Entanglement in Finite Spin Systems

Transcription

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0.2
0
0
−0.2
3
0
0.5
1
1.5
Magnetic field, B
2
2.5
−0.2
3
0
0.5
1
Q"7EV # X/@!PL:!X
1.5
Magnetic field, B
Q"F?V
1.2
-
2
2.5
3
2
2.5
3
X,@!PL:NX
1.2
1
1
0.8
0.8
10 spin
0.6
0.4
0.6
0.4
0.2
0.2
0
0
−0.2
0
0.5
&F
1
1.5
Magnetic field, B
Q[>EVZR
X,@!PL:NX
2
2.5
3
−0.2
0
0.5
1
&F
1.5
Magnetic field, B
QYV\` 5/C=R
X,@NPL:!X
X $32H:<#' $=N,} lp65#<(7HK5L 7>M'5#4/2/?,'@A'H>V7H<k("' 1%G$,2@A79I9'?>V7H<cI9$3hi'<('5[%GL9%G('@ %($3S'%NCOJ"'wI9';y
I9'2/'='<`4
($,7/y"F'?&Ig$,%B%(';Z#7 4 " }^2\1}-,H, }-,H,H, _RNg("='%('u8;:=<(-H'%Z4/?3?D:9E("'u("=<(''q?&4/%GE%GK=$3=%
7/q("'8`"=4/$3u"=4U-/'xDi''u(<`4/8'I4U)J4ULA)W"'4 84/?,8:?&4
6 ($,2Z("'P'5#<(7HK5LM7/>'5`42H?3'@B'5n.';Q8;'K9J>V7/<("='
Uyq4/=IT$y.%(K$,e8`"=4/$3F%J)W"'<#'x1)]7M%(K$3F%J"=4U-/'xDF''kH'K90N
e :; 6,0m;Q1+ s]
9;5
-;zW1B;Q
0
12
8
3 spin
5 spin
7
Derivative od Entropy of Entanglement, dE/dB
10
8
6
4
2
6
5
4
3
2
1
0
0
−2
0
0.5
1
1.5
Magnetic field, B
2
2.5
−1
3
0
0.5
1
Q68V\` X,@NPL:!X
1.5
Magnetic field, B
2
2.5
Q"<?V" X,@!PL:NX
8
8
9 spin
7
7
6
6
7 spin
5
4
3
2
1
5
4
3
2
1
0
0
−1
−1
−2
3
0
0.5
1
1.5
Magnetic field, B
2
2.5
−2
3
0
0.5
1
Q"7EV # X/@!PL:!X
1.5
Magnetic field, B
Q"F?V
-
2
2.5
3
2
2.5
3
X,@!PL:NX
12
6
10 spin
10
5
4
3
2
1
0
6
4
2
0
−1
−2
8
0
0.5
&F
1
1.5
Magnetic field, B
Q[>EVZR
X,@!PL:NX
2
2.5
−2
0
0.5
1
&F
1.5
Magnetic field, B
QYV\` 5/C=R
X,@NPL:!X
X$32H:<#'[$FN C'<($,-
4
#$3-H'W7>=#"'W'5#<(7HK5LZ7> '5#4/2/?,'@A'5>V7/<#"'W1%($32@A79I9'?9>V7/<6I9$3h '<#'5+%GL9%G('@ %($3S'%N
OJ"'AI9'Ry9I9'2H''<#4($,7/E"='?,Ic$,%P%(';x(7 4 " }M^D\G}/,H, }-,H,W, _RN z#"'%('A8;:<#-/'0%C4?,?D:9x("'E("=<(''Z?&4/%G
%(K$3F%{7Hk("='A8`"F4$,z"=4U-/'EDF''c(<`4/8;'0Iz4U)J4ULu)W"'[4 84?&8;:?&4
6#$32q("'B'5#<(7HK5L7/>+'5#4/2/?,'@A'H0nF'Q8;'K
>V7/<J#"'OUyq4/=IT$y.%(K$,e8`"=4/$3F%J)W"'<#'x1)]7M%(K$3F%J"=4U-/'PDi''H'K90N
I
e
f - z f 0 <q> 4 k >
100
100
4 spin
6 spin
80
Second Derivative of Entropy of Entanglement, dE/dB
80
60
40
20
0
−20
−40
−60
−80
60
40
20
0
−20
−40
−60
0
0.5
1
1.5
Magnetic field, B
Q68V
2
2.5
−80
3
0
0.5
1
1.5
Magnetic field, B
X,@NPL:!X
2
2.5
Q"<?V\U X,@!PL:NX
80
80
7 spin
8 spin
60
60
40
20
0
−20
−40
−60
−80
40
20
0
−20
−40
−60
0
0.5
1
1.5
Magnetic field, B
2
2.5
−80
3
0
0.5
1
1.5
Magnetic field, B
Q"7EV # X/@!PL:!X
Q"F?V
100
,
2
2.5
100
11 spin
80
60
40
20
0
−20
−40
−60
−80
−100
3
X,@!PL:NX
9 spin
80
3
60
40
20
0
−20
−40
−60
−80
0
0.5
1
1.5
Magnetic field, B
Q">EV
-
X/@!PL:!X
2
2.5
3
−100
0
0.5
1
1.5
Magnetic field, B
2
2.5
3
QY$VZRTRZX,@!PL:!X
X $32H:<#'$=N ].'08;7H=IrI9'<#$3-
4($,-/'k7>x("='z'5(<#7/K.Lw7/>'5#4/2/?,'@A'5A>V7/<("'z1%($,2g@A7.I'?l>V7H<qI9$3hi'<('5
%(L.%G('@ %G$,S'0%N5OJ"'{I9';y.I9'2/''<`4
#$37H"F'?&I$&%+%(';+#74D"m}M^D\1}-,H, }/,W,H, _RN
M#"'%('{8:<(-H'%64/?3?iD:9("='
("=<(''W?,4H%1+%(K$,=%67HE#"'l8`"=4/$3A"=4U-H'Di''A(<`4/8'IA4U)4UL)W"='84/4 ?,8:?&4
($,2x6 ("='l'5(<#7/K.L!7/>b'5#4/2/?,'@A'50n
';Q8'K9W>V7H<J("'O
yM4/=IE$
y.%(K$,e8`"F4$,=%)W"'<#'x1)]7M%(K$3F%W"=4U-/'xDi''H'K90N
e :; 6,0m;Q1+ s]
9;5
-;zW1B;Q
I
0.8
0.9
0.75
Entropy of Entanglement top width, ∆B
Magnetic field at entanglement maximum
0.85
1
0.8
0.7
0.6
0.5
0.4
0.7
0.65
0.6
0.55
0.5
0.45
0.3
0.4
0.2
0
5
10
15
Number of spins, n
20
25
Q[8
V C7E8H5/PLCT:C;Y(5/G!> ^ 8H]NPL^ M!^ C;Y(5/G!> >9:5/B,CT@ $
;C Y >4:5$8;: TKL>9^J>4:5&8;:!F 8;:J>]5/B$8;@?CTKO8H5/>9F74M!B/D>\Y[CTB
5/G!>A5,B/>9:NFS
0.35
3
4
5
6
7
Number of spins, n
8
9
10
11
Q"<?V.:5,B/CT@$ C;Y >9:5,8;:TKL>4^J>9:55/CT@ %&PLFN5/G Q5,GN>
PLX F!>&":!>4F < $5,GN> >9:5/B,CT@ $ C;Y 4> :5$8;:TKL> ^J>4:5G!89DPL: F!>474B,>98;X,>4F 5/C R C;Y P ,5 X ^ 8H]P ^ M!^ D;8;KLM!>EVS
%&PLFN5/G
X$32H:<#' =$ N $5OJ"'+DF'"=4U-.$37H<b7>5("'6?,7984($,7/P7>.("='6@M4
Q9$3@E:@t4/=Ix("='(7HKx7/>.("'6'5(<#7/K.Ll7>9'H`42H?3'@A'5
>V7/<lI$Yh '#< '5C%(L.%G('@ %($3S'%N
10
Spin expectations value along x, <σx>
9
8
7
6
5
4
3
2
1
0
0
0.5
1
1.5
Magnetic field, B
2
2.5
3
X$32H:<#' $FN *
Y.K$,k';Q9Ki'8R`4
#$37Hu-
4?,:' K B >V7H<lI9$Yh '<#'5l1%($3=2%GL9%G('@ %G$,S'0%N=OJ"='P%(L9%1#'@ %G$,S'$,=8;<#'4H%G'0%
D.L7/='P>V<(7H@ A%GK$,=%#7k}/,E>V<(7H@ ?,';> W(7M<#$,2/"5N
7 #"'<C-/'<#L$,5('<#'%G($,2B("=$325%W#7M7#'!4Di7/:9 "=2H:<#'l$=N %NYX$,<#%GW("'@M4Q.$,@Z:=@ yu87/
X1'08R(:=<('0Iq#7M7988;:=<
4
]("'{K"=4H%G'W#<#4/=%($Y#$37HM@B7
-H'%64U)J4UL!>V<#7/@ "m}/N.OJ"5:F%("='{8;7H
X1'8;(:<#'l$&%+7HD.-5$,7/:F%G?,L!>4/?,%('l>V7H< 4/=I
("='1%G$,2B@B79I9'? NFOJ"$,%W4H8R#:=4?,?3L4/2/<#''0%])W$3("u6 ("='<('0%G:?3#%J$,S& W : )W"'<#'P$Yl$&%W8;7H
X1'8;(:<#'Iq("F4
W$Yl$&%("='
I9'<($,-
4
($,-/'Hn9<#4("'<("'u#"''5`42H?3'@B'57/>*>V7H<(@M4
#$37Hq$3#%('?3>1n9("F4
C%("7/:?&IuDF'!4W4B@M4
Q9$,@Z:@ 4
W("='
+OPNY9'8;7H=I9?,LM("='@A4Q9$3@E:@ $&%W@Z:=8`"%(@A4/?3?,'<J#"=4e>V7H<J("''H#<(7HK.LM7>'5#4/2/?,'@A'50N
4E; ]; P; H; /:P;uP; ( P;Q69; ;H
90 Z;Q1
OJ"'JI9'Ki'FI9'=8']7H("='I9';y.I9'2/''<`4
#$37H "='?&IP>V7/<*("8 =']1%($3=2{@A79I9'?5)W$3("%{%GK=$3=%$&%*%("7
)W!$,O"F2/:<#'+$FN(9N
a\'+)J45T(7{K<#'%('<#-/'("='+';Q9Ki'8;('IZ%G"=4/<(KKi'4/ n04/=I%1#$3?,?57I9'K<#'%#%b#"'+@M4Q.$,@Z:=@ -
4/?3:'HNOJ"$&%*@A4//'0%
"m}^2\1}/,W, }-,H,W4, _J4E2/7.79Iq8`"=7/$&8;'/N.u>4/8;#"'4/8;(:=4/?i"F4=I9?,$,2A7>#"'I9'Ry9I9'2H''<#4($,7/"F'?&Iq$&%J4
8D$3l@A7/<#'PI'4 ?,$,84
('x#"=6 4BX1:=%GW(7%G($&8`M(7B#"$,%l-U4/?3:='/N l=4(:<`4?,?,LqI9'Ki'FI%W7/u("='!%GL9%G('@ 1L5Ki'xy$Y>)'
"=4U-H'W7ZI9'2/'='<`4/8;LZ$3A#"'C2H<(7H:=IA%1`4
#'l7l"='?,IA$&%6=''I'IEyED:4?&%(77/B("'{.:@ZDi'<]7>T%GK=$3=%6(<`4/8'I
4U)J4UL/NX=7/<!'Q94/@AK?3'Hn$3>l)]'q"=4U-/'4u%| %($3('q%(K$,w1%G$,2z8`"=4/$3n*4FIg7/?,L[(<`4/8;'4U)J4ULc7H'q%(K$,("'T"='?&I
I0
f - z f 0 <q> 4 k >
e
0.16
Entanglement of Formation, EF
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
0.8
1
1.2
Magnetic Field, B
1.4
1.6
1.8
2
X$32H:<(' $FN %5p65#4/2/?,'@A'5J7>>V7/<#@M4
($,7/eDi';1)''u1)7B='4<#'%GW'$,2/".Di7/<`%]>V7/<J("='1%G$,2A@A7.I'?N
4Di7
-/'C$,%](7.7B%1#<(7H2PyM$YWI9'K=<('0%(%('%6("'xKF'04EyA4/=I " }M^D\1}-,H,W, }-,H,H,W, _]$&%]4ZDF'G#'<W8`"7/$&8;'HNHa".L
("='84q)]'x$3u>4/8;:F%G'{("' "=<`%G-
4?,:'x7>T#"'PI9'Ry9I98 '2H''<#4($,7/=% 4 "='?&I Pq6 ("'4Di7
-/'x84/?,8:?,4($,7/=%+("='
I9';y.I9'2/'='<`4
($,7/F%P"='?&IA$,%+4?,)4UL9%%(@A4/?3?='7H:2/"B("F4
+$3+)W$,?3?F7]I'K<#'%#%("='l'5#4/2/?,'@A'H6@M4Q.$,@Z:=@eN
OJ".:=%J$3C%G"7H:?&IuDi'!4A%(4>V'8`"7/$&8;'HN
1.2
ε = 1/(10 + 100B)
ε = 1/(10 + 100B)
ε = 1/(100 + 1000B)
ε = 1/(1000 + 10000B)
ε = 1/(10000 + 100000B)
1
0.8
0.6
0.4
0.2
0
−0.2
0
0.5
1
1.5
Magnetic field, B.
2
2.5
3
X$32H:<#'l$=N.(FOJ"='!'5(<#7/K.Lu7>'5#4/2/?,'@A'HW>V7H<{I9$3h '<#'5{I9';y.I9'2/'='<`4
($,7/F%["='?&I%NFl'<('!("'Z%(L9%1#'@ $,%
4e1%($32B1L.Ki')W$Y#"a%A%(K$,=%l4FI#"' "=<`%1W#"<(''x(<`4/8;'0Iu4U)J4UL/N
X7H<+#"'{>V:?3? kkt@A79I9'?i#"'x%($3(:=4($,7/q$&%I9$Yh '<#'50N.C'<#'l#"'P2/<#7/:=IM%G#4('x7>b#"'x%(L9%1#'@ &¦x:$&^H:='
¡ z'5#4=2/?,'I \ %G''B("'A'Q5!%('8;($,7/9_;TOJ"=4
$&%n #"$,%P@A79I9'?)W$3?,?"=4U-/'A4u7/9| S'<(7u'5#4/2/?,'@A'HP>V7H<
("='M2/<#7/:=I[%1`4
#'/NTOJ"'M<#'%(:?3P7/>4I9';y.I9'2/'='<`4
($,7/u"='?&I\"'<#'A$,%P7H?3Lz(7kI9'%G(<#7
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%("=4?,?T%G''P?,4('<0N
'84:=%('+7/>=("='W<#4("'<:=9>V7/<((:F4
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I =n
("='x<#'%(:?3#%]7/q("'P1%($32B@A79I9'? "=4U-H'l(7BDi'P^5:'0%1#$37H'I(7A%G7H@A'{'Q5#'5N9C7
)]'-/'<+("'<('P4<#'P4!.:@ZDi'<
7>>4/8;(7H<#%J%(:2/2H'%G($,2Z#"'<#'%(:?YJ#7MDF'K?&4:F%G$,D?,'/NO*7BDi'2H$3)W$3("Tn>V7/<J("='<('2/$,7/ B )'P84%G.$,K
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)W"=' B 8 N9.'08;7/FI9?3LHn=%($3@A$,?,4/<W$3.-/'0%1#$3254
#$37H=%]>V7/<W#"'1%G$,2A@A79I9'?T)'<#'84/<(<#$3'0Iq7H:9l$,S
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e 0
:;s<T< 0mLP;Q1+ +
9;5
/;szW;Q1
I7I
1.4
With de−−degenerating filed.
Without de−−degenerating field.
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
Magnetic field, B
2
2.5
3
X$32H:<#'l$=N q
FOJ"='!'5(<#7/K.Lu7>'5#4/2/?,'@A'HW>V7H<{I9$3h '<#'5{I9';y.I9'2/'='<`4
($,7/F%["='?&I%NFl'<('!("'Z%(L9%1#'@ $,%
4EkGk @B79I9'?b)W$Y#"%M%GK$,=%W4/=I#"' "=<`%1l7H'x(<`4/8'Ie4U)4ULHN
5 5L 5 >)]'!%('; "{,A$3k'0^5:=4
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I97e7P"=4U-H'Z>V7H<C#"'A7("='< k)o @A79I9'?&%NbOJ"'AC4/@A$3?3(7H$,4/[=7
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1.5
Spin 1 to 4 traced.
Spin 1 to 5 traced.
Spin 1 to 6 traced.
1
0.5
0
0
0.5
1
1.5
Magnetic field, B.
2
2.5
3
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1
2 spins
3 spins
4 spins
5 spins
6 spins
7 spins
8 spins
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
Magnetic field, B
2
2.5
3
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Eigenvalues
e 0
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Difference between finite XX systems and infinite systems, D(N)
5
4.5
4
3.5
3
2.5
2
1.5
1
2
3
4
5
System size, N
6
7
8
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1
Entanglement, E
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Magnetic field, B
3
1.5
B field
2
3.5
4
4.5
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6
1
0.9
0.8
Entanglement, E
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
2.5
3
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(7\2/'Z#"'eI'=%($Y1Lg@M4(<#$YQN\V$,f4\%($3@A$,?,4/<E)J4UL\4H%EI7/'e$3/
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("='[<#'%(:?3e%("7
)Ws$,N"=2H:<('s$=N,}/%=N+ >Z)'[87/@AK=4<#'u"F2/:<#'$FN3}*: )W$Y#"N"=2H:<#'$=N,}/%w)]'\%G''\%(7/@A';#"$,2
$,H#'<#'%G($,2=NJOJ"'s"=<`%1I9$,%#8;7H5($,5:=$Y1Lc\87/<#<('0%GKi7/FI9$3=2g(7 ("'\?3'-/'?x8<(75%(%($3=2w)W"'<#'c)'[%()W$3#8`">V<#7/@
I
f - z f 0 <q> 4 k >
e
0.35
0.3
0.25
0.2
0.15
0.1
Result from 2 E12 = E1 + E2 − E12
Approximate result from Arnesen et al.
Their plot is done for kT = 0.1
quant−ph/0009060
0.05
0
0
0.5
1
1.5
2
2.5
Magnetic field, B
3
3.5
4
4.5
X$,2/:<#' $=N,}-* b '4<#'%G1y.'$,2/".Di7/<'5#4/2/?,'@A'HJ>V7H<J("'lkkGk @A79I9'?b)W$3("%M%(K$,=%N
0.3
0.25
Entanglement of Formation, E
F
0.2
0.15
0.1
0.05
Entanglement of Formation, kB T = 0
Entanglement of Formation, kB T = 0.1
0
Entanglement of Formation, Arnesen et al.
−0.05
0
0.5
1
1.5
2
2.5
Magnetic field, B
3
3.5
4
4.5
X$32H:<#' $FN3}-
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7/K=KF75%G$3('HNQ&x$,-/'M("$&%)]'x84/K<#'I9$&8R("'P%(K$3u%GL9%G('@M%8;7H=%G$&I9'<('0IM"'<#'{4/?3?iD:9("='{1%($3=2Z:$,-/'<#%#4?,$31L
8;?&4/%#%(7ADi'@M4/%#%G?,'%#%N
0
−1
10 Lowest lying Eigen values
−2
−3
−4
−5
−6
−7
−8
0
0.2
0.4
0.6
0.8
Magnetic Field, B
1
1.2
1.4
X$32H:<#' $FN3}M(OJ"'A}-,E?,7
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4?,:'%]>V7/<J#"'lkGkm@A79I9'? N
0
−1
10 Lowest lying Eigen values
−2
−3
−4
−5
−6
−7
−8
−9
0
0.2
0.4
0.6
0.8
Magnetic Field, B
1
1.2
1.4
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E
EF
1
2
3
4
5
6
Traced
Keept
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Entropy of Entanglement, EE
1.5
1.4
1.3
1
1.2
0.4
0.3
0.2
0.1
0
−0.1
1.5
1.1
1
1
0.5
1.5
1.5
0.5
1
∆
0.5
Γ
1
0
0.5
Γ
0
Q[8V .
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:5,8;:TKL>4^J>4:5(C;Y CTB,^ 8H5/PLCT:
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0.3
1.4
1.3
1.2
1.1
1
0.25
0.2
0.15
0.1
0.05
0
−0.05
3
0.9
3
2
3
2
2.5
2
1.5
1
1
1
0
Γ
0.5
0
Γ
∆
Q[8V .
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2.5
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0.5
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1.5
1
0.5
0
0.1
0.05
0
−0.05
0.5
3
1
1.5
2
2
2.5
Γ
3
2.5
3
1.5
2
1
0.5
0
1
Γ
Q[8V .
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1.5
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2.5
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0.12
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1.4
1.2
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3
0.1
0.08
0.06
0.04
0.02
0
3
3
2
3
2
2.5
2.5
2
2
1.5
1
1
0
Γ
1.5
1
1
0.5
0
0
Γ
∆
Q[8V .
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Entanglement in Finite Spin Systems

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