 )+'AB&'A!5ABI=8^&)x>R <&6 +'A ! A =01D<&&<,<&<,7!8P',T3N/!8P7!8[?rDP/1 +<,_; 8+-z`5'Z;B5 8 D= &)O=V?qNM5 8&<,AB L=8#)+'AB&'A5ABE ,3G"#&<+<,5(= <,C&'*=5 +<,C/B !&V?q &&2=8#+' 0,<&<+')+<,C/1*=8#/1 +<,_; 8+ &(=%5 C8,b=("4'i01i2)O 8&(j &<+'G01 +'%);+(8 F5 ,C+'5 8,C&:565 83J% `&' "-).)OABC! AB"-,3 <,C"E83&' 8+6N6/158 )+'5 ,<,AB "4<8+'G);+.3 AB+<"4%&<+B5|+<J,C&<+,C+' "4 9&:0+ =Q(,C ',V2)+'AB8/1 8+ =AB .85 8 f,C UuAB+<"#&<,AB P8-/1 +<,_; 8+6)8,798f5= +< , +8+,C&8'<,J, $)8D01 +' +(C8,V"4j "4G3N/!8:7!8[? ,CbI=8$)+'AB&'A5ABQ
& !&E=AB 48 #5ABC5 &<,AB ρy=-"#&<+<,5= <,C&'4=-&,CC
ky[AB d=,C+F798-5 &<&'
5ABC5 &<,AB `<&;'+"2':*B+*#PNkNK'-F')/PXB?S*/Y6? ,CtNM,<&'8 Uu"-,CCE=%5 ,C+'5 8,C&'(7!8; 9&<,7!8Qy
=&)(ny, ky)
8 ,UhAB+<"4 "4 !&G01 +'I &' "-)*)OABC AB"-,3Y&' C7!8)AB8+>&'AB8&y
ρy = Qy|0〉Q
>
& !&=AB s8 -/1 +<,_; 8+V
&Y8 I)+'AB8/1 8+PBAB -=N_ ,C&Y/!8
V,P (x, j)5AB"-"43*"#&<+<,5
= <,C&':=5 +<,C/B !&fV?q &&+'=8,C&f=8X/1 +<,_; 8+ &=8P+' 0,'&<+'=("4'i01f)+'H7!8j"4'01
, 9&( &'%5'Z; 01.AB+'*=:V?qNM5 8&<,AB ^=8D)+'AB&'A!5AB(V, P )
<8+V?q !&<+'xQ
UT /VN;V k/"N( 2 4
P R'+ = *B;WS= GC 2 $/h;= (V, P )(
^<<&'H "4(=()+' 8/1T,C !&' +B5 &<,C/1f798; !&<,798 6 +'A A =01:<&&<,<&<,7!8K': K%<*B* 9 L( (V, P )
( Z(X4N N#Q$9;'' B(Q ;N+*1W (Q= *B;WS= (;': L5T
g+!/N N) 83C σx,j$!G8*hu(?$9 )(hB";'+"2':*B+*#(4 9'F':/dXB>W*/P52
3WSB δ$]"< +/n(3*BI' M ?= Q\& N(?$9%. s')/PXB45TP1=
‖σx,j − V,P (x, j)‖ < δ(|x|)
': . x ∈ L T. T#(,G* 9
jC
UT 2Z$#"h6<*43<*4,6 ^ c,s
,BI41\& NN1$9%<*B* 9(O*$B# u.*B (X4N 4[$9E'' B( ;N+*1W (R= *B;WS= (| L/i|;+$ 
e[b,C <& 5g)A',C&<,C/1b5AB+<+'<)OAB = 9&8MG8 ,C&,C+'Y9 !&8%"4AB,C g8 / 8+g)+'AB+'&<+'H=,O +' !&'*=
eiφ ! &T5()AB8+T&'AB8&φ95*798,+' /!, !&fz%=,C+'*7!8
UxG= 8M4/ 8+'2)+'AB)+'&<+'H
=,O +' !&'6Q9et,C <& 5T 09&<,C/1T5AB+<+'')AB = !&fz=8 ,C&,C+'269 9&T&'AB8&' 8+'/B 8+')+'AB)+'G)+'A5'ZE=
eiφ )OAB8+G8 5 +<&,C φ@=AB 5-&'AB8&'%)+'A!5ZG8 E=%8&<+'6Q[
4)+' "-, +-56B6b<,V?qAB =,<)OA'#=#/15 &' 8+'E)+'AB)+'I5AB+<+'<)OAB = 9&4z`= 8M/B 8+'E)+'AB)+'=,O +' !&'- ,C* AB8>'AB !&:=AB >);+:` +<C,C NOAB+'>AB `)O 8&*8&<,CC, +*3#)+'A!5=8+'<& =+'==@?3(NW 8*WSB $!Y'; *46E)AB8+('&<,C"4 +(./ 8+'.)+'AB)+'>5AB+<+'<)OAB = 9&'> &(/1 +<,_; +>798[?q C'AB 9&=,<&<,C 5 &'6Qg %P= 8M,H "4P56B6t55 ,f<&G,C"-)OA'<,CJQge[4)+'ABJH "4 R = 9&<,C&p5'Z5 <&.=AB 5G= *x>^Q
et%5AB"-) &<8=>'&T8 P) 8P)C8T=, #5 ,CQe[>56B=+'.01 +O)OAB8+fx>^ <&3%=AB >=@? 8 5 ,C+'5 8,C&b7!8; 9&<,7!8
U| &t,C6?r0,C&b=T= &' +<"-,C +Y6? ,C!NM,<&'Y8 E/15 &' 8+ |Ψ〉 &' !7!8V?q &&
U(|Ψ〉⊗|0 . . . 0〉) 'AB,C&&' ;7!8.'AB 4)+' "-, +T798J,C&: .798J,C&T=.'AB+<&<,|Y,8 .0+ =.)+'ABJ;J,CC,C&'(=@?qj &<+'= :V?q && |1〉 Qte? ,=-5AB <,<&'4zL5AB <&<+<8,C+'-8 k5 ,C+'5 8,C&
Z = U ∗V UA
V<&:V?r))C,56&<,AB
=@? 8 >)Z;B'φ
<8+f8 D798J,C&8M,CC,3,C+'5AB !&<+'AB:);+f>7!8J,C&=*AB+<&<,>=U
Q L/1 +<,_;GAB+'798
Z<&YAB,C #=
eiφ Id)OAB8+Y&'AB8&
φ<,
U6/B,C&=.0+ =)+'ABJ;J,CC,C&'=@?rB55 )&' +6! &Y)+'A!5Z
=@? 8 eiφ Id
= G56B(5AB 9&<+,C+'Q
nT ~ d& > x*8; !&<8" l!&&'G>,<&<,C 08,<Z;J,CC,C&S1Q lAB, !&
0 ≤ α < β2 ≤ 1 Q
E p : 8MX5 ,C+'5 8,C&'Q0
&Q1
="4j "4G&)(m, k)
Q ‖(Q0 − Q1)|0〉‖tr ≥ β
Q ‖(Q0 − Q1)|0〉‖tr ≤ α
Q
RSI>8 I)+'AB&'A!5AB.7!8; 9&<,7!8,C !&' +B5 &<,U@',C"-)f)AB8+Y5f)+'ABJH "4 B2/1 +<,_; 8+)+' );+'(8Z;B+'=
Q0|0〉AB8
Q1|0〉 &t 9/1AB,T3f"#&<+<,52= <,C&'5AB+<+'<)OAB = 9&'8G)+'AB8/1 8+6QR&<&' =%
+' )AB =8-)+'AB8/1 8+8 -8 ,798fJ,C&6Q9RB55 )&'.<, & 8 "4 9&<,5fJ,C&5AB+<+'')AB =4845 ,C+'5 8,C&798[? ,Ct6/B,C&*NM5 8&'Q v G)+'AB&'A!5AB% [?q<&.5 ) = !&();B 6 +'A A =0156+.G',C"%83&' 8+( ) 8&f)+'=,C+':>J,C&f+' 9/1A|1);+f*)+'AB8/1 8+.7!8[?r/15:)+'ABJ;J,CC,C&'
1/2(1 − β + α)Q D"4A!=,_;
=AB 5)+'AB&'A!5ABf5 ,=''8 Bz);+<&<,C+Y=g5 ,C+'5 8,C&'Q0
&Q1
T/1 +<,_; 8+Y5AB"-"4 52);+g565 8 += 8MX5 ,C+'5 8,C&'
R0 &
R1&' (798
‖(Q0 − Q1)|0〉‖tr ≥ β =⇒ ‖(R0 − R1)|0〉‖tr ≥ 1 − 2−n
& ‖(Q0 − Q1)|0〉‖tr ≤ α =⇒ ‖(R0 − R1)|0〉‖tr ≤ 2−n
5:798,O)O 8&f(UV,C+': X&' "-))ABC AB"-,3VV? ,= & 9&.=:5AB"EJ,C +(= 8ML&S!)O:=>+' ) &<,C&<,AB );+CH G=.5 ,C+'5 8,C&'
Q0 &
Q1Q
e[>/1 +<,_; 8+.NM5 8&':AB+'f*)+'AB&'A5AB5 ,='<8./15>5 ,C+'5 8,C&'R0
&R1
AB8; !&*+ B=
Q0 &:=
Q1QOe[<,C"%83&' 8+ELz#)+'' 9&8 )+'ABJ;J,CC,C&'-NM)AB !&<, C "4 9&UV,CJI=E'
&<+'AB"-) +6; &)+'AB&'A!5AB%<& 6 +'A ! A =01Qet:5AB"-) &<8=*<&s)C8=, #5 ,C AB8 4=AB AB ',C"-) "4 9&V? ,C !&<8,C&<,AB [Q9RS6?r0,C& IUu,C&
=8F"4j "4E&)=G)+' 8/1I7!8GV?qAB ^+' 5AB 9&<+'AB+':=GV?q &<8==>)+'AB&'A!5AB 6 +'A A =01 β2 p.w~(.w]zSup p.k Gx k ;kx # pow4kI~ 3 β − α 3wy poupzIv z
> >
5 3B'<,7!86QlAB,C&L
8 %3 09012B="4 &<& 9&s8 E'<&'H "4=)+' 8/1b,C 9&' +B5 &<,C/1s798; !&<,798 6 +'A! A =01t &%AB,C&
V4/1 +<,_; 8+E5AB+<+'<)OAB = 9&6Q v AB"-"44-)+'AB&'A5AB#'& 6 +'A ! A =01
5?q<&798[? ,CsNM,<&'8 ',C"%83&' 8+%7!8; 9&<,7!8)+'A!=8, !&G=:"#&<+<,5G= <,C&'ρ0, . . . , ρk−1
&σ1, . . . , σk
OAρi
=<,C0 IV?q &&E=798J,C&'G=:+' 0,'&<+'G=8k/1 +<,_; 8+% &G="4'i01 8<&'/ 9&:798%%)+'AB8/1 8+G,EO5 &<8E34&<+ <UhAB+<"#&<,AB
Pi &
σi=<,C0 %V?q &&:=%5>"4j "4
+' 0,<&<+' 8<&'G)+'H.V?rB5 &<,AB `=8$)+'AB8/1 8+6Q D)O 8&(=AB 5G565 8 +6 D&' "-).)OABC! AB"-,3t 3AB 08 8+*=:V?q !&<+'
x= 8M$5 ,C+'5 8,C&'
Q0 &
Q1/1 +<,_ !&
Q0|0〉 = ρ1 ⊗ . . . ⊗ ρk−1 &
Q1|0〉 = σ1 ⊗ . . . ⊗ σk−1e? ,=<&(AB+'7!8<,@5= 8MX &&'('AB 9&)+'A5'Z% )OAB8+3E AB+<"4&<+B5|N5?q<&.7!8[? ,C@NM,<&'8 )+'AB8/1 8+E7!8,5AB+<+'<)OAB =d8M &&'%=8K<,C"%83&' 8+6g &%=AB 547!8-V?q 9&<+'
x= /+,C&Ej &<+'
B55 )&';);+>5AB+<+'5 &<,AB k=8`<,C"E83&' 8+6Q@l!,g5*= 8MF &&'>'AB !&(&<+'H*AB,C [5?q<&>798%34'AB+<&<,=8`<,C"E83&' 8+5AB+<+'<)AB =z-8 '7!8 5%=E&<+ UuAB+<"#&<,AB *,C"-)OA'<,CJ:z-+'6C, +*)Z!<,798 "4 !&6 &(=AB 5G7!8:V?q 9&<+'
x= /+,C&(j &<+'+' < &'Q
@
~ [ ]|RS"#01)+'A!5Z6Q lAB, !&c, s ∈ [0, 1]
&' 47!8s < c
Q \=N_ ,C&PX)+'ABJH "4`z)+'AB"4''
CIc,s=:3I"# ,H +'G'8,C/B !&'
E p : 8MX5 ,C+'5 8,C&'*798; !&<,798Q0
Q1
2="4j "4G&)4n,m
NQ R@NM,<&'G= 8MD &&'."-,M&'.<8+
n7!8J,C&'
ρ0 &
ρ1&' .7!8
F (Q0(ρ0), Q1(ρ1)) ≥ c bAB8+(&'AB8.. &&'"-,M&'('8+
n7!8J,C&'
ρ0 &
ρ1
F (Q0(ρ0), Q1(ρ1)) ≤ s
:AB AB 8 X)+'AB&'A!5AB=>)+' 8/1,C !&' +B5 &<,C/1G7!8; 9&<,7!8Gz3"4'i01f)AB8+5:)+'ABJH "4Q
e[)+'AB8/1 8+G5AB"-"4 5-);+ 9/1A|1 +G3P"#&<+<,5-= <,C&'ρ0
8/1 +<,_; 8+6@798,sC8,))C,798I5 ,C+'5 8,C&
Q0Q1RS 9/1AB,*AB+'T84)+'AB8/1 8+T798J,C&'798, .AB 9&6':*/(=<,C0 5AB"-"4.=798J,C&'
=.'AB+<&<,.= T.5 ,C+'5 8,C&Q0
Q1e[()+'AB8/1 8+2))C,7!8.8 .&<+ <UhAB+<"#&<,AB [! &+' 9/1AB,(+' 0,<&<+'=L"4'01#8 /1 +<,_; 8+67!8,())C,798L_ ; "4 !&
Q∗1
<8+-'I+' 0,'&<+'6Y &-/1 +<,_;D798L798J,C&'*8M,CC,3,C+'(8&<,CC,');+.5 ,C+'5 8,C&
Q1'AB 9&(J, $&'AB8.= .V?q && |0〉 Q 56B(AD,C[NM,<&'8 :"#&<+<,5E== <,C&'
ρ1&' CG798
F (Q0(ρ0), Q1(ρ1)) ≥ c);+
=N_ ,C&<,AB D=*3G_;= C,C&'>()+'AB8/1 8+)O 8&6 X0,'i 9&f8 ,798 "4 !&.<8+227!8J,C&'f798, :'AB 9&);B227!8J,C&'=:'AB+<&<,>=5 ,C+'5 8,C&'6&<+ 'AB+<"4 +
Q0(ρ0) L8 X &&7!8,[,._;= C,C&'G8X"4AB,C
c6/15
Q1(ρ1)Q v )+'AB8/1 8+*'&(B55 )&'%/15)+'ABJ;J,CC,C&'
cQ
(56B:AF (Q0(ρ0), Q1(ρ1)) ≤ s
)AB8+>&'AB8ρ0
&ρ1
;8 E&<+ UuAB+<"#&<,AB ^3, 9&,C 9/+<,3 9&'798J,C&'E=P'AB+<&<,#&<+ UuAB+<"4
Q0(ρ0) K8 p &&E7!8,T<&%AB,C p=
Q1(ρ1))AB8+
&'AB8&ρ1
&7!8,:=AB 5f8 f)+'ABJ;J,CC,C&'>8-)C8s=f+' "4 &<&<+'7!8J,C&'8M,CC,3,C+'= sV?q &&
|0〉 )+'H())C,56&<,AB ^=Q∗
1
Qetp5AB"-) &<8=k=^5`)+'ABJH "4^)OAB8+Fx>R =5AB8=^3)+' 8/1k=kx>R ⊂ =
~q:Q;lAB,C&(V1, V2)
>/1 +<,_; 8+(=@? 8 X)+'AB&'A5ABEx>R (3)
)OAB8+8 X3 0901LQe? ,=<&7!8
P5 ,C+'5 8,C&Q0
/F=<,C0 +I3F)+' "-,H +'DB5 &<,AB V1
=8 /1 +<,_; 8+6s &P5 ,C+'5 8,C&Q1
/$=<,C0 +V? ,C 9/1 +''D=X^= 8M,H "4$B5 &<,AB
V ∗2
Qs*,C <,VQ0(ρ0)
=<,C0 XV?q &&-=8W+' 0,<&<+'D=8 /1 +<,_; 8+ 8'&'E/ 9&>V?rB5 &<,AB =8`)+'AB8/1 8+6 &
Q1(ρ1)V?q &&:=E5%"4j "4%+' 0,'&<+' 8<&'E)+'H*V?rB5 &<,AB
>w
=8F)+'AB8/1 8+6QOe[%Uu,C&:7!8%5> &&':, !&:0+ =G_;= C,C&'I<,C0 ,_;798%%)+'AB8/1 8+>)O 8&* AB8Uu,C+'L);B' +4=PV? 8 \z^V?r8&<+'Y &4=AB 5D7!8[? ,C.NM,<&'L8 W)+'AB8/1 8+4798,.'PUV,C&4B55 )&' +P/150+ =)+'ABJ;J,CC,C&'Q
Y n( :b : Y bY bO. Y
:>3%"4j "4:"# ,H +'7!8>3I5 3B''Gx>R E &':,C !&<+'A!=8,C&''8+2:"4A=H :=:35 3B'':R (!,C<& ;&<8+' O=(5AB ',= +' +T8 *NM&' <,AB X7!8; 9&<,7!8*=*`R .Q9ebG01 +C,&<,AB L3%)C8T<,C"-)5AB <,<&'IzP8&'AB+<, +>/1 +<,_; 8+ &:>)+'AB8/1 8+'Gz4)OA''= +:=>"#B5'Z,C G7!8; 9&<,7!86O &:z5AB"-"%8 ,7!8 +/15T/1 +<,_; 8+g);+bV? ,C !&' +<"4=,3,C+'=@? 8 56 ;7!8; 9&<,7!8"#,b ,C 9&' +'=,i 9&&'AB8 AB8+'$&'AB8&'^UhAB+<"4k=k5AB"-"E8 ,56&<,AB !&<+'kX)+'AB8/1 8+'6Q.et 5 3B'k5AB+<+'<)OAB = 9&'x>`R .|. &'Y,C 9&<+'A=8,C&');+b(ABJ;9B'Z, &t&'<8"4AB&'A* > o ~q w Q v AB"-"4= t56Bt=x>R (,C [? ();BY=T+' 3&<,AB #()+<,AB+<, !&<+'`R &x>^R .Q v ) = !&6AB -(/8D &<ZAB+'H "4 oo 7982^R ( "$
=^R * > o 5?q'&Sz=,C+'79823*)8,' 5f=g)+'AB&'A!5ABY5 3B<,798Y|?qNM)+<,C"4
) ,C "4 9&G<,sV?qAB +'<&<+' ,C !&% AB"EJ+'4=%)+'AB8/1 8+'EzP= 8M@ &:',b/1 +<,_; 8+)OA'8)C88 X798<&<,AB z^5'Z;B5 8 =@?q !&<+'D 8MOQ +6sV?qAB )O 8&I',C"%8 +-8 )+'AB&'A!5ABX5 3B'',798Dz^= 8M"4'01*);+*8 ^)+'AB&'A5ABE7!8; 9&<,7!8-z4= 8MF"4'i01* >=, #5 8C&' = >556B6;GUu,C&=@?qj &<+'G7!8; 9&<,7!8 [?q<&(=@?r85 8 8&<,CC,C&'8D)+'AB8/1 8+6; &.=AB 5G^R
=`R ( > o ⊂ x>`R (Q
:G>Y =
^R \AB F&<,C+'%AB+'(,C"-"4=,3&' "4 9&* ⊂ x>^R (QbAB8+>/1AB,C+(7!8V? ,C 5 C8<,AB d+'5 ,C)+'A7!8P<&E/!+,b,CT'8 4&=#/1 +<,_; +7!8#&'AB8&'L<&<+&' 0,L=%)+'AB8/1 8+')O 8&j &<+',C"-) "4 9&'E D8&<,CC, 9&.8 $');B5G8D)C8 09tzI= 8MPUuAB,fV?q<);B5:+'' +</1%8ML"4'016798,O<&2JAB+< >);+f8 X)ABC B"4QRO<&.AB+'f)OA'<,CJ>=>= /,C +f28 ,C&,C+'())C,798f);+5'Z;B5 8 =.)+'AB8/1 8+'6 &*=%565 8 +* 8+*)+'ABJ;J,CC,C&'E=E<855Hh7!8,t6?qNM)+<,C"4%',C"-) "4 9&5AB"-"4G3I&<+B5=@? 8 FAB)O +&' 8+>=G&,CC%NM)AB !&<, C|( `<);B5G+'=8,C&6Q v ?q<&*5G7!8UuAB 9&^&''8"4AB&'A4 &.(ABJ;69B<Z,g= %~q^ w )+'AB8/ 9&x>`R
=>Y ( &(,C <,gx>^R
=^R .Q
v 3B'<,7!8 "4 9&69AB 4) 8&Y,C"#0,C +8 4+' B5'Z "4 !&2=fV? Z!!)OAB&<ZH'.=f AB 5AB"-"E8 ,56&<,AB 9&<+'*)+'AB8/1 8+'| P28&'AB+<,i 9&.zG);+<&01 +28 >'AB8+'5*=@?r6B2)8JC,5QY XO &6AB P,C&798[?q k5AB"-)NM,C&'I=E3L5AB"-"E8 ,56&<,AB [@);+<&01-=@?r6B:) 8&G/1AB,C+:8 k@ &>,C"-)OAB+<& 9& );+2NM "-)!*565 8@=*V?q 09C,C&':%8 >5AB"-)NM,C&':=>5AB"-"%8 ,56&<,AB F=
O(1)J,C&'f= 2
"4A!=H 6/15G6Bf);+<&01AB+'798> .6Bf);+<&01>,CO 5'<,C&'Ω(
√n)
J,C&'2=:5AB"-"E8 ,56&<,AB [Q v )O = 9&6= (%56B=+'%=><'&'H "4.=G)+' 8/1*,C 9&' +B5 &<,Uu6;AB `/1AB,C&.UVB5 ,C "4 9&798P#Uu,C&-=P);+<&01 +4=#V?r6BI [?q<&-=@?r85 8 L8&<,CC,C&'$8Md)+'AB8/1 8+' g<,f5?q &,C&-P56B65?q<&G7!8[? ,CNM,'&' +,C&8 45'Z; P=@?r6BG&' C47!8@<,Y)+'AB8/1 8+'%);+<&01 9&I5 &<&'45'Z; AB+' 8+G)+'ABJ;J,CC,C&'^h<8+%B5 &<,AB %=8d/1 +<,_; 8+i:=-09,C d' +,C&G'8) +<, 8+'#zX5 C#798[? ,C8+, 9& g);+<&01 +Y=f5'Z; QB,65AB"-"4fg)+'AB8/1 8+'Y'AB !&s&'AB8&V)8,' 9&'Y &s)O 8/1 9&5AB"-"%8 ,7!8 +26/B !&T7!8f)+'AB&'A!5AB. .5AB"-"4 51,CY)O 8/1 9&2= &' +<"-,C +35Z; (=@?r6B798,Y"#M,C"-,'- 8+'%5'Z; 5%=#<855H%zLV?r/ 5t &8&<,CC, +%5 &<&'45'Z; Qb(,C ',VOIUu,C&G=<8))OA' +g7!8[? ,Cg);+<&01 9&=TV?r6Bs "4A=,_;f E+<, I 8+Y56);B5 ,C&'z>5AB 9/,C 5 +'T2/1 +<,_; 8+6Q
:2"# ,H +'. ;AB08sAB,Y=23("456 ,7!8.7!8; 9&<,7!8f AB8s) +<"4 &<&' !&=2+' B5'Z +="# ,H +'%,C !&' +'' !&'EV? Z!!)OAB&<ZH'=G AB 5AB"-"E8 ,56&<,AB d 9&<+'E()+'AB8/1 8+': `>8&'AB+<,
> o
9&.zE);+<&01 +.=>V?q 5'Z /1j &<+' "4 !&6Q; 5:56B6V?r+<08"4 9&(=>5AB 9/1NM,C&'G=AB :5 ,='<8 "#+'5'Z-)C86O56+5'Z;B5 8 K=:)+'AB8/1 8+') 8&G0,C+G<8+P);+<&<,-=V?q 5'Z /1j &<+' "4 !& UhAB 5 &<,AB =I798'&<,AB 7!8[? ,C+'AB,C&-=8p/1 +<,_; 8+6sABJ&' ; 9&4=+'<8C&&'I5AB+<+' I 9&<+'X 8M=-"# ,H +'4 AB = &' +<"-,C ,'&'Qge?qNM)+''',C/!,C&'P=45 &<&'4 AB8/1 CP5 3B''@ AB&'Xx>`R ∗ @ [?q<&);Bf5 3,C+' =@? 8 >);+<&69V?q 5Z /1j &<+' "4 9&()O 8&2) +<"4 &<&<+'8MP= 8M#)+'AB8/1 8+'=*&<+<,5'Z +6 5A!AB+'=AB ; !&I 8+'E+' )OAB '6b &,C <,2+'=8,C+'PV?qNM)+''',C/!,C&'L=Lx>^R ∗ );++))AB+<&-zF5 C=Ix>^R (Q?r8&<+');+<&6O5"4j "4I 5'Z /1j &<+' "4 9&E) 8&:)O +<"4 &<&<+'-8^/1 +<,_; 8+G=@?qj &<+')C8NM,C016 !&G);++))OAB+<&%8M^)+'AB8/1 8+'6t &,C ',Y) 8&j &<+'-+'AB8=+'-=)+'ABJH "4G)C85AB"I)NM|Qet:)OA<,C&<,AB 4=*x>^R ∗ );++))OAB+<&Tz:5 C.=(*Y k [?q<&=AB 5.);B=8-&'AB8& /,= 9&')+<,AB+<,VQ
*J8&f=@?q &<8=, +TV?qNM)+''<,C/,C&'*=>5. AB8/168X"4A=H *=*)+' 8/12,C 9&' +B5 &<,C/16 AB8CAB -5AB <,= +' +4)C8<, 8+'I"4A=H I+'=8,C&'6Y W,C"-)OA 9&4=IC,C"-,C&&<,AB #<8+I-5AB"-"%8 ,56&<,AB F !&<+'/1 +<,_; 8+` &D)+'AB8/1 8+6( &$<8+$3p ;&<8+'d=$+' )OAB '$=8)+'AB8/1 8+6Qv 3B'<,7!8 "4 9&6s3^<,C&<8;&<,AB W<&I5 3,C+'s)8,'7!8b"4j "4X<,f+' )OAB 'I=I)+'AB8/1 8+'4'AB 9&C,C"-,C&'Ez
1J,C&6t &%<,-/1 +<,_; 8+% #J;B'4D=5 ,',AB d798#<8+G4 =%= 8MkJ,C&'E798[? ,C
+'AB,C&. X+' )AB 'z'7!8<&<,AB 6V?qNM)+''<,C/,C&'*+''&'> 09Gz>Y ~ v2 af oQY XUV,C&6!+'<8C&&AB+<,C0,C ;VJ;B<8+Y8 v K= B'&B=D~ o 6/158 .5AB"-)NM,C&'( 47!8<&<,AB Y&<+'HUu,CJ1"4AB 9&<+'>7983%5 3B''.=3 09012B="4 &<& !&T8 L<<&'H "4.=)+' 8/1(,C 9&' +B5 &<,Ub5AB"-"45 C8,7!8V?qAB #/, 9&2=(=5 +<,C+'/15(3+'<&<+<,5 &<,AB L<8)) "4 9&,C+':7!8.3:AB 08 8+2&'AB&(="4'015Z; 01('AB,C&AB09+<,C&<Z"-,7!8<&. .QRSO'&fAB8/1 9&*B="-,.= .3C,C&<&' +&<8+'G798f+'<8C&&'.<8+f v .6?q &' = !&*zI* .6/15G=.80"4 9&&<,AB > $7!8; 9&<,C&'G=@?r6B &. $ AB"%J+'E=798'&<,AB ())+'AB)+<,6;5 )O = 9&>55 ,[ [?q<&);B(&'AB8 AB8+',C"-"4=,3&6Q
<3 ⊕ - ∗(2)et%5 3B'' ⊕ ^R ∗(2)
G &'.,C !&<+'A!=8,C&'>= (~ v2 af oQ1RS;6?r0,C&T=(35 3B''*zG= 8M-)+'AB8/1 8+'7!8; 9&<,7!83>)C8T<,C"-).7!8,'AB,C& B/1 +<,_; 8+T)A'f8 (798<&<,AB Fh5 3B<,798|z5'Z;B5 8 =I= 8Mp)+'AB8/1 8+'6Y7!8,2+' )AB = !&4);+I8 82J,C&6QYetP/1 +<,_; 8+-)+' =Wi`=5 ,',AB W'J;B 9&*8 ,798 "4 !&<8+.G =>= 8M$J,C&'(7!8[? ,Cg-+' 8|Q;>AB8(5AB"-"4 AB >);+*=5 +<,C+'5 &<&'D5 3B''g)8,- AB8-=AB AB -8 NM "-)D=X)+'AB&'A5ABD= "4AB 9&<+ 9&PX+ BD=LV?q 5'ZN/1j &<+' "4 9&#= 4-<'&'H "4-=X)+' 8/1#z`)C8<, 8+'-)+'AB8/1 8+'6 &- _ W AB8-)+' 9&'AB 8 d&<ZAB+'H "4X=#ab<,C+' 'AB 798,T AB8) +<"4 &I=4"4AB !&<+' +V? ,C 5 C8',AB ⊕ ^R ∗(2) ⊂ *t <8,C/ 9&.3)+' 8/1G=lQ Z +E~ Z; Q
lAB, 9&S
&T
= 8MF ' "EJ*_ ,6π
8 =,<&<+<,CJ8&<,AB =G)+'ABJ;J,CC,C&':<8+S × T
&V8 >UhAB 5 &<,AB `=
S × T= 0, 1 ))O 3WSBZ$9%p* *WSB;Q*AB+'
π &
V=N_ ,' 9&
8 < 8F AB `A!56g=G3-"# ,H +'I<8,C/ 9&' 8 E);,C+'%=E798<&<,AB (s, t) ∈ S × T
<&>5'ZAB,<,=("# ,H +'G6&'AB,C+''8,C/B !&23=,<&<+<,CJ8&<,AB
πQs<& 9/1A|1G8 AB8 8+
1h798>V?qAB X AB"-"4 +
(8+' C,-= G&'AB8&'-3X<8,C&'|> &t'&G 9/1A61L8 AB8 8+
2 )+' AB"-"4 f A &iNQ v Z;B5 8 d=
= 8M AB8 8+'G=AB,C&E+' 9/1A|1 +E8 J,C&a, b ∈ 0, 1 +' )AB Q[(8+' C,# &f A &% -)O 8/1 9&
);B5AB"-"%8 ,7!8 +(8 :UuAB,7!8: < 8FI5AB"-"4 5"#,,C.) 8/1 !&('>"4 &<&<+'G=@?rB55AB+'=$<8+8 <&<+&' 0,[*m'W(hQRf090 !&( < 8$<,[ &.' 8 "4 !&*<,
a ⊕ b = V (s, t)Q
4 AB&'ωc(G)
3:/ 8+5 3B'<,7!8.=@? 8 4&' < 8[15?q<&Sz=,C+'(3:)+'ABJ;J,CC,C&'."#M,C"#*6/153B798 CG.= 8M <AB8 8+'.)O 8/1 9&(090 +
> ]
ωc(G) = maxa,b
∑
(s,t)∈S×T
π(s, t)(
1 ⊕ a(s) ⊕ b(t) ⊕ V (s, t))
AD"#M,C"%8" <&()+<,.<8+&'AB8&'.UhAB 5 &<,AB a, b : S × T → 0, 1 Q
v AB <,=H +'AB zX)+'' 9&E=)+'AB8/1 8+'E7!8; 9&<,7!8E8&'AB+<,EzL);+<&01 +E=4V?q 5'Z /1j &<+'N"4 9&6Qg(8+' C,P & f A &I'4+' 5AB !&<+' 9&4/ 9&-7!8- < 8 P5AB"-"4 5g &5 +' 9&8 p &&798; !&<,798 |Ψ〉 J,C);+<&<, 5Z /1j &<+'>=AB !&,C09+'= 9&f5'Z;B5 8 L3:"4AB,C&<,(=T798J,C&'6QbAB8+25'Z;B7!8798'&<,AB
s ∈ SO*8+' C,4=,<)OA'I=@? 8 -"4<8+'I)+'A<5 &<,C/1
Xs = X0s − X1
s
<8+GP);+<&<,4=V?q && |Ψ〉 Qg:#"4j "4 f A &-=,<)OA'P=P"4<8+'I)+'A<5 &<,C/1
Yt = Y 0t − Y 1
t
)OAB8+5Z;B798798'&<,AB
t ∈ TQbetAB+''798[? ,C+'AB,C/1 !&- 8+I7!8<&<,AB [bI= 8M AB8 8+'I@5 &<8 9&I3`"4<8+'
5AB+<+'<)OAB = 9&':<8+2 8+T);+<&<,>=(V?q && |Ψ〉 Q!ebG)+'ABJ;J,CC,C&'=(+' )AB =+'(a, b)
zG3E798'&<,AB (s, t)
'&(,C <,t=AB );+P (a, b|s, t) = 〈Ψ|Xa
s ⊗ Y bt |Ψ〉
$=N_ ,C&(3/ 8+(7!8; 9&<,7!8G=8 < 8G
);+
ωq(G) = maxXs,Yt
∑
(s,t)∈S×T
π(s, t)∑
a,b∈0,1
(
1 ⊕ a ⊕ b ⊕ V (s, t))
〈Ψ|Xas ⊗ Y b
t |Ψ〉AD>"#M,C"E8" <&)+<,<8+&'AB8&'5ABC5 &<,AB *=:"4<8+')+'A<5 &<,C/1 Xs, Yt(s,t)∈S×T
QbAAB
τ(G) =1
2
∑
(s,t)∈S×T
∑
c∈0,1
π(s, t)(
c ⊕ 1 ⊕ V (s, t))
v ?q<&3G/B 8+=(3<&<+&' 0,(&<+<,C/!,3*5AB <,<& 9&z:+' )OAB =+'0AB8
1/15()+'ABJ;J,CC,C&'
1/2QY
+' 0+'AB8); 9&f5AB8)a &
b9 9&2."4j "4* %!AB PABJ&<, !&fAB+'2V?qNM)+''<,AB P7!8,C/ 9&'
<8,C/ 9&')OAB8+ωq(G)
ωq(G) = τ(G) +1
2maxXs,Yt
∑
(s,t)∈S×T
π(s, t) (−1)V (s,t) 〈Ψ|Xs ⊗ Yt|Ψ〉 (1)
2 $/ L*[= & Ql6 GN Ol6 ⊕ ') r ∗(2)
)& [( `$ WX('Q$#"< f3ik$9, GC
W W @W WM %8 +<&<,5 #,C !&<,C&<8 d&<ZLY,C <&' ,C bA=AB (A' K);+B=AM Bt)8JC,L o m o
Q lQf CY)+'AB8/147!8[<,YV?qAB dB="4 &E3L/BC,=,C&'4=8Z'( 3o'R$9(#\* _|\*/(Nb &,C = ) = "-"4 9&$=&'AB8&'^&<ZAB+<,)Z9<,7!8f,C>NM,<&'^8 JOAB+< ^'8) +<, 8+'<8+XD5AB+<+' 3&<,AB =@?q /1H "4 9&'L=,<& 9&'|Qe[X)+<,C 5 ,C)O$=4568'-A56# 4+<"4$7!8D=4 /1H "4 !&'L69 9&C, 8F= .8 %5 +<&,C G+' 0,AB F=V?q');B5NV&' "-)*'AB !&(,C = )O = 9&'>=);+"4H &<+'>NM&' +< 798,@)OAB8+<+, 9&:j &<+'%5AB !&<+ B68F"4j "4G,C <& !&6);+(=>01 9&'><,C&<8(= (8 +' 0,AB `=,& 9&'P=IV?q<);B5NV&' "-)6QtebD)+' 8/14= f C=8Uu,C&G7!845-)+<,C 5 ,C)O4<&,C 5AB"-);&<,CJX/153-"456 ,7!8E7!8; 9&<,7!8 &* &'%7!8;C,_;= N)C8*)+'AUhAB ==5AB8/1 +<&'E=%3#5 , 5# Q bQl!&))[ o m !B]1NQ
e[-&<ZAB+'H "4P= f C6?r))C,7!8#zX&'AB8&%'<&'H "4I)Z!',798#69 9&I=/+<,3JGJ,C ;,C+'6=AB 9&*./ 8+'('AB !&*+<J,C&<+,C+' "4 9&: AB&'
1 & −1
Q;* /1 +'<,AB $)C8*01 +%=%5&<ZA+'H "4@)) IV? ,C 09C,C&' v2 l @L &'I= "4AB 9&<+'I);+ QqfQ v 38' +6OpQ AB+< OEQ[l!Z,C"4AB ! & %QqQ ABC&6Q
>
# I g% '#':43)R<*k4*/PS$/hB" $ '( 3o' $!(D,* 6(_|\*/(GCKa/ ':*/ <[$9%'; .B8"N>>$*)7$!(7$ '<WSB)8('':/G"'(MC !Z#(6NNp* ,[$n .*B;NN;Y_*':/<*BnG*WW d a"+*/ <?$ ' .B = & P1;' CYgb *9 *CN:/_!$ ;h<?$9 !X( ,5 p*B;$9 !q*B /C7$/NW (5fN:*= 4( Y*/X*B; +" .* 1
−1C f
a
c(E4*B(S*#(
*/@"+*./ < $B*;mD\"( .*?$!(q$9 !KNq$ ''N>u #(6Ni4* ,54 b
d
C( $ $9 !/ N#>#( ip*1 MC K<B(
|〈ab〉 + 〈bc〉 + 〈cd〉 − 〈da〉| ≤ 20g% FA
/J 〈x〉 $#"(( BO<*Pp*/ E8/X9NQ$9O_*#p*BS*#>*/<"\*./ <xC
>AB8>CAB :=AB +:8 %,C !&' +<)+' &&<,AB =%3#/,AB3&<,AB =E5 &<&',C 09C,C&'E);+>*AB,>=3"456 ,798E798; !&<,798%= G56B=+'G= < 8M$zI= 8M <AB8 8+'6Q
lAB, 9&S = T = 0, 1 π 3*=,<&<+<,CJ8&<,AB I8 ,UuAB+"42<8+
S×T &
V3UuAB 5 &<,AB =T/C8;&<,AB
=N_ ,);+
∀(s, t) ∈ S × T V (s, t) = s ∧ tl!,b:= 8M <AB8 8+':(8+' C,I &(f A &* );+<&01 9&);B:=@?q 5Z /1j &<+' "4 9&6tAB ^)O 8&>=5 +<,C+' 8+%5AB"-)OAB+<&' "4 9&I);+E7!8;&<+'-/+<,3J6&'AB,C+'
a(0), a(1)5AB+<+'<)OAB = 9&-8M+' )OAB '
=@?q(8+' C,8MP7!8<&<,AB 0 &
1+'<)O5 &<,C/1 "4 9&6 &
b(0)b(1)
5AB+<+'<)OAB = 9&(8MP+' )AB 'f=f A &f8M#7!8<&<,AB
0 &
1Q!e[T/+<,3Jf6&'AB,C+'
(−1)a(0) (−1)a(1) (−1)b(0) &(−1)b(1)
&<,UuAB 9&Z!!)OAB&<ZH'*=:V? ,C 09C,C&' v2 l &.=AB 5
|〈(−1)a(0)+b(0)〉 + 〈(−1)a(0)+b(1)〉 + 〈(−1)a(1)+b(0)〉 − 〈(−1)a(1)+b(1)〉| ≤ 2 +63I/ 8+(5 3B'<,7!8=
G<&.=AB );+
ωc(G) = maxa,b
∑
(s,t)∈0,12
1
4
(
1 ⊕ a(s) ⊕ b(t) ⊕ (s ∧ t))
=1
2− 1
8maxa,b
(
(−1)a(0)+b(0) + (−1)a(0)+b(1) + (−1)a(1)+b(0) − (−1)a(1)+b(1))
$ABJ&<, 9&:,C <,ωc(G) ≤ 3
4 &@V?qAB %/1 +<,_;, "4 9&b7!8[? ,C9NM,<&'Y8 '&<+&' 0,5 3B'<,7!8= &' +<"-,C ,<&') +<"4 &<& !&b=Y090 +/15)+'ABJ;J,CC,C&'
3/4Q
etE)+' 8/1=8XUV,C&f7!8>3E&<ZAB+<,=:3%"456 ,7!87!8; 9&<,7!8: [?rB="4 &<&,C&.);B=:"4A!=H zK/B+<,3JF6&'AB,C+'$A!56$56B5ZF<&X);B'`);+D3p/1 +<,_;56&<,AB nNM)O +<,C"4 9&=3/!,AB3&<,AB X=*V? ,C 09C,C&' v2 l );+f=2/B+<,3J6&'AB,C+'f+'<8C& !&f=*&'<&'2=()OAB3+<,&<,AB O5 &<8(<8+(=);,C+'*=)ZAB&'AB (5AB+<+' 6Q;Y $8&<,CC, !&*5>UV,C&6 AB8.)OAB8/1AB .=5 +<,C+'G8 <&<+&' 0,$7!8; 9&<,7!8F9 ! X)+'ABJ;J,CC,C&'`=$<855H4<&<+<,5 &' "4 !&P<8)O +<, 8+'Fz5 C$=X3"4 ,CC 8+'G<&<+&' 0,E5 3B'<,7!8Q
lAB,C& |Ψ〉 =1√2
(
|00〉 + |11〉)
V?q &&( 5'Z /1j &<+'%);+<&01G);+.*8+' C,G &+f A &6Q:N_ ,''AB
> !
|ϕ0(θ)〉 = cos θ|0〉 + sin θ|1〉|ϕ1(θ)〉 = − sin θ|0〉 + cos θ|1〉
e["4<8+'*=@?q(8+' C,% &.= f A &('AB 9&>AB+'.=AB .);+
Xa0 = |ϕa(0)〉〈ϕa(0)|
Xa1 = |ϕa(π/4)〉〈ϕa(π/4)|
Y b0 = |ϕb(π/8)〉〈ϕb(π/8)|
Y b1 = |ϕb(−π/8)〉〈ϕb(−π/8)|
)AB8+a, b ∈ 0, 1 Q
& 9&>=AB AB&<+'G5'ZAB,M$= |Ψ〉 AB `
〈Ψ|X ⊗ Y |Ψ〉 =1
2Tr( tX Y
)
)AB8+:&'AB8X
&Y
Q k/1 +<,_;4AB+'G,' "4 9&%7!8O= &'AB8:56B63PJAB I+' )OAB 'I<&=AB 4/15-)+'ABJ;J,CC,C&'
cos2(π/8) ' 0.85 > 3/4[ &3L"#8/,'I+' )OAB '-<&=AB #6/15
)+'ABJ;J,CC,C&'sin2(π/8)
Q
W nT q # M >AB84=AB AB #= P5 &<&'$'5 &<,AB 8 D)+' 8/1F<,C"-)$=@? 8 &<ZAB+'H "4`=$at<,C+' AB \798,
/BK AB8F) +<"4 &<&<+'=+' C, +F3p/B 8+$7!8; 9&<,7!8=@? 8 < 8G
=8 &) ⊕ `R ∗(2)zp8
"#M,C"-,i&<,AB F<8+.=)+'A=8,C&'('563,C+'*=:/15 &' 8+'.+' * D0+ =%=,C"4 <,AB [Q
# T 0 ( <P3 A[/VN;S
T$9(#N) 83C(;-n(5 FG/h |Ψ〉 ".*
= *B;WS= E' 8$9? '')BNY G 7$#*B) .':*1i$9 E<NNEo':*B W H = A ⊗ B ': 7+= dimA = dimB = n
C r; 4N *= s ∈ SG/
Xs 7](NNp*#3C8 A $94*/C (%'L'<±1
5 `') $N:*= t ∈ T59G/h
Yt ](NNp*#3C B $9Xp*C (O'+(''( ±1C K<B([
+!/N?$9(GB< ,E+"'3o h.*/ <( xs*B$
yt$9
R2n2 P1= 〈Ψ|Xs ⊗ Yt|Ψ〉 = xs · yt
5a': . s ∈ S
t ∈ TC
Y"<G~'+M= N#N;5h3/VN;S
T$9(%N)6NC -n(5Y
xs
yt$9(*B< , h.*/ <(1$9
RN ') -. s ∈ S
t ∈ T
C/VN; A B $!(4.':*1i($9m@'N K$9$/ 4 )(SB2dN/2e 5
H = A⊗B 5 G/ |Ψ〉 " .*2 NNB h+">= C\B= >$! H C K<B(%@\!/n O$!(O#( ip*]CXs
A Yt
B $9Ep*C (9'+(''( ±1(7= 〈Ψ|Xs ⊗ Yt|Ψ〉 = xs · yt
5 ': . s ∈ S
t ∈ T
C
" 7)Nh+*WSBCo Q gAB8+%&'AB8s, t
@AB,C&xs = Xs ⊗ I|Ψ〉 &
yt = I ⊗ Yt|Ψ〉 Q v AB"-"4Xs, Yt
AB 9& ±1)AB8+
' 8./B 8+')+'AB)+'|; & |Ψ〉 <&. AB+<"4xs
&yt
'AB 9&.= AB+<"41; &
∀(s, t) ∈ S × T xs · yt = 〈Ψ|Xs ⊗ I · I ⊗ Yt|Ψ〉 = 〈Ψ|Xs ⊗ Yt|Ψ〉> Q.>AB8)+'AB8/1AB (5%' = .8 $56B=+'% 01H +' "4 9&(+''&<+' ,C 9& AB8(<8))AAB .798
|Ψ〉 = |ΨN 〉 = |Ψ1〉⊗N A |Ψ1〉 =1√2
(
|00〉 + |11〉)
<&8 G);,C+'G EQ*AB8*CAB 8&<,CC,' +(5 +<&,C ()+'AB)+<, &'*="#&<+<,5*= s8C,
>
σx =
(
0 11 0
)
σy =
(
0 i−i 0
) &σz =
(
1 00 −1
)
>AB8.5AB"-"4 AB *);+.798 7!8(ABJ' +</&<,AB .<,C"-)|Qu1
σ2x = σ2
y = σ2z = Id
Jσxσy = −σyσx, σxσz = −σxσz
&σzσy = −σyσzh5|
σx ⊗ σx|Ψ1〉 = σy ⊗ σy|Ψ1〉 = σz ⊗ σz|Ψ1〉 = |Ψ1〉h=;σx ⊗ σy|Ψ1〉, σx ⊗ σz|Ψ1〉
&σy ⊗ σz|Ψ1〉
'AB 9&(&'AB8.AB+<&<ZAB01AB ;8M`z |Ψ1〉Q
lAB, !&>"#,C !&' ; 9&x = (x1, x2, x3)
&y = (y1, y2, y3)
= 8MF/15 &' 8+':=R
3 8 ,C&,C+'6 &)OA'AB
X = x1 σx + x2 σy + x3 σz &
Y = y1 σx + y2 σy + y3 σz
e[b798;&<+'TABJ' +</&<,AB t5 ,='<8t"4AB 9&<+' !&s7!8X
&Y
'AB 9&g=["#&<+<,5bZ +<"-,C&<, =L56+<+'#V? ,= !&<,C&'Y=AB 5P=IABJ' +</J-z$/ 8+'E)+'AB)+'-= ±1 Qg:#)C86gAB /1 +<,_;:UVB5 ,C "4 9&>798
〈Ψ1|X ⊗ Y |Ψ1〉 = x1y1 + x2y2 + x3y3 = x · yRS@ AB8.+'<&'EzI"4AB 9&<+' +>5AB"-"4 9&:5 &<&'G5AB <&<+<85 &<,AB ':01 +C,'% `=)C80+ ==,C"4 <,AB |QbAB8+*&'AB8&
k ∈ 1, . . . , dN/2e O'AB, 9&
X2k = I⊗k2 ⊗ σx ⊗ σ
⊗dN/2e−k−1z
&X2k−1 = I⊗k
2 ⊗ σy ⊗ σ⊗dN/2e−k−1z
>AB8./1AB (,C ',O=N_ ,@8 5ABC5 &<,AB `=N
"#&<+<,5Z +<"-,C&<, (<8+ dN/2e 7!8J,C&'6=L56+<+'X 09zFV? ,= 9&<,C&'D &I7!8,f !&<,5AB"-"%8&' 9&6Q bAAB
xs = (x1s, . . . , x
Ns )
&yt =
(y1t , . . . , y
Nt )
Q| %8&<,CC, !&g=)C8tUu,C&g7986',i 6= j
AB+'Xi⊗Xj |ΨN 〉 '&tAB+<&<ZAB01AB ;z |ΨN 〉 ,C@ [?q<&.);B.=, #5 ,CE=:/1AB,C+(7!8[?q $)Ai 9&
Xs =N∑
i=1xi
s Xi &
Yt =N∑
i=1yi
t Yi
Xs &
Yt'AB 9&fJ, X=2ABJ +</BJf<8+ dN/2e 7!8J,C&'6zG/B 8+'2)+'AB)+'f= ±1 & C/1 +<,_; !&
∀(s, t) ∈ S × T 〈ΨN |Xs ⊗ Yt|ΨN 〉 =N∑
i=1xi
s yit = xs · yt
& !&=AB f8 8G
z>= 8M <AB8 8+'s);+<&016 9&T=fV?q 5'Z /1j &<+' "4 9&652&<ZAB+'H "4 AB8) +<"4 &.=@?qNM)+<,C"4 +(=:"# ,H +'E<,C"-)C,_;3/ 8+.=8D09,C $"#M,C"#tzI5 < 8[ $+'5 +<,C/ 9&V?qNM)+'<,AB
(1)
ωq(G) = τ(G) +1
2maxxs,yt
∑
s,tπ(s, t)(−1)V (s,t)xs · yt (2)
AE3."#M,C"-,i&<,AB IYUV,C&g<8+b&'AB8gt/15 &' 8+'xs, yt ∈ R
min(|S|,|T |) Q /1AB,C&sAB+'s));+ &<+'8 5AB NM,AB ` ;&<8+' C%/15%)+'ABJH "4(=G)+'AB0+"-"#&<,AB ^' "-,=N_ ,Q
> m
- ⊕ ') r ∗c,s(2)
$#"( B%\& )N1$!(K<*B* 9(L
PE= (5': . s8x$!
_ n5Y2\!/n l6 j $9 !>lG ( Gx
5:\*/C <*#F 9'E':/dXB>W*/fNn5(3
= -
x ∈ L5`*/<B(
ωq(Gx) ≥ c(|x|) C -
x ∈ L5`*/<B(
ωq(Gx) ≤ s(|x|) Ce?qNM)+''<,AB
(2),C"-)C,7!8IV? ,C 5 C8<,AB ⊕ ^R ∗(2) ⊂ Y *QOY Uu,C&6tlQ Z +EL"4AB !&<+'
V? ,C 5 C8<,AB $)C8)+'5 ,
# N rY G. c, s : N → [0, 1]PQ= 0 ≤ s < c ≤ 1
5⊕ ') r ∗
c,s(2) ⊂ r
c,s(2)
$#"'>$9Q<*O'' MCtlAB,C& L ∈ ⊕ ^R ∗c,s(2)
&6;)AB8+(&'AB8&x ∈ Σ∗ Gx
< 8`5AB+<+'<)OAB = 9&6Q );+<&<,C+=
Gx AB8CAB Y=N_ ,C+Y8 -/1 +<,_; 8+
V (x)&' 79822)+'AB&'A5AB(x>R
(2)5AB+<+'<)OAB = 9&
,I"4j "4);+"4H &<+'=-5AB+<+'5 &<,AB K &G=-5AB"-) &<8=P798-5 8Mk=Gx
QGx
<&G=AB );+T=T ' "%Jf=(7!8<&<,AB
S &
T98 (=,<&<+<,CJ8&<,AB X=)+'ABJ;J,CC,C&'
π<8+
S×T1 &T8
UhAB 5 &<,AB `=:/BC8;&<,AB V
Qe[`/1 +<,_; 8+
V (x)5AB"-"4 5);+X&<,C+' +P8 5AB8)
(s, t) ∈ S × T8 Z;B+'=' AB 3
=,<&<+<,CJ8&<,AB π)8,(,Cb)+' );+'%V?q && |0〉|0, s〉 + |1〉|1, t〉 OAF%)+' "-, +>7!8J,C&(5AB+<+'<)OAB =^8
+' 0,<&<+')+<,C/1 =8/1 +<,_; 8+6@ &:G8&<+'G8+' 0,<&<+'-=>"4'01 LQe[I/1 +<,_; 8+G 9/1AB,(+' 0,'&<+' 8X)+'AB8/1 8+6798,O@5 &<8:8 >5 +<&,C :&<+ <UhAB+<"#&<,AB $ &f>C8,O+' 9/1AB,QY _ [:/1 +<,_; 8+*@5 &<83"4<8+'
P0 = |Ψ+st〉〈Ψ+
st|P1 = |Ψ−
st〉〈Ψ−st|
P2 = Id −P0 − P1
A |Ψ±st〉 = |0〉|0, s〉±|1〉|1, t〉 <8++' 0,<&<+' d & #Ql? ,C;ABJ&<, 9&28 4J,C&
c ∈ 0, 1 9,C)+' =3I=5 ,<,AB F7!8[? ,Cb8+,C&.)+<,'= (:)+'AB&'A!5ABG,C ,C&<,3V;<,C AB D,C[+' < &<&':,C"-"4=,3&' "4 9&6QY I<8))OA 9&s798TT)+'AB8/1 8+s@5 &<8T8 &<+ UuAB+<"#&<,AB 4+<J,C&<+,C+' E3*=5AB"-)OA 9&
<8+[+' 0,<&<+' &g<8+b'AB %+' 0,<&<+')+<,C/1 b| &b E565 83 !&b3.)+'ABJ;J,CC,C&'2"#M,C"#T=<855H=8k)+'AB8/1 8+GUVB5#8k/1 +<,_; 8+
V (x)[AB dABJ&<, !&%8 4NM)+''<,AB d&'AB8&G)AB,C !&%' "EJ3JPz
> N[/15-3D"4j "4-"#M,C"-,i&<,AB d<8+G=/15 &' 8+'%8 ,C&,C+'6[7!8,5AB+<+'<)AB = !&%,5 ,zLV?q &&=%V?q');B5%)+<,C/1=8`)+'AB8/1 8+6Q@*,C <,V;34)+'ABJ;J,CC,C&'"#M,C"#=<855H:=@? 8 ^)+'AB8/1 8+8 8
Gx<&*3P"4j "4I7!8%3#)+'ABJ;J,CC,C&'I"#M,C"#I=I<855H:=@? 8 )+'AB8/1 8+>UuB5-z
V (x)O5
798,@)+'AB8/1G&<ZAB+'H "4Q DABJ' +</1*_ ; "4 9&(7!8[? 8 <&<+&' 0,:AB)&<,C"#:)OAB8+2>)+'AB8/1 8+.5AB <,'&'>',C"-) "4 9&
zE"4 &<&<+'8 :)Z;B'(−1)a(s) <8+fV?q && |0, s〉 &
(−1)b(t) <8+V?q && |1, t〉 ;A a(s) &
b(t)AB 9&
+' )OAB '.=AB *z34798<&<,AB (s, t)
);+(=.)+'AB8/1 8+'(AB)&<,C"#8MD)AB8+( 8Gx
Q
W @
>AB8%/1AB %/!8d= %3X'5 &<,AB )+'5= !&'gzX&<+6/1 +'%I&<ZAB+'H "4P=4at',C+' 'AB [t7!8[? ,C/,C&(8 $C, ` ;&<8+' b !&<+'%<<&'H "4(=G)+' 8/1(,C !&' +B5 &<,Uh: &.)+'AB0+"-"#&<,AB k' "-,=N_ ,Q Y5 &<&'f);+<&<,f AB8CAB Y)+'AB8/1 +YV?qNM,'&' 5=@? 8 .5AB NM,AB -)+'5 , 9&<+'fV?rC01AB+<,C&<Z"4=@?r))+'AM,C"#&<,AB D)AB8+2`* v >a\ &8 X<<&'H "4(=()+' 8/1> ;&<8+' =8P&) ⊕ `R ∗(2)
)OAB8+
w
5^"4j "4k)+'ABJH "4Q v &<&'k5AB NM,AB <&X AB8/1 C. &D AB8D)O 'AB D7!8`F<!&'H "4D=)+' 8/1I,C 9&' +B5 &<,Uu47!8; 9&<,7!8-) 8/1 !&IUhAB8+< ,C+I8 W56B=+'D<,C"-)L)OAB8+V?qNM)+''<,AB W=@?rC01A+<,C&<Z"4%=@?r))+'A6M,C"#&<,AB [b) +<"4 &<& !&%)O 8&j &<+'#=I&<+'AB8/1 +=-"4 ,CC 8+'EC01AB+<,C&<Z"4E)AB8+5 +<&,C )+'ABJH "46QsY p@ &6b8 d)+'AB&'A5ABP ;&<8+' = E4UuAB+<"#C,<"4L798; !&<,798L)O 8&E'&<+B=8,C+'P p8 pC01AB+<,C&<Z"4X=@?r))+'A6M,C"#&<,AB UVB<&<,=, 8MAB+''7!8[? ,C2'&ENM)+<,C"4P K&' +<"4="#M,C"-,i&<,AB F="#&<+<,5.)OA<,C&<,C/16Q
lAB,C&G = (V,E)
8 $0+)ZG AB FAB+<, 9&'Q D AB&'
`* v >a(G) = max
W⊂V(W × (V \W )) ∩ E
e[.)+'ABJH "4>=@?qAB)&<,C"-,&<,AB X5AB <,<& !&fz565 8 +f^( v >a(G)
)OAB8+8 #0+)ZG
=&,CCn = |V | '&$ s5AB"-) &6Q A "# $ &$ ,CCC,3"4'AB AB !&F"4AB 9&<+'K= $8 n+<&<,5 d5 H J+'~ m]g7!8GV?qAB 9&:)OAB8/B,C&>= &' +<"-,C +:8 %/ 8+>))+'A5'ZDuz-8 `UVB5 &' 8+
α ' 0.878)+'HN
=%^( v >a(G)
F&' "-)>)ABC AB"-,3g+ =AB"-,'O $UV, 9&>+'5AB8+'z-3#)+'AB0+"-"#&<,AB ' "-,=N_ ,Q1e[ 8+2))+'A!5Z(5AB <,<&'.z:NM)+<,C"4 +3>&,CC.=f3:5AB8)O"#M,C"#.=
G5AB"-"4
>"#M,C"E8" =@? 8 D)+'AB0+"-"4%7!8;B=+&<,798'8+2( 9&<, +'6)8,(z5AB <,= +' +(8 :+' 3M&<,AB =E5E)+'AB0+"-"4@ k8&'AB+<, 9&:>/B+<,3JzPj &<+'I=*/15 &' 8+':=
Rn Qet#+' 3M&<,AB k<&
8 )+'AB0+"-"4-' "-,=N_ , &G'E+''AB8= ^&' "-)G)ABC AB"-,3VQO( I'ABC8&<,AB d))+'A5'Z-<&AB+'.ABJ&' 98E $)+'A< & 9&.(/15 &' 8+'(ABJ&' 98*<8+8 F<);B5E6&'AB,C+'G=G=,C"4 <,AB
1Q
*)+'H=: AB"%J+' 8'(&' 9&&<,C/1*)AB8+."4 C,AB+' +(3I5AB '& 9&'G=@?r))+'A6M,C"#&<,AB α:ZAB&6
:,C = +6;^A'' [ & ?q:AB C2~q ogAB 9&_ ; "4 !&*"4AB !&<+'% > ]#798<,t3-5AB 5N&<8+'* ,798 "4F &,C&$/!+,*AB+'$)OAB8+D&'AB8&
ε > 0,C% &,C&F,C"-)A'',CJk=@?r))+'A!5Z +
`* v >a(G)
z8 DUuB5 &' 8+α + ε
)+'H6Q>AB82CAB 2+' &<+'AB8/1 +f3"4j "4:5AB <& !&'>=@?r))+'A6M,C"#&<,AB $7!8 A! "# f & ,CCC,3"4'AB
zE);+<&<,C+.=@? 8 D)+'AB&'A!5AB=:)+' 8/1:,C !&' +B5 &<,C/1%z= 8ML)+'AB8/1 8+')OAB8+`* v >a(G)
Qe? ,=<&-798XL/1 +<,_; 8+4/^&<,C+' +-8 $+'j &'D=8W0+)ZF8WZ;B+'=@Y &6YAB,C&I !/1A61 +L5'Z;B5 8 \== 8MF'AB"-"4 &'>345AB <&<,C&<8; !&Gz45'Z;B5 8 k=>= 8M$)+'AB8/1 8+'6O'AB,C&* !/1A61 +:G"4j "4'AB"-"4 &8M4= 8M4)+'AB8/1 8+'6Q!et.J8&)OAB8+T)+'AB8/1 8+'T +G=.+' !/1A61 +f8 .+' )OAB '*=,O +' !&'*6? ,CAB 9&.+' 8`= 8ML A 8=(=,@ +' 9&'6; &.8 :+' )AB ,= 9&<,7!8%6? ,CAB !&(+' 8$:"4j "4G A! 8=@Ql!,%)+'AB&'A5AB- &,C&5 3B'<,7!88 '&<+&' 0,AB)&<,C"#I)OAB8+>:)+'AB8/1 8+'' +,C&=%"4 &<&<+'=@?rB55AB+'=FzV?r/ 5G<8+8 5AB8)O"#M,C"# &.=@?r&<&<+<,CJ8 +*=
1z&'AB8.'AB"-"4 &'(=:3
5AB8)=0
8M`8&<+'6Q v )O = 9&6O= *%56B>AFV?qAB k8&'AB+<,'G*)+'AB8/1 8+'z-);+<&01 +=V?q 5'Z /1j &<+' "4 !&6,C)O 8/1 9&&' !&' +T=@? 8&<,CC,' +25 & 5'Z /1j &<+' "4 !&2)OAB8+5AAB+'=AB + 8+'+' )AB 6Q>AB8(CAB (/1AB,C+*5 ,=''AB8>7983-"#+<01%="# A 8/!+'E798,t 8+*<&)O +<"-,'G);+V?q 5'Z /1j &<+' "4 !& 8+b)O +<"4 &s=&<+<,5'Z +Y=@? 8 EUuB5 &' 8+sNMB5 &' "4 9&Y 09z*5 C8,=V?rC01AB+<,C&<Z"4=@?r))+'AM,C"#&<,AB ^= A! "# . &( ,CCC,3"4'AB [Q
>AB8T5AB"-"4 AB 2);+T=5 +<,C+'>()+'AB&'A!5AB ⊕ `R ∗(2))8+2`* v >a\7!8. AB825AB <,=N
+'AB 6)8,. AB8.)+'AB8/1AB (I5AB+<+'5 &<,AB [QlAB,C&
m = |E| Q gAB8+ i ∈ VAB ` AB&'
∆iG= 0+'E=8F A 8=
i; &
∆ = Σi ∆iG= 0+'G&'AB&
=G
Q:N_ ,''AB 8 G"4<8+'G=:)+'ABJ;J,CC,C&'%'8+V 2 = 1, . . . , n2 );+
wIo
∀(i, j) ∈ 1, . . . , n2 π(i, j) =
1
∆ + m
<,(i, j) ∈ E
∆i
∆ + m
<,i = j ∈ V
0<,C AB
$5AB <,=H +'G8 D/1 +<,_; 8+>798,@)+' =$.=5 ,<,AB ('8,C/B !&'
∀(i, j) ∈ 1, . . . , n2 V (i, j) =
1<,
i 6= j0
<,i = j
v 5'ZAB,M$<&&' t7!8
τ(G) =1
2
∑
(s,t)∈S×T
∑
c∈0,1
π(s, t)(
c ⊕ 1 ⊕ V (s, t))
=1
2lAB,C&
(X,Y ) = (xii∈V , yii∈V ) ∈ Sn×S
n 8 2'&<+&' 0,27!8 5AB 7!82= <AB8 8+'Y(8+' C, & f A &G= >)+'AB&'A!5ABI=5 +<,C&G5 ,=''865AB"-"4I=AB I);+:E&<ZAB+'H "4-=ab<,C+' 'AB [QlAB,C&
f(X,Y ) =1
2+
1
2
∑
(i,j)∈1,...n2
π(i, j)(−1)V (i,j)xi · yj
3f)+'ABJ;J,CC,C&'f=<855Hg=5 &<&'T<&<+&' 0,| &g'AB,C&X0, Y0
8 <&<+&' 0,2AB)&<,C"# f(X0, Y0) =
maxX,Y f(X,Y )Q1>AB8CAB Y"4AB !&<+' +T7!81 & 9&=AB . AB&<+'.5'ZAB,M4=)OAB,=
π(i, j) @5 &'
z45Z;B5 8 =*5AB8)='AB"-"4 &'63#'&<+&' 0,%090 ; !&'-<&(8 I<&<+&' 0,<"4 &<+<,798 AB ) 8&.'8))A' +.7!8
X0 = Y0Q v AB"-"4
f<&J,CC,C 6,C+'%'!"4 &<+<,7!8
1
2
(
f(X,Y ) + f(Y,X)
)
− f
(
X + Y
2,X + Y
2
)
=1
2(∆ + m)
∑
i∈V∆i(xi · yi − 1)
+1
4(∆ + m)
∑
(i,j)∈E
(
− xi · yj − xj · yi + xi · xj + yi · yj
)
=1
4(∆ + m)
∑
(i,j)∈V
(
− 4 + 2xi · xj + 2yi · yj + (xi − yi) · (xj − yj))
+6)OAB8+(&'AB8./15 &' 8+'*8 ,C&,C+'62xi · yi + 2xj · yj + (xi − yi) · (xj − yj) ≤ 4
h55 ,t)O 8&('/1AB,C+(zIV?r,=G=@? 8 F565 8b=G=,O +' !&<, C|NQ v AB"-"4
f(X,Y ) = f(Y,X)AB FABJ&<, 9&
f
(
X + Y
2,X + Y
2
)
≥ f(X,Y )
v AB"-"4"#M,C"E8" =f
<8+[b/15 &' 8+'g8 ,C&,C+'s<&b 09 "4 !&sAB G"#M,C"E8" <8+t3.JAB88 ,C&'1AB 4 4==8,C&7!8ff"#M,C"E8" =
f<& 09 "4 9&f&<&' ,C 9&2 (
(X +Y )/2, (X +Y )/2)
5?q<&Sz=,C+'%7!8:V?qAB $)O 8&(<8))OA' +(7!8:3-<&<+&' 0,090 ; !&'E'&.<!"4 &<+<,7!8QlI/ 8+(<&AB+'
f(X,X) =∆
∆ + m− ∑
(i,j)∈V
π(i, j)xi · xj
+6<,k =
^( v >a(G)
A "# ( & ,CCC,3"4'AB FAB !&."4AB 9&<+'E798
w >
k ≤ maxxi,yi∈Sn
1
2
∑
(i,j)∈V
(1 − xi · xj) ≤ α · k5798,[=AB %,5 ,V5AB"-"4
∆ = m/2
1
3+
k
3m≤ max
X,Yf(X,Y ) ≤ 1
3+
αk
3m
:N_ ,''AB :)+'ABJH "4z)+'AB"4''G *^( v >a(k, ε)
E p ( D0+)Z
G = (V,E)8 F 9&<, +
k &.8 D+'
ε > 0Q
1
|E|^* v *a
(G) ≥ kQ
1
|E|^* v *a
(G) ≤ k
α− ε
Q
v 798,@)+'5H=%)+'AB8/1&<ZAB+'H "4G'8,C/B !&6Q
# r; %. N;WVNk
T. \"'ε > 0
59r r ,mi ' ,7g%D
(k, ε) ∈ ⊕ ') r ∗c,s(2)/J
c = (1 + k)/3
s = (1 + k − αε)/3C
et)+'ABJ;J,CC,C&'G=:090 +(zI5 < 8 ⊕ `R ∗(2)=AB =AB 58 G))+'AM,C"#&<,AB F=:3&,CC=
345AB8)O%"#M,C"#=G
z-8 `UuB5 &' 8+α
)+'H6Q: *%56B>=`* v *a55 ,g'&*AB)&<,C"#VQv ) = !&6Y AB8-<)O +'AB 47!8L= 4L56B=+'D=@?r8&<+'4)+'ABJH "46Y)OAB8+-'798 I8 C01A+<,C&<Z"4-=@?r))+'AM,C"#&<,AB dAB)&<,C"# [?q<&:);B5AB !8[@5 &<&'4))+'A!5Z-)8,''I=AB + ;,' 5zI=(C01AB+<,C&<Z"4*) +UuAB+<"# 9&'* &.,C A6/ 9&'6Q
<V, ⊕ - 7JH*:2% "8:2; PX#$:2;!() *&;I8 *Z$/2%W:2;N;O79()!:246:2k *8 8791):2;
etI5 3B' ⊕ `R ∗(2) 5'Z; 01%);B.<,@V?qAB $5AB <,=H +'%798(= 8ML)+'AB8/1 8+'*'AB 9&(7!8;
&<,798| &.798>:/1 +<,_; 8+. 8+( !/1AB,8 7!8<&<,AB `8$5'ZAB,MX);+<"-,[8 $ ' "EJ:_ ,t=@?q &&'798; !&<,798AB+<&<ZAB01AB ;8M tG)+'AB8/1 8+') 8/1 !&E"4'8+' +%V?q &&I798,T 8+E<&% !/1A61X &E ==8,C+':3I798'&<,AB $5 3B'<,7!8798,O 8+.<&)OA'Q g+*5AB 9&<+'3I<,C&<8;&<,AB $5Z; 01G<,@ &&' 'AB 9&();B.AB+<&<ZAB01AB ;8M = (556B6,C[<&,C"-)A<,CJG=:.=,@ +' 5 , +();+UV,C&' "4 9&6Q
Dt?l ⊕ = ') r ∗ (N`_*R$B;"'>$9 $9 !$NN(
S5T5
': . s ∈ S
t ∈ T5`$9(Q" .*F' ( |ϕs〉
|ϕt〉0BC(O= ( WWBBA5
?$/NhS WSB $!F'L#\*#PhB"(π S × T
5 3WSB V
$9S × T
$*B 0, 1 _* GBWSB $9%p*/ *WSBC /VN;
c, s : N → [0, 1]$9 ! GBWSB)I(8= s(n) < c(n)
': -. nC
NI*L3<*4,6 ⊕ = ') r ∗ \B;WVN;`+& N)6NQ$!(1_** 9L
PQ= r; L. x ∈ L
592+!n l6 Gx$ WX(' ⊕ = '* r ∗ 5>\*/C _*]C j8':*B W $9
xN
NK'-^':PXBB>S*/tN |x| 5`$9 !1'+ B ( A
B') %i l6 $;<*%'+#+*#3d B">$! ' (-( ^* 7 )
c(|x|) C
ww
r; L. x /∈ L592+!n l6 Gx
$ WX(' ⊕ = '* r ∗ 5>\*/C _*]C j8':*B W $9x
NNK'-9':/dXBB>W*YN |x| 5= ><*Q'+#+*#d B"$9? ' ($9a& 9')BN>= \ '[$9'+ , (A,B)
jL`l6 3 h M"Nu '?j s(|x|) C* 8 ⊕ `R ∗(2)
<&5 3,C+' "4 9&I8 < 8 ⊕ 7^R ∗ ["#,GV? ,C !/1 +''P [?q<&%);BE/!+,VQt*AB8CAB I)+'AB8/1 +#7!8s"#C0+'XX)AB&' !&<, .09,C =L)8,' 5$B55AB+'=F8W/1 +<,_; 8+67!8,f) 8&'61 +(=:), 01 +.)+'AB8/1 8+'6;V? ,C 5 C8<,AB F<8,C/ 9&'G+'<&'/+,
# ⊕ = '* r ∗c,s ⊂
rc,s(2)
') *. 6')*BL* h<(E$9>\BK'@" $!G 6$9>\BN<<6WWBc
sC
" 7)Nh+*WSBCtetD)+' 8/1P=454&<ZAB+'H "4P'IUV,C&% d&<+'AB,E &)6Qt*AB8E5AB"-"4 AB E);+=AB +GV?qNM)+''<,AB =I3L/ 8+P ,I3L)+'ABJ;J,CC,C&%"#M,C"#4=-<855HG=-= 8M)+'AB8/1 8+'i=@? 8 < 8P7!8; 9&<,7!8
G=8#&)O ⊕ 7`R ∗ 9)8, AB8T=N_ ,''AB T8 < 8
G′ )OAB8+fx>R (2)
Q9>AB8"4AB 9&<+'AB : <8,C&'798<,b*8+' C,% &(f A &>AB 9&*= 8M$)+'AB8/1 8+':)OAB8+
GAB+'>,CbNM,<&'G8
)+'AB8/1 8+ v Z;+<>)OAB8+G′ =AB 9&>3-)+'ABJ;J,CC,C&'I=%<855H><&>NMB5 &' "4 !&>34"4j "4E7!8%5 C
=8L5AB8)Ih*8+' C,f A &iT8 < 8G
Q!>AB8"4AB !&<+'AB f_ ; "4 9&f3%+'5 ,C)+'A!7!85?q<&Sz=,C+'798;<, v Z;+<><&.8 $)+'AB8/1 8+>)AB8+
G′ ;AB+'(,CbNM,<&'%= 8M$)+'AB8/1 8+'()OAB8+G
&<&' ,C0 ; !&3"4j "4)+'ABJ;J,CC,C&'%=%<855H6Q v 5 ,t<8 4+#zI &JC,C+.&<ZAB+'H "4Q
O G′
lAB, 9&%*8+' C,- &f A &G= 8M)+'AB8/1 8+'G)OAB8+ 8GQ@*)+'H+'5 )&<,AB d=I 8+%7!8<&<,AB
798; !&<,798[,CG))C,798 !&E5'Z;B5 8 d8 I"4<8+'I)+'A5 &<,C/14J,C ;,C+'[798IV?qAB k AB&'I+'')5 &<,C/1N"4 9&
A = A0 − A1 &
B = B0 − B1O<8+:V?q &&G7!8[? ,CAB 9&+' 8[@,C <,Y798I<8+: 8+:+' 0,<&<+'
)+'AB)+'O /1 9&<8 C "4 9& 5'Z /1j &<+'46/155 C8,Y=V?r8&<+')+'AB8/1 8+6@ &> ==8,' !& 8+>+'N)AB Q +' "#+'798^);+D,CC 8+'D798
A &
B'AB 9&X=L"#&<+<,5XZ +<"-,C&<, $z/ 8+'
)+'AB)+'-= ±1 g=AB 5X&' C-7!8A2 = B2 = Id
s5AB"-"4A∗ = A
&B∗ = B
AB W ==8,C&>7!8
A &
B'AB !&> 09 "4 !&8 ,C&,C+'6O &()O 8/1 9&=AB 5j &<+'%/8:5AB"-"4E=>&<+
UhAB+<"#&<,AB 7!8; 9&<,7!8%)C8& B&798P=G"4<8+'6Qb*AB8ENM)AB,C&' +'AB I5IUV,C&E W|Qb(8+' C, & 2 A &();+<&01 9&.8 $ &&. 5Z /1j &<+' |Ψ〉 +<J,C&<+,C+'Y; & 9&*=AB .=798'&<,AB |ϕs〉
&|ϕt〉
,C+' )AB = !&(a, b)
6/15)+'ABJ;J,CC,C&'
PG(a, b|s, t) = 〈ϕs ⊗ Ψ ⊗ ϕt|Aa ⊗ Bb|ϕs ⊗ Ψ ⊗ ϕt〉et/ 8+(=8 < 8
G)AB8+.)+'AB8/1 8+'(*8+' C,G &+f A &(<&.=N_ ,);+
ωA,B(G) =∑
s,t,cπ(s, t)V (c|s, t)PG(c|s, t)
APG(0|s, t) = PG(0, 0|s, t) + PG(1, 1|s, t) PG(1|s, t) = PG(0, 1|s, t) + PG(1, 0|s, t) &
V (c|s, t) = c ⊕ 1 ⊕ V (s, t)Q `-,C ',
PG(0|s, t) − PG(1|s, t) = 〈ϕs ⊗ Ψ ⊗ ϕt|(A0 ⊗ B0 + A1 ⊗ B1
−A1 ⊗ B0 − A0 ⊗ B1)|ϕs ⊗ Ψ ⊗ ϕt〉= 〈ϕs ⊗ Ψ ⊗ ϕt|A ⊗ B|ϕs ⊗ Ψ ⊗ ϕt〉
v AB"-"4PG(c|s, t) =
1
2
(
1 + (−1)c(P (0|s, t) − P (1|s, t))) .)+'ABJ;J,CC,C&':'AB"-"4 !&>z
1N3
/B 8+.=G
<&
w o
ωA,B(G) =1
2
(
∑
s,t,cπ(s, t)V (c|s, t)
(
1 + (−1)c〈ϕs ⊗ Ψ ⊗ ϕt|A ⊗ B|ϕs ⊗ Ψ ⊗ ϕt〉)
)
:N_ ,''AB zX)+'' 9&E8 < 8G′ )AB8+Ix>R
(2)Qtet4/1 +<,_; 8+E=,')A'-=#798;&<+'#+' 0,<&<+'
798; !&<,798 1
2
1 &
2QBe[s= 8ME)+' "-, +'Y'AB !&g'g+' 0,<&<+'s)+<,C/1s &ss= 8ME= +< , +'
UhAB+<"4 !&%I+' 0,<&<+'P=-"4'01|Q@e[-/1 +<,_; 8+%&<,C+'#8 47!8<&<,AB (s, t)
8Z;B+'='8,C/B !&G3=,<&<+<,CJ8&<,AB F=)+'ABJ;J,CC,C&'
π &)+' );+'%'(798;&<+':+' 0,<&<+'*= V?q &&
|0〉|ϕt〉|0〉|ϕs〉 + |1〉|ϕs〉|1〉|ϕt〉Rb 9/1AB,IAB+'(*= 8M$+' 0,<&<+'
1 &
28$)+'AB8/1 8+6O798,t@5 &<8%i-&<+ UuAB+<"#&<,AB [ &
C8,O+' 9/1AB,G= 8MX+' 0,<&<+'=:"4'i016Qe[:/1 +<,_; 8+.@5 &<8%AB+'=.5'Z; 01* 9&<+':=,O +' !&'I+' 0,<&<+'4 WF)A''<,AB [g W5AB 9&<+'ABC3 9&L<8+V?q &&4=LAB K)+' "-, +-+' 0,<&<+'Ql!,5)+' "-, +G+' 0,<&<+'-<&= V?q && |0〉 [AB+'I/1 +<,_; 8+GAB+'=AB 4':+' 0,'&<+'G= :V?qAB+'=+'
1,
1,
2,
2Qtl? ,C<&= V?q && |1〉 tAB+':I/1 +<,_; 8+%AB+'=AB 4':+' 0,'&<+'= V?qAB+'=+'
1,
1,
2,
2Qe[D/1 +<,_; 8+#"4<8+'X_ ; "4 9&X'-= 8MW)+' "-, +'4+' 0,'&<+'Pz^V?r,=$=D3
"4<8+')+'A<5 &<,C/1
P0 = |Ψ+〉〈Ψ+|P1 = |Ψ−〉〈Ψ−|P2 = I − P0 − P1
A |Ψ±〉 =1√2
(
|00〉 ± |11〉)
l? ,CbABJ&<, !&*8 ^+'<8C&&c ∈ 0, 1 AB+'(,Cg5AB 5 C8&>7!8G% n=>= 8M$J,C&'*= * < 8
G &,C&c &.,C@)+' =D3-=5 ,<,AB `5AB+<+'<)AB = !&';<,C AB F,C@+' &<&',C"-"4=,3&' "4 !&6Q
WO*G G′
5 &<&'-'5 &<,AB AB8:)+'AB8/1AB G7!8@<,g*8+' C,- & f A &G'AB 9&G= 8M^)+'AB8/1 8+')AB8+ 8
GAB+',CtNM,<&':8 $)+'AB8/1 8+ v Z;+<()AB8+
G′ &' [798
∀(s, t) ∈ S × T PG(0|s, t) − PG(1|s, t) = PG′(0|s, t) − PG′(1|s, t)v AB"-"4XV?qNM)+'<,AB W=
ω(G) PUV,C&I,C 9&' +</1 ,C+#7!8
PG(0|s, t) − PG(1|s, t) Y55 ,.<8 4+z)+'AB8/1 +-V? ,C 09C,C&'
ω(G′) ≥ ω(G)YAWV?qAB =N_ ,C&
ω(G′)5AB"-"4D3`)+'ABJ;J,CC,C&'D"#M,C"#
798[?rF8 p)+'AB8/1 8+4=P'#UV,C+'LB55 )&' +I);+P/1 +<,_; 8+-=8 8G′ Qge? ,C 09C,C&'X,C 9/1 +''X' +
)+'AB8/1G (iii)
QlAB, 9&,C <,*8+' C,T &$f A &Y= 8M%)+'AB8/1 8+'s)AB8+
G798,);+<&01 9&Y8 - && |Ψ〉 Q>N_ ,'AB
8 X)+'AB8/1 8+ v Z;+<)OAB8+G′ Q v Z;+<+'AB,C&f+' 0,'&<+'
1 &
2=:3E);+<&.=8D/1 +<,_; 8+6Q
Rg)OA''H=E8 ^&<+'AB,',H "4E+' 0,'&<+' 1798[? ,Cs,C ,C&<,3C,'I= :V?q && |Ψ〉 QetE)+'AB8/1 8+%L=AB 5I
)OA'''<,AB X&<+'AB,*= +< , +'.+' 0,<&<+'(=V?q &&.&'AB&
|0〉|ϕt〉|0〉|ϕs〉|Ψ〉 + |1〉|ϕs〉|1〉|ϕt〉|Ψ〉Y P5AB !&<+'AB3 9&<8+(+' 0,<&<+'
1 v Z;+<f))C,798(8 ,C&,C+'
A⊗ I &
B⊗ I9A#V? ,= 9&<,C&'
0,C&.<8+.3);+<&<,G= |Ψ〉 7!8,t));+<&' ;,C&zV?r8&<+')+'AB8/1 8+6Q;e?q &&(&'AB&t= /!, !&
|0〉|ϕt〉|0〉A ⊗ I(|ϕs〉|Ψ〉) + |1〉|ϕs〉|1〉B ⊗ I(|ϕt〉|Ψ〉)
w ]
v Z;+<#+' !/1AB,$#+' 0,<&<+' 1
&2
8 /1 +<,_; 8+6Q v $= +< , +LO5 &<8F#5Z; 01L=+' 0,<&<+'.7!8V?qAB F-=5 +<,C&'.5 ,=<86ABJ&' ; !&(V?q &&
|ϕ〉 = |0〉|0〉A ⊗ I(|ϕs〉|Ψ〉|ϕt〉) + |1〉|1〉I ⊗ B(|ϕs〉|Ψ〉|ϕt〉)AV? ,= 9&<,C&'#P &'- &' =8I)AB8+G0,C+<8+ |ϕt〉
= :)+' "-, +G56B6O & |ϕs〉= :I'5AB =@Q
Y $"4<8+ !&(= .3J;B |Ψ±〉 /1 +<,_; 8+(ABJ&<, 9&*+'<8C&&'0, 1
/15G)+'ABJ;J,CC,C&'
PG′(0|s, t) = 〈ϕ|P0|ϕ〉 =1
2
(
1 + 〈ϕs ⊗ Ψ ⊗ ϕt|(A ⊗ I) · (I ⊗ B)|ϕs ⊗ Ψ ⊗ ϕt〉)
1/!, !&(=
AA∗ = BB∗ = 1f &
PG′(1|s, t) = 〈ϕ|P1|ϕ〉 =1
2
(
1 − 〈ϕs ⊗ Ψ ⊗ ϕt|(A ⊗ I) · (I ⊗ B)|ϕs ⊗ Ψ ⊗ ϕt〉)
(,C ',
PG′(0|s, t) − PG′(1|s, t) = 〈ϕs ⊗ Ψ ⊗ ϕt|A ⊗ B|ϕs ⊗ Ψ ⊗ ϕt〉 = PG(0|s, t) − PG(1|s, t)5798,[ &JC,C&>5' .=3)+' 8/1Q
WO*G′ G
(5 ,C)+'A7!8 "4 9&6 AB8s)+'AB8/1AB T= Y5 &<&''5 &<,AB 4798', v Z;+<<&s8 -)+'AB8/1 8+)AB8+ 8
G′ AB+'(,C[NM,'&'=.)+'AB8/1 8+'(*8+' C,G & f A &.)OAB8+G
&' (7!8
PG(0|s, t) − PG(1|s, t) = PG′(0|s, t) − PG′(1|s, t)lAB,C& v Z;+<*8 `)+'AB8/1 8+:)AB8+> < 8
G′ Q v AB"-"4 AB :);+* /C8 +*3#)+'ABJ;J,CC,C&'I798[?rv Z;+<.=@?qj &<+'%B55 )&');+:/1 +<,_; 8+6QY D&'AB8&'01 +C,C&' v Z;+<>))C,798:8 G&<+ UhAB+"#&<,AB 8 ,C&,C+'
C8M`+' 0,<&<+'
1 &
27!8[? ,Cb+'AB,C&=8/1 +<,_; 8+6[,C <,Y7!8[?rz#'AB ^+' 0,<&<+'
)+'AB)+' 1,C ,C&<,3C,%z |0〉⊗p Q>AB8.=5 +<,C/1AB .V?rB5 &<,AB `=
C=3"# ,H +'%'8,C/B !&'
C|0〉|ϕs〉|0〉⊗p = |0〉|α0s〉 + |1〉|α1
s〉C|1〉|ϕt〉|0〉⊗p = |0〉|β0
t 〉 + |1〉|β1t 〉v AB"-"4/1 +<,_; 8+>O5 &<8G=*5'Z; 01>=+' 0,<&<+'>)+'H*/1AB,C+*+' 8$3I+' )OAB '=8$)+'AB8
/1 8+6 AB8*CAB *= /1AB,C+*=5AB"-)OA' +(* &&'."# ,C)8><8+.5(=,@ +' 9&'*+' 0,<&<+'6;5G798,5AB"-)C,798^8 ) 8 L AB&&<,AB 6Qf*AB8P)OAB8/1AB D<8))A +6T P)O +<&'`=^01 +C,C&'f798|α1
s〉 = |β0t 〉 = 0
)8,'7!8P5 &<&'X);+<&<,D=LV?q &&-' +`+' < &'P);+IL/1 +<,_; 8+ s5 3` #UV,C&798[?r80"4 !&' +(.5Z; 5*='855H.=8$)+'AB8/1 8+6Q;:5AB"-)OA'AB
|α0s〉 =
∑
i|i〉|αi
s〉 & |β1
t 〉 =∑
i|i〉|βi
t〉A |i〉 <&g8 J;BT=2V?q<);B52 01 =+'f);+b |ϕs〉, |ϕt〉
B5AB+<+'<)OAB = 9&8+' 0,<&<+' 2
&. |αis〉, |βi
t〉'AB 9&:=./15 &' 8+'+<J,C&<+,C+'I ,t8 ,C&,C+'6 ,bAB+<&<ZAB01AB ;8M;N5AB+<+'<)AB = !&
8\+' 0,<&<+' 1QT( F&' CF=5AB"-)OA<,C&<,AB )O 8&P|?qABJ&' ,C+Xz);+<&<,C+L=F3=5AB"-)OA<,C&<,AB =
l5'Z"-,=&.= &&' |α0s〉
& |β1t 〉
Qet*)+'AB8/1 8+(+' 9/1AB,:f+' 0,<&<+' 1 &
28X/1 +<,_; 8+6798,
O5 &<8%'<8J<&<,C&<8&<,AB ()AB8+(ABJ&' ,C+(V?q &&∑
i
(
|0〉|0〉|i〉|ϕt〉|αis〉 + |1〉|1〉|ϕs〉|i〉|βi
t〉)
R@"4'8+' &(ABJ&<, 9&*:+'<8C&&0
/15G)+'ABJ;J,CC,C&'
w
PG′(0|s, t) =1
2
(
1 +∑
i,j〈i|ϕs〉〈ϕt|j〉〈αi
s|βjt 〉)
:N_ ,''AB (z)+' 9&(= 8MX)+'AB8/1 8+'(*8+' C, & f A &.)AB8+GQ*8+' C,G & f A &.);+
&01 9&48 W AB"EJ+'$=X);,C+'# 09(z^3=,C"4 <,AB =XV?q<);B5$ 01 =+'D);+--/15N&' 8+' |α0
s〉, |β1t 〉
[7!8IV?qAB k AB&'Σk |k〉 ⊗ |k〉 tAkI)+' "-, + |k〉 ));+<&<, 9&IzX(8+' C,4 &
= 8M,H "4>z f A &6Qt/ 9&2798 8P (5AB"-"4 5(8+' C,* & f A &2,C ,C&<,3C,' 9&28 P5 +<&,C AB"%J+'D=X+' 0,<&<+'4z |0〉 5AB), 9&F );+- :% 8+ |k〉 '8+8 8&<+'X+' 0,<&<+'s)+'A=8, !&V?q &&
Σk |k, k〉 ⊗ |k, k〉 &f)+' );+' 9&. 09 "4 !&8 L+' 0,'&<+'>=>3%"4j "4=,C"4 <,AB D798>5 C8,5AB 9&' ; !&G |k〉 = >3L<8) +<)OA<,C&<,AB ^8 ,UhAB+<"4
Σj |j〉Q@b/B !&>=+'5 /1AB,C+ 8+':7!8<&<,AB
=:3I);+<&(=8D)+'AB8/1 8+6(8+' C,G & f A B,C&('AB !&*,C <,[= .V?q &&* 5'Z /1j &<+'∑
i,j,k
|0〉|0〉|0p〉|i〉|k〉|i, k〉 ⊗ |0〉|1〉|0p〉|j〉|k〉|j, k〉e[AB+''7!8[? ,CY+'AB,C/1 9&E 8+G798<&<,AB K=8k)+'AB8/1 8+6@,C3L)3B5 9&E 9&<+'4 8+'%= 8M,H "4- &G&<+'AB,<,H "4+' 0,<&<+'|5798,[=AB %V?q &&
∑
i,j,k
|0〉|0〉|ϕs〉|0p〉|i〉|k〉|i, k〉 ⊗ |0〉|1〉|ϕt〉|0p〉|j〉|k〉|j, k〉(8+' C,D & f A &#))C,7!8 9&L5'Z;B5 8 8 X&<+ UuAB+<"#&<,AB \= B="#+'= <8+- 8+-)+' "-, +798J,C&6)8,G))C,7!8 9&
C8M`+' 0,'&<+'= 8MO&<+'AB, &798;&<+'@ k5AB !&<+'AB3 9&%<8+:)+' "-, +
+' 0,<&<+'Q v 5 ,t=AB GV?q &&∑
i,j,k
(
|0〉|0〉|ϕs〉|0p〉|i〉|k〉 + |1〉|0〉|α0s〉|i〉|k〉
)
|i, k〉
⊗(
|0〉|1〉|ϕt〉|0p〉|j〉|k〉 + |1〉|1〉|β0t 〉|j〉|k〉
)
|j, k〉Y8,6b p5AB !&<+'AB3 9&4<8+'AB d)+' "-, +I798J,C&6g*8+' C,P5AB), >G'AB K<,M,H "44+' 0,'&<+'P<8+'AB 7!8;&<+<,H "44<,Y547!8J,C&G<&G= V?q && |0〉 [ &G5'Z; 01#+' 0,<&<+'&<+'AB,% &%798;&<+'46/15G+' 0,<&<+'5 ,C 7k &E',Mk',T5#7!8J,C&E'&E= GV?q && |1〉 Q f A &GUV,C&E=#"4j "4Qbg,C ; "4 !&6,C.))C,798 !&*zE AB8/168$8 :&<+ UuAB+<"#&<,AB $= B="#+'=F<8+f 8+)+' "-, +.798J,C&6QRS'AB 9&"#,C 9&' ; !&>= V?q &&
∑
i,j,k
(
|0〉[
|0〉|ϕs〉|k〉|i〉|k〉 + |0〉|i〉|k〉|α0s〉]
+ |1〉[
|0〉|ϕs〉|k〉|i〉|k〉 − |0〉|i〉|k〉|α0s〉])
|i, k〉
⊗(
|0〉[
|1〉|ϕt〉|k〉|j〉|k〉 + |1〉|j〉|k〉|β0t 〉]
+ |1〉[
|1〉|ϕt〉|k〉|j〉|k〉 − |1〉|j〉|k〉|β0t 〉])
|j, k〉RT"4<8+' !&f_ ; "4 9& 8+f)+' "-, +798J,C&f= 23%J;B*<& =+'=X &f 9/1AB, !&.(+''8C&&8/1 +<,_; 8+6Qet-=,'&<+<,CJ8&<,AB `=:)+'ABJ;J,CC,C&' AB,C !&'G<8+ 8+'.+' )OAB '/1 +<,_;
PG(0, 0|s, t) =∑
i,j,k
(
1 + 〈ϕs|i〉〈αis|k〉
)(
1 + 〈ϕt|j〉〈βjt |k〉
)
PG(1, 1|s, t) =∑
i,j,k
(
1 − 〈ϕs|i〉〈αis|k〉
)(
1 − 〈ϕt|j〉〈βjt |k〉
)
Y8,'7!8:)AB8+(&'AB8i, j
∑
k
〈αis|k〉〈βj
t |k〉 = 〈αis|βj
t 〉
w !
$'AB"-"# 9&PG(0|s, t) = PG(0, 0|s, t) + PG(1, 1|s, t) AB `ABJ&<, 9&6;5AB"-"4%AB $:/1AB83,C&6
PG(0|s, t) − PG(1|s, t) = PG′(0|s, t) − PG′(1|s, t)
<VG ⊕ - 7JH*:2% "8:2; 4 7n0(5%W:2;L"8:2&;N(K/
>AB85AB 9&<,C 98AB zL01 +C,' +<<&'H "4=I)+' 8/1G7!8; 9&<,7!8G5AB <,= +'6Q 18'798[?rz)+'' 9&6sL/1 +<,_; 8+4 !/1A69,C&48 D7!8<&<,AB [s5 3B'<,7!8XAB8 7!8; 9&<,7!8Y5'ZAB,',X);+<"-,8 ' "%JF)+'= &' +<"-,C 2zd5'Z;B5 8 =P= 8MW)+'AB8/1 8+'6"#,P5P798<&<,AB P5AB+<+'<)OAB =, 9&&'AB8 AB8+'$zp=D &&'X)8+'6Qf>AB8D5AB <,= +'AB X"#,C 9&' ; !&F56BDAD798'&<,AB X)A') 8/1 !&G5AB+<+'<)OAB =+'4zP=:"#&<+<,5= ',C&'@5?q<&Sz=,C+'4zL= &&'798,s+'<&' !&G 5'Z /1jN&<+'>6/15V?q<);B5%)+<,C/1=8F/1 +<,_; 8+6O5798,b) +<"4 &<&<+X4)+<,AB+<,s8^/1 +<,_; 8+>=@?r6/1AB,C+:8 `)C80+ =$5AB 9&<+ BE<8+)+'AB8/1 8+'6QAB+<"4 C "4 9&6
Dt?l ⊕ '* r ∗ ( h<*R$B"Q$! $9 !$NN(
S5T5
': . s ∈ S
t ∈ T5`$9(8*hhQ$9N) B"
ρs
γt5
?$/NhS WSB $!F'L#\*#PhB"(π S × T
5 3WSB V
$9S × T
$*B 0, 1 _* GBWSB $9%p*/ *WSBC /VN;
c, s : N → [0, 1]$9 ! GBWSB)I(8= s(n) < c(n)
': -. nC
NI*L3<*4,6 ⊕ ') r ∗ \BWuNh\& )N>$9(m<*B* 9(L
PQ= r; #. x ∈ L
5;T+!n Rl Gx$ WX,' ⊕ ') r ∗ 5.,*/ <*#j7':*B W [$! x
NNK'-^':PXBB>S*/tN |x| 5`$9 !1'+ B ( A
B') %i l6 $;<*%'+#+*#3d B">$! ' (-( ^* 7 )
c(|x|) C r; #. x /∈ L
5;T+!n Rl Gx$ WX,' ⊕ ') r ∗ 5.,*/ <*#j7':*B W [$! x
NNK'-9':/dXBB>W*YN |x| 5= ><*Q'+#+*#d B"$9? ' ($9a& 9')BN>= \ '[$9'+ , (A,B)
jL`l6 3 h M"Nu '?j s(|x|) CY X01 +V8 >"#&<+<,5>= ',C&'
ρs'(=5AB"-)OA':
ρs =∑
ipi|Ψi〉〈Ψi|
&2,C' "EJ +,C&=AB 5#7!8[? 8 < 8 ⊕ x>^R ∗ '-+'=8,Lz$8 < 8 ⊕ 7!^R ∗ QtY dUu,C&6b55 , [?q<&E);B%P56B6t56+: AB"%J+'E=@?q &&' |Ψi〉
));+,' 9&*= 3-=5AB"-)OA<,C&<,AB F5 ,='<8.) 8&.j &<+'GNM)OAB 9&<, `E AB"%J+'I=798J,C&'| &:=AB 5% AB 565 83JI);+>%/1 +<,_; 8+6QOe[E+'5AB8+'Gz#=*"#&<+<,5= <,C&') +<"4 &G=AB 5-=@? 8 -5 +<&,C 4"# ,H +'48^/1 +<,_; 8+E=E)OA' +:8 k AB"%J+'#NM)OAB 9&<, =798'&<,AB (zI3EUuAB,6 &(=AB 5G=@?qj &<+'JO685AB8)$)C8f_ `= .'AB ` ;C'Q
e[%09,C F=G)8,'i 5%ABJ&' 98^<&(=, #5 ,CIz- /C8 +6Q;>AB8.)OAB8/1AB ."4AB 9&<+' +:7!8G3I)8, 5=8E/1 +<,_; 8+g= b5&)O=)+'AB&'A5AB<&g8%"4AB,C g5 C=@? 8 %/1 +<,_; 8+ 6 +'A A =01Q
# ^ c,s ⊂ ⊕ ') r ∗
1−2c(1−c),(1+s)/2
': G. ^':*B+*B h<Q$9G\9'<" $9- $9\BN<'WW
c
s/"N( `*B
(1 + s)/2 < 1 − 2c(1 − c)C
"N8B)Nh+*WSBCRSO<8 4&2=>"4AB !&<+' +.798*>)+'ABJH "4xlI5AB"-) &)OAB8+xl =@?r)+'Hw Q oQ > <&2= 235 3B'' ⊕ x>^R ∗ QlAB, 9&.=AB 5:= 8MX5 ,C+'5 8,C&'27!8; 9&<,7!8
Q0 &
Q1=("4j "4
&)(n, k)
&' .7!8'AB,C&
‖(Q0 − Q1)|0〉‖tr ≥ c
w
'AB,C& ‖(Q0 − Q1)|0〉‖tr ≤ s:N_ ,''AB 8 < 8 ⊕ x>^R ∗ '8+
S = T = 0, 1 );+
∀(s, t) ∈ S × T π(s, t) =1
4
&V (s, t) = s 6= t
e[X798'&<,AB L5AB+<+'')AB = !&'D'AB 9&ρs = Qs|0〉
&γt = Qt|0〉
Qf L`56BXA ‖(Q0 −Q1)|0〉‖tr ≥ c
34<&<+&' 0,=()+'AB8/1 8+':5AB <,'& 9&z4@5 &<8 +*8 %"4<8+'%)O +<"4 &<& 9&==,<&<,C 08 +
Q0|0〉=
Q1|0〉=."# ,H +'(AB)&<,C"# &Tz:+' !/1A61 +f+'<8C&&2=.5 &<&'."4'8+'*8
)+'AB8/1 8+)+'A=8,C&3>JAB f+' )OAB '.6/15)+'ABJ;J,CC,C&'c2 +(1−c)2 = 1−2c(1−c)
Q Y56BA ‖(Q0−Q1)|0〉‖tr ≤ s
s)+'AB8/1 8+'Y 2) 8/1 !&=AB +Y3(JOAB 2+' )AB f6/15f)+'ABJ;J,CC,C&')C8G0+ =L798
(1 + s)/2b56+%,C% 4)O 8/1 9&%+' )OAB =+'X/154)+'ABJ;J,CC,C&'P)C8E0+ =#7!8
s8M 7!8<&<,AB
(Q0|0〉, Q1|0〉) &
(Q1|0〉, Q0|0〉)T<,C AB ,C#)AB8+<+, !&Xj &<+'$8&<,CC,'#)OAB8+
=,<&<,C 08 +Q0|0〉
=Q1|0〉
/15G)+'ABJ;J,CC,C&'%<8)O +<, 8+'%zs5G798,['&,C"-)A'',CJQ
< + - ∗
>AB82 &<8=,AB .zE)+'' !&ff)+'AB&'A!5AB=*)+' 8/1,C 9&' +B5 &<,C/1(798; !&<,798.z= 8M#)+'AB8/1 8+'f);+<&016 9&(=>V?q 5Z /1j &<+' "4 9&(f)C8201 +8MD798,@'AB, !&6J &<,<8+f*"4j "4>)+<,C 5 ,C)(7983%5 3B''*x>R h5NUO3);+<&<,.oQ o NQ9at+'H)O 8P=(5'ZA'2'AB 9&25AB 982<8+T5 &<&'*5 3B''. 01 + ' 8b+''8C&&>5AB '7!8 9&<&*%&<ZAB+'H "4o w 5 ,=''AB86Q>AB8>5AB"-"4 AB :);+ $=AB +.8 =N_ ,C&<,AB $UuAB+<"4 C)8, AB8.)+'AB8/1AB (7!8:3I5 3B''%x>^R ∗ AB+'798[?q C<&+'<&<+' ,C 9&'EzI8 D&'AB8+(' 8 "4 !&6;5AB 9&<, !&x>R .Q
W #6 etD=N_ ,C&<,AB =-3D5 3B'4=45AB"-)NM,C&'Lx>^R ∗ <&GV?qNM&' <,AB ;&<8+' CL=-3D=N_ ,
&<,AB k=-x>R 8k56B=)C8<, 8+':)+'AB8/1 8+'6Q@>AB8:3P= &,CCAB %5 ,=''AB86O +' )+' ; 9&G3&' +<"-,C ABAB0,G,C !&<+'A!=8,C&'%)AB8+*x>R (Q
lAB,C&kf AB"EJ+'=f)+'AB8/1 8+'6Q9l!8+s8 9&<+'
x ∈ Σ∗ =f&,CCn = |x| BV?q<);B5f&'AB&<8+
798 g0,C+-G<<&'H "4G=%)+' 8/1G,C !&' +B5 &<,C/1E' +45AB <&<,C&<8E=q(n) = qV (n) + k (qM (n) +
qP (n))798J,C&'6Q|e[
qV (n))+' "-, +'g798J,C&'b'AB 9&bt7!8J,C&'[)+<,C/1g=8%/1 +<,_; 8+6AB+'g7!85'Z;B7!8
+' 0,<&<+'=qM (n) + qP (n)
798J,C&'5AB+<+'')AB =$zV?q<);B5=f"401. &.zV?q<);B5:)+<,C/1=5'Z;B5 8 F=
k)+'AB8/1 8+'6Qe[()+'AB8/1 8+'. G'AB !&.);B(8&'AB+<,'>zI5AB"-"%8 ,7!8 +* !&<+'% 8M@Q
& !&>=AB >=.UuAB 5 &<,AB :z45 +'AB,' 5%)ABC AB"-,3m, qV , qM : N → N
8 K4"N 2 jm
#((3* 9(qV , qM )
iS<( h<3 ;E<&%8 #UuAB 5 &<,AB V : Σ∗ → Σ∗ 565 83JL K&' "-)E)A
C! AB"-,32 K3$AB 08 8+=P'AB K !&<+'QbAB8+I5'Z;B7!8x ∈ Σ∗ V (x)
<&E,C 9&' +<)+' &'X5AB"-"48 bm(n)/2 + 1c V&<8) (V1(x), . . . , Vbm(n)/2+1c(x))
Q bAB8+>5'Z;B7!8iVi(x)
<&34='5 +<,C)&<,AB =@? 8 5 ,C+'5 8,C&%7!8; 9&<,7!840,' 9&E<8+
qV (n) + kqM (n)7!8J,C&'6QOb,C ; "4 !&6[I)+' "-, +%7!8J,C&
=:V?q<);B5)+<,C/1G=8F/1 +<,_; 8+(<&(=<,C0 5AB"-"4E & 9&.%798J,C&.=GAB+<&<,Q & !&#=AB 4=UuAB 5 &<,AB #z^5 +'AB,' 5X)OABC! AB"-,3
m, qM : N → Ns &8 LUuAB 5N
&<,AB qP : N → N
@8 '+ jm
4,3* 9((qM , qP )
i'(Nh' ;Pi
)AB8+i = 1, . . . , k
<&=N_ ,=P"# ,H +'$ ;AB08 s5?q<&I8 PUuAB 5 &<,AB
Pi : Σ∗ → Σ∗ g'8+E3B7!8 CDAB [? ,C"-)A'85 8 L+'<&<+<,5 &<,AB =P565 83J,CC,C&'Q bAB8+-5'Z;B7!8
x ∈ Σ∗ Pi(x)<&,C 9&' +<)+' &'D5AB"-"4X8
w m
bm(n)/2+1/2c V&<8) (Pi,1(x), . . . , Pi,bm(n)/2+1/2c(x))Q gAB8+5Z;B798
jPi,j(x)
<&s3>='5 +<,C)&<,AB `=@? 8 5 ,C+'5 8,C&>7!8; 9&<,7!8I0,' 9&:<8+
qP (n) + qM(n)7!8J,C&'6Q;et
qP (n))+' "-, +':7!8J,C&'
'AB 9&2,C 9&' +<)+' &'(5AB"-"4(V?q<);B5>)+'AB)+':=8L/1 +<,_; 8+6AB+'f7!8(qM(n)
8&<+'f5AB <&<,C&<8 !&zG AB8/168XV?q<);B5:=2"4'01|Q!:*)C86!',
qe : N → N<&T8 *UuAB 5 &<,AB P&' C:7!8
qe ≤ qP
5'Z;B7!8Pi
) 8&6/1AB,C+8-)C8qe(n)
7!8J,C&Y);+<"-,'s7!8J,C&'s)+<,C/1Y798,'AB !& 5'Z /1j &<+'696/B !&798.)+'AB&'A!5AB( (5AB"-"4 5/15*7!8J,C&')+<,C/1=28&<+')+'AB8/1 8+'|Q9* 4&' ;)+'AB8/1 8+<&.=,C&
qeS)+<,AB+<, 5'Z /1j &<+'Q
* Xp N#8$9%'' B8 ;N+*1W 7= *;WW= [j m#(,G* 9(#
k'L ( (qV , qM , qP )
+'<&<+' ,C 9&G'&G5AB 9&<,C&<84=@? 8 k/1 +<,_; 8+
Vz
m"401
(qV , qM )V+'<&<+' ,C !&% &G=
k)+'AB8/1 8+'
zm
"401(qM , qP )
V+'<&<+' ,C 9&'P1, . . . , Pk
Q* F&' s<'&'H "4%<&:=,C&qeS4)+<,AB+<, 5Z /1j &<+'
<,*5?q'&#^56B#)OAB8+PPi
Q =N_ ,C&L8 5 ,C+'5 8,C&(V, P1, . . . , Pk)(x)
0,' !&X'8+q(n) =
qV (n) + k (qM (n) + qP (n))7!8J,C&'.=3EUVBAB F<8,C/ 9&'Ql!,
m(n)<&,C"-);,C+6(5 ,C+'5 8,C&'
P1,1(x), . . . , Pk,1(x)V1(x), . . . , P1,(m(|x|)+1)/2(x), . . . , Pk,(m(|x|)+1)/2(x)V(m(|x|)+1)/2(x)'AB 9&f))C,798TT8 Tz3%<8,C&'(=f8&<+'6!'AB,C&28M
qV (n)+ qM(n)798J,C&'2=8#/1 +<,_; 8+6'AB,C&
8MqP (n) + qM(n)
=%5'Z;B7!8G)+'AB8/1 8+6Ql!,m(n)
'&);,C+6;5?q<&.G/1 +<,_; 8+*7!8,t))C,798%)+' "-, +(5 ,C+'5 8,C&AB $NM5 8&'3-798 5
V1(x), P1,1(x), . . . , Pk,1(x), . . . , P1,m(|x|)/2(x), . . . , P1,m(|x|)/2(x), Vm(|x|)/2+1)(x)
g,C ; "4 9&6b)AB8+E8 # !&<+'xt3X)+'ABJ;J,CC,C&'P7!84#5AB8)
(V, P1, . . . , Pk)B55 )&'
x<&G=N_ ,45AB"-"4P & 9&G3X)+'ABJ;J,CC,C&'P=@?qABJ&' ,C+ |1〉 AB+'G=@? 8 -"4'8+'4=8d7!8J,C&G=4'AB+<&<,=8$/1 +<,_; 8+*= (3IJ;B'56 AB ,7!8
(|0〉, |1〉) @)+'H(NM5 8&<,AB `=8`5 ,C+'5 8,C&(V, P1, . . . , Pk)(x)<8+V?q &&.,C ,C&<,3 |0〉⊗q(n) Q
>AB8)AB8/1AB *zI)+'' 9&*=N_ ,C+.3-5 3B''=5AB"-)NM,C&'x>^R ∗
T/VN;k,m, qe : N → N
c, s : N → [0, 1]
$9( GBWSB)Cc= s(n) < c(n)
': `. nC 21 ') r
c,s(k,m, qe)<* 3_*/, $!( _** 9
LP = & +!/N>$9 GBWSB)
qV , qM : N → NjD Ln(3*Bi9':/dXBB>W*C- 4"N f V
jm
#((i3* !(
(qV , qM )i'( h< ;f = (5 ': %. ;h+"
x$9E.*/
n5
-x ∈ L
5+!/N GBWSB qP : N → N = qP ≥ qe
$9('+ i ( P1, . . . , Pkj
m#(,G* 9(
(qM , qP )iS<( h<3 ;
qeiL*'(SBi<NN h\"(3o =
(V, P1, . . . , Pk)(x)*1'2'8*BB<9'+#+*#d B">* 7 )
c(n)C
-x /∈ L
5Y': `. GBWSB qP : N → N = qP ≥ qe
. 7'L (P1, . . . , Pk
jm
#((3* 9(qM , qP )
i'(Nh' ;qeik* 'SB(i NNB h+",5
(V, P1, . . . , Pk)(x)*1B'*B<9'+#+*]dhB">* ' s(n)C
fkN+* $9K' ') r ∗(m)\& SB $9( '* r ∗
1,1/2(k,m, qe) . (?C( 3WSB)
k,mjX Ln(3*BiY':/dXB>W*/ T. (1( GBWSB) qe
Ce[+''8C&&.)+<,C 5 ,C);t=4~q w b<&<8,C/ 9&
# NQ& SB $!( ') r ∗1,1/2(k,m, qe)
#. ([C( 3BWW) k,m, qej +/ni
3*B;':/dXB>W*/I(Nh 3 ?$*) +$,%r9C
o9
v ?q<&Sz=,C+'E798<,@()+'AB8/1 8+'.);+<&01 !&:8$)C8.8 G798; !&<,C&'G)OABC! AB"-,3%=@?q 5'ZN/1j &<+' "4 9&.:)+<,AB+<,VAB+'3G)8,' 5*=8P<<&'H "4(=()+' 8/1(,C !&' +B5 &<,C/1> (= );B''();BT5 C=*>Y bQe? ,=:=>3G)+' 8/1><&f=(+' "#+'7!8 +Gh &2=*)+'AB8/1 + 27!86/15*5 &<&'>+'<&<+<,5 &<,AB <8+:V?q 5'Z /1j &<+' "4 !&6bAB k)O 8&<8))A +:7!83L&,CC-=V?q<);B5I)+'AB)+'4=:)+'AB8/1 8+'G<&)ABC AB"-,3 tAB "4AB 9&<+'L798[? ,C2<8 4&%7!8[? ,CE, !&B'' P=4)3B54)OAB8+G"# ,C)8 +I=G)8+<,_56&<,AB *=("4'01*798 8+> 9/1AB,EG/1 +<,_; 8+6Qe[(&<+ <UhAB+<"#&<,AB *8 ,C&,C+'>))C,7!8);+)+'AB8/1 8+'2AB 9&2AB+'2=.&,CC*NM)OAB 9&<, C &TAB P) 8&T = /,C +( &2565 8 +T 8&)+'ABJ;J,CC,C&'%=@?qj &<+'%B55 )&'* $&' "-).NM)AB !&<, t AB = &' +<"-,C ,<&'Q
nT : ] >AB82CAB f8&<,CC,' +2*)+'AB&'A!5ABGx>^* )OAB8+.x>R =AB 8P&<ZAB+'H "4 > ]%)OAB8+T)+'AB8/1 +
79843D5 3B''Px>^R ∗(2)5AB 9&<, !&4x>R .Q v &<&'4,C 5 C8<,AB p<&G)+''7!8-,C"-"4=,3&'s5 ) = !&
AB82+' "#+'798AB f7!8(*)+'AB&'A!5AB:=5 +<,C&5 ,=''AB8f<&2&' O798*(/1 +<,_; 8+f8&<,CC,'*(Uu,C&f798#)+'AB8/1 8+'L);+<&01 !&X=FV?q 5'Z /1j &<+' "4 !&$)AB8+Xj &<+'$);+<&<,5 8C,H +' "4 !&FNM,C016 9&X 9/1 +' 8M 9= 8M#)+'AB8/1 8+'7!8,; *);+<&01 9&f);B2=@?q 5'Z /1j &<+' "4 !&( .)OAB8+<+, 9&);BT+' 8'<,C+fz%5)+'AB&'A!5AB J, 4+6B<,s)+'AB8/1 8+'Y 2);+<&01 9&);BY=@?q 5'Z /1j &<+' "4 9&6AB -ABJ&<, 9&Y3>5 3B''* .;798,t5AB !&<, 9&Gx>R ⊂ Y (;=AB 58 F)+'AB&'A5AB%NM,<&'>"#,*5G [?q<&.);B*5 C8,t=5 +<,C&,5 , NQ v 5 ,5AB !&<+B<&':/15.3)C8);+<&T=)+'AB&'A5ABTZ;J,C&<8 69= T'7!8 YV?q 5'Z /1j &<+' "4 !&<&)C8& B&.8 :+''AB8+'5G798,@)O +<"4 &*8MX)+'AB8/1 8+'(=&<+<,5Z + AB8.AB"-"4(,C <,[,5 ,[= 8 $56B(&),7!8Q
# r ⊂ ') r ∗(2)
" 7)Nh+*WSBCglAB,C& (V1, V2)8 /1 +<,_; 8+%)OAB8+8 3 0901
L ∈ x>R (Q@>AB8%5AB ',= +'AB G/1 +<,_; 8+
V<8,C/B !&.)OAB8+*x>`R ∗(2)
o Q 1 &' +(8 ),H5%8$Z;B+'=@ &* 9/1A61 +*+'<8C&&*8D)+'AB8/1 8+
2Q
> Q *5 /1AB,C+F=$+' 0,<&<+' 1
=3 );+<&F=8 )+' "-, +`)+'AB8/1 8+6* & 2
=k3 );+<&`=8= 8M,H "4Q.l!,(3d),H5`=`V?q &)`)+'5= 9&'k &,C&L&'AB"%JO<8+#),C))C,7!8 +
V2z
(
1,
2) &>B55 )&' +*<,[ &( 8 "4 9&*',@+'<8C&&(=@? 8 "4'8+'%=8D)+' "-, +*7!8J,C&(=
1=AB |1〉 Ql!,@3I),H5G &,C&(&'AB"%JOG<8+fUVB5;))C,798 +
V ∗1
z(
1,
2) &*B55 )&' +
<,[ &. 8 "4 9&*',@&'AB8(7!8J,C&'.= 1'AB 9&(= (V?q && |0〉 Q
et:5AB+<+'5 &<,AB 4=5f)+'AB&'A5AB=5AB8f)+''798f,C"-"4=,3&' "4 !&T=5 C=8I)+'AB&'A5AB*x>`>=5 +<,C&*8D&<ZAB+'H "4 > ]Q;*AB8.=AB AB *798 7!8,C =,56&<,AB *<8)) "4 9&,C+':5 ,=''AB8|Q
v AB"-"4 AB G);+)+'AB8/1 +G7!8@6? ,CYNM,<&'I8 )+'AB8/1 8+%'UV, 9&%B55 )&' +);+(V1, V2)/152)+'ABJ;J,CC,C&'8I"4AB,C
cBAB+'b,CNM,<&'T= 8ME)+'AB8/1 8+'
P1 &
P2'UV, !&sB55 )&' +Y);+
V/15)+'ABJ;J,CC,C&'cQ9e[)+'AB8/1 8+
P1)OA''H=f8 48 ,7!8+' 0,<&<+'
1 &
P2)OA''H=f8 -+' 0,<&<+'
2,C ',[798[? 8 $+' 0,'&<+')+<,C/1
2Qb/ 9&.7!8:)+'AB&'A!5ABG G5AB"-"4 5.= 8MX)+'AB8/1 8+'
,C ,C&<,3C,' 9& 8+'f+' 0,'&<+'(
1,
2)= TV?q &&. 5Z /1j &<+'
V1|0〉6/15:*+' 0,'&<+'
15AB+<+'
)AB = !&I8p+' 0,<&<+' =8p)+'AB&'A!5ABDx>R (t &4+' 01',C&<+' 2
5AB+<+'')AB = !&-8p+' 0,<&<+' #QY LUhAB 5 &<,AB $=8DJ,C&7!8[? ,C@+'AB,C&6:= 8M,H "4)+'AB8/1 8+.@5 &<8:3E"4j "4&<+ UuAB+<"#&<,AB $7985 C-798[? ,C8+,C&G@5 &<84= I)+'AB&'A5AB#x>^*pQ@e[I)+' "-, +G)+'AB8/1 8+G UV,C&:+<, [Q[RS<&(5 3,C+>79834)+'ABJ;J,CC,C&'E=%09,C `=*= 8M$)+'AB8/1 8+'*<&*3I"4j "4798G5 CE=8F)+'AB8/1 8+=8D)+'AB&'A!5ABx>`>K CNV"4j "4E 09%z-5 CG=8D)+'AB8/1 8+*)AB8+G)+'AB&'A!5ABEx>R .Q
o o
(5 ,C)+'A7!8 "4 9&6O= 8MF)+'AB8/1 8+':)OAB8+>E)+'AB&'A5AB-x>^R ∗(2) E) 8/1 !&&<+<,5'Z +:)C8
798[? 8 P)+'AB8/1 8+T)OAB8+()+'AB&'A!5ABx>R .Q9Y LO &6P1
.+'AB,C&T);BT=*798<&<,AB [! &V?q &&f798[? ,C 9/1AB,8L/1 +<,_; 8+f'&2=AB 5*,C = ) = !&.=(3E7!8<&<,AB L7!8
P2G+' 8Q:("4j "4>7!8*=
>)+'AB&'A!5AB%x>^*p55 ,[' +<&(zE09+ 9&<,C+(7!8>:+' 0,<&<+' 1 =
= )O =X);B=8X+'<8C&&=8D&<,C+01%6&'AB,C+'%@5 &<8);+.:/1 +<,_; 8+6Q
X''
e[AB+'2=(5*<&01. AB8f/1AB 2 &<8=,(V?qNM)+''<,C/,C&'(=*=,O +' !&'"4A=H 2=.)+' 8/12,C 9&' +B5 &<,C/1G7!8; 9&<,7!8%zP)C8<, 8+')+'AB8/1 8+'6Q[>AB8G/1AB %=N_ ,Y)C8<, 8+'G5 3B''=-'<&'H "4=L)+' 8/1I,C 9&' +<"4=,3,C+'L 9&<+'D5 C8,=~ v2 af o. &-5 C8,.=^~q w s &#/1AB - &<8=,V?qO &.=: A."4A!=,_;56&<,AB 6Q;*AB8(6/1AB . 09 "4 9&> &JC,[8 5AB NM,AB F<8+<)+' ; 9&'E 9&<+'C01AB+<,C&<Z"4(=@?r))+'A6M,C"#&<,AB ^ &2)+' 8/1(,C 9&' +B5 &<,C/1*798; !&<,798(zI= 8MX)+'AB8/1 8+'6 & AB8/1AB * &<8=,G5 &<&'G5AB NM,AB F= .56B=+'G=8F)+'ABJH "4%^( v >aQ
etX)+<,C 5 ,C);P798'&<,AB AB8/1 +<&'4C,#8Mk<<&'H "4G=I)+' 8/1%7!8; 9&<,7!8Ez$= 8M^)+'AB8/1 8+'<&Y356+B5 &' +<,i&<,AB P=3:5 3B'=.5AB"-)NM,C&'(`R ∗ = Y3B7!8 C&'AB85'Z; 01'AB 9&T5 3B'<,7!86B"#,Y)+'AB8/1 8+'T=,<)OA' 9&=@?qAB+'=,C ;&' 8+'27!8; 9&<,7!8T &Y);+<&01 !&2=V?q 5'Z /1j &<+' "4 !&4)+<,AB+<,VQ ` G,C&( ,b<,[V?q 5'Z /1j &<+' "4 !&) +<"4 &:8MD)+'AB8/1 8+'>=G&<+<,5'Z +6=,C"-,C 98; !&GV?qNM)+'<,C/!,C&'=I5 &<&'I5 3B'' =''AB8:=I*Y . ,Y6? ,Cb 8+:)O +<"4 &=%)+'AB8/1 +%V?r));+<&' ; 5$z`=%3 0901)C85AB"-)C,798|b3$)3B6 9&48='<8I=P*Y (t ,T<,<,C"-) "4 9&>`R ∗ =
>Y *5G7!8,[<&) 8&j &<+'G+'<8C&&.:)C8)+'ABJ;JQ 3G_ L=:'><&01! AB8./1AB &<8=,>8 >798<&<,AB X)+'A5'Z: L+' B5'Z; 9&*5 &<&'>5 3B'*=
5AB"-)NM,C&'z6/1AB,C+ 1<&5.7!8*x>^R ∗ 5AB 9&<, !&2>Y *e[+'<8C&&'T=@? ,C ;))+'AM,C"#J,CC,C&''7!8,''t= b3);+<&<, w Q > AB8g=AB 9&s8 EJAB )AB,C !&b=2= );+<&6 &b AB8g6/1AB s5AB <&<+<8,C&b8 )+'AB&'A!5ABI=)+' 8/1I,C 9&' +B5 &<,C/1-7!8; 9&<,7!8I)OAB8+>)+'ABJH "4 w : 7!8,Y' "%J)+'AB"4 &<&' 8+6Qv ) = !&6=45AB !&<+,C 9&'X=D&' "-)- AB8PAB 9&# "-)Oj5'Z`=D&' +<"-,C +43)+' 8/1`/ 9+' "-,'.=5"4 "4AB,C+'Q v 5 ,;5AB <&<,C&<8(3>)+<,C 5 ,C);(),<&'.zGNM)AB+' +61 &)+<,C 5 ,C);;)+'ABJH "4AB8/1 +<&*zI5AB <,= +' +6<8,C&'Gz-5<&01Q
o >
~q>+] l5AB&<& :+'AB 'AB [Q x*8; !&<8" 5AB"-)8&<,C 0L)A<&' 5 &<,AB [X = )+'ABJ;J,CC,<&<,5)ABC AB"-,3V&<,C"4Q ^<h+Bu g/<M= gBK' .*WSB*/ gBK'+!/hWX0mg;g;gABOu w N > ]Q
~q*et + m > l /*+'AB+ v +''&' pe@8 =@ * < /k`AB& 2 ,VsB=Z98 l!8= [g =p^+<,Al 6 01=1Q +'A!AU%/1 +<,_;56&<,AB = Z;+'= 'XAU))+'AM,C"#&<,AB )+'ABJ "46Q R r2L|C1L$ " 1g);01 o o >w o mm > Q
~q:lm > l /p(+'AB+ = l!Z"%8 >lUh+Q Y+'ABJ;J,CC,'&<,5D5'Z5,C 0dAU()+'AAUu[YF 5'Z;+B5 &' +<, |&<,AB kAUg .QRS r2+6C1L$ " 1g);01 > o w o mm > Q
~J;] eB *J;,VQ%at+B=,C 0D0+'AB8)k&<ZAB+<FUuAB+*+ =AB"- 6QERS r2L|C 1gs);01.o >Io o > m o m]Q
~fgebm o eB J;,V[et 5PAB+<&< A Gg = v +'<&' de[8 =@Q#>AB = &' +<"-,C ,<&<,5#NM)A 9&<,3&<,C"4Z;Bt& TAV)+'A|/1 +Y,C 9&' +B5 &<,C/12)+'AB&'A5AB6QgBK' .*WSB*/agBK'+!/hWXBo w o9 o mm o Q
~ :F,5Z;B f +6gl!Z;_ AB=fB'' +6 9A!-:,CC,3 [g =kt/,Y ,C01= +'AB [Q4F8C&<,)+'A6/1 +D,C 9&' +B5 &<,C/1)+'AAUu A &'Ad+' "4A|/1,C 9&<+B5 &J,CC,C&S B<8"-)&<,AB 6Q R r2L|C [ 1gs);01 oo w o wIo o mQ
~ v2 l : m 1ABZ dfQ v 38' +6[`,5'Z;B Q AB+< @(J +l!Z,C"4AB !1b = (,5'Z;+'=Q ABC&6QY+'AB)A=#NM) +<,C"4 !&T&'A&''&A56Z,== V/B+<,3J>&<ZAB+<,6QIrT]XpMC . 4C-Nb uBC>w o ]1 r o 5 & o m mQ
~ v2 af oB .,5'Z;+'= v /1 b &' + 1 +6$f <"-,C abAB +6 = 1ABZ Wd&<+'AB86Q v AB 'N798 5. =XC,C"-,C&'AUt AB A56[<&<+&' 0,6QsRS % g <NO gBK' .*/iWWB*/hgBK'+!/hWXB);01 >w > o9m > oQ
~qsetm > (+<, Y ,C014 =etB ADetA6/B 1QGa 2AV)+'A6/1 +IAB NV+'AB8 =)+'A!AUf<<&' "4 ;&<Z ,C+)A T +G =F&<Z ,C+>)+'ABJ "46Q*R r2L|C 1gs);01! ww !oo* ncAB+ .c-:l o mm > Q v Y+''6Q
~ !Bm F,5Z;B %Q +' =/,=lQ 1ABZ 'AB [Q W$9.W +BX Kr^i\BK' N((NQ Q +' "# ^ =F5AB"-); 91 o m !BmQ
~ *] l!Z;_ AB=2B' +6bl!,CC/!,AX`,56C,V[ = v Z;+<*B51A sQafZ A =0145AB"I)NM,C&SpAU>,C 9&' +B5 &<,C/1$)+'AAU <'&' "46QRS rfL6pC mgsb);01 > m o w oo m]Q
~ l l!Z;_ AB=2B'' +- = F,5Z;B (l!,C)' +6Q +<,C/B&'X5AB,C /1 +'<8)8JC,5X5AB,C E,C ,C 9&' +B5 &<,C/1E)+'A!AUY<<&' "46QsR R'L6pC 1 R 1gs);01*]m o m Q
~ m] F,5Z GQ A! "# ^ =/,= bQ. ,CCC,3"4'AB [Q R"-)+'A|/1=n))+'AM,C"#&<,AB C01AB+<,C&<Z"4UhAB+E"#M,C"E8" 5 8&- =Ki&<,_J,CC,C&)+'ABJ "48<,C 0' "-,=N_ ,C&')+'AB0+"-"-,C 0Q C>g ' o > ooo ] oo o9] o mm]Q
~ i o 1ABZ; <&B=@QlAB"4AB)&<,C"#,C ;))+'A6M,C"#J,CC,C&S4+'<8C&'6Q C >g ' o9 o! !Bm ]m > o Q~ f w >AB"-,C , 9 ,C 0 s T A5 < [9 =PafZAB"#B f &<Z[QORS= !&<,C&45Z5-,Tx>^.
5AB"-) &' > w Q
o w
~q=oB RSAB+'= ,X +' ,=,$ = (AB ;==FABU<Q sM)OAB 9&<,3:A T +DJOAB8 = UuAB+ > 798 +<kA56CCd=5A=JP5A=%/,3^$7!8; 9&<8" +<08"4 9&6Q COgBK' BC^-X4NBC3.C m w w m] o > > oQ
~q oB l!8JZ;B<Z%:ZAB&6 8>:,C = +665'Z; ; I`A' V6 =.9 ?q:AB CVQ )&<,C"#,C ;))+'AM,C"#J,CC,C&%+''8C&'@UuAB+[`*( v >ad =GAB&<Z + > V/B+<,3J v l *R QrfL6pC " 1gO);01 o o o ]otdB<Z,C 0&'AB [t v [lE > oQtRS v AB"-)8&' +lA!5 , &S1Q
~q^ w ,C+'AB&B=F(ABJ;9B'Z,2 = , ,&'<8"4AB&'AQLx*8; !&<8" "E8C&<,V)+'A6/1 +,C 9&' +B5N&<,C/1I)+'A!AU<<&' "4 ,C&<ZC,C"-,C&'=)+<,AB+% 9& 0 "4 !&6Q CEg9' BC-X4NBC 3.C3 w o > m o9] > w Q
~q$cG w ,C+'AB&B=E(ABJ;69B<Z,V , ,^&''8"4AB&'A =LatAB"4A|!8 ,cT"# B"-,VQsx*8; 9&<8"` +<C,C *+<&<Z98+2)+'A!AU[<<&' "4 1(+'("%8C&<,C)>` +<C,C T"4AB+'.Z C)Uh8;&'A%(+<&<Z!8+ R >gs);01 o m o m > w Q
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