Seminar@Osaka, April 10, 2017 Projection Measurement and Holography Tokiro Numasawa Osaka University Particle Physics Theory Based on JHEP08(2016)077 arxiv:1604.01772 [hep-th] Collaboration with N.Shiba(Harvard) T.Takayanagi and K.Watanabe(YITP)
(1)Introduction
i = "i "i H A i (1) H B j (2) h i (1) j (2) i = h 0 i = p 1 ( "i "i + #i #i) 2 i (1) i h j (2) i h 0 i (1) j (2) 0 i h 0 i (1) 0 i h 0 i = "i "i S A =0 A =Tr B [ ih ] S A = Tr A [ A log A ] 0 i = 1 p 2 ( "i "i + #i #i) S A = log 2 j (2) 0 i6=0
h0 (x) (y) 0i 6=0 S A = c 3 log l c l
AB! X (A i B j ) AB (A i B j ) i,j X A i A i =1 X B j B j =1 A i i j Bj
S A S A
(2) Local Projection measurement in a CFT P = Y x2p x ih x Y x2p c I x P C P i i i i
Bi H = X i z i z i+1 + x i 0i 1 2 i 1 16 i "i "i "i #i #i #i!i!i!i
hb Bi O(x 1 ) O(x 2 ) O(x n )! 0 hb e H O(x 1 )O(x 2 ) O(x n )e H Bi hb e 2 H Bi ny i=1 ho(x i )i Y U x Bi x
i = e ph P 0i = ih = h i h O(x 1 )O(x 2 ) i = O(x 1 ) O(x 2 )
X( ) =2ip K( / p )+K( p ) 1 2 t 2q p-it p+it X 1 ζ K( ) = q p 50 1 1 + X 2k 2k 2k 1 2k k=1 q p 2 log 40 30 1 2 e 2 ξ Identify 0 y -2π log(ρ) w Identify 0 x 20 10 0.2 0.4 0.6 0.8
P q 0 q x 0.5 x +0.5 =0.6 p =0.5 q =5.3 S A (x) S A (x) 10 5 5 10 S ground A (x) x 0.05 0.10 0.15
(t)i = e iht e ph P 0i "i "i "i (t)i = e iht e 4 H Bi S A (t) = ( 2 c 3 c 3 t (t <l/2) l (t >l/2)
q P 0 0.5 0.5 q =0.6 p =0.5 q =5.3 S A (t) S A (t) S ground A (t) 0.15 0.10 0.05 5 10 15 t 0.05 0.10 0.15
O 1 (w 1 ) O 2 (w 2 ) w O 2 ( 2 ) O 1 ( 1 ) P 2 = P = r q + w q w
2q l n w n(w 1 ) S A = c 6 log 2l(l +2q) q b + b
2q l 1 l 2 w n(w 2 ) n(w 1 ) S A = 1 6 log 2l 1(l 1 +2q) q + 1 6 log 2l 2(l 2 +2q) q +2 b + 1 3 log q l 1 +2q l 1 q l 1 +2q l 1 + q q l 2 +2q l 2 l 2 +2q l 2
S A = ( c 6 log 2l 1(l 1 +2q) + c q 6 log 2l 2(l 2 +2q) +2 (l q 1 q), b c 3 log l 2 l 1 (l 1 q) 2q l 1 l 2 l 1 q 2q l 1 q l 1 l 2
(3) Partial Entangling and Swapping in two CFTs P = Y X X n x i 1 n x i 2 hm x 1 hm x 2 Y (Ix 1 Ix) 2 x2p n x m x x2p c P C P
CFT1 CFT2 CFT1 CFT2 P P
= 1 + i 2 2 1 S ent = c 3 2
CFT1 CFT2 CFT1 CFT2 w -q+ip Cy q+ip P P Cx -q-ip q-ip
-q+ip Cy q+ip w Cx -q-ip q-ip y 2 =(x ip q)(x ip + q)(x + ip q)(x + ip + q) 2 = R R C y C x dx y dx y S ent 2c 3 log q p 2 S interval S interval = c 3 log l
Ā 1 Ā 2 A 1 A 2 Ā 1 Ā 2 A 2 A 1 S A1 = c 3 log q p S A2 = c 3 log q p S ent = S A1 + S A2
(4)Holographic duals of Local projections O 1 (w 1 ) O 2 (w 2 ) w O 2 ( 2 ) O 1 ( 1 ) = r q + w q w
ds 2 = RAdS 2 d 2 +2d d 2 = f(w) = f( w) 2z 2 (f 0 ) 2 00 f 8f 0 f 0 + z 2 f 00 f 00 2z 2 ( f 0 ) 2 f 00 8f 0 f 0 + z 2 f 00 f 00 = 8z(f 0 f 0 ) 3 2 8f 0 f 0 + z 2 f 00 f 00 ds 2 = R 2 AdS dz 2 z 2 + L(w)dw2 + L( w) 2 d w 2 + 2 z 2 + z2 2 L(w) L( w) dwd w T ww L(w) = 3(f 00 ) 2 2f 0 f 000 4(f 0 ) 2
A A A A
w 2q l ξ Im[w]=0 A ξ=f(w) -q P q l f(p) i A A S A = Area( A) 4G N = c 6 log 2l 1(l 1 +2q) q
2q l 1 l 2 3.4 S A l 2 l 1 q Phase-1 Phase-2 3.2 A A 3.0 2.8 2.6 P Q P Q 2.4 2.2 2.0 0.00 0.02 0.04 0.06 0.08 0.10 l 1 q A A S A = c 6 log 2l 1(l 1 +2q) q + c 6 log 2l 2(l 2 +2q) q + c 3 log q l 1 +2q l 1 q l 2 +2q l 2 2( l 1+2q l 1 l 2 +2q l 2 ) 1 4 A A S A = c 6 log 2l 1(l 1 +2q) q + c 6 log 2l 2(l 2 +2q) q
ds 2 = R2 AdS z 2 h dz 2 + 1 2 z 2 2 2dx 2 + 1+ 2 z 2 2 2dy 2 i Disconnected Geodesics Connected Geodesic BH Horizon
=0.6 p =0.5 q =5.3 q 0 q x 0.5 x +0.5 2.0 S A 1.5 1.0 0.5 10 5 5 10 0.5 x 1.0
q 0 q 0.5 +0.5 S A (t) 3 2 0.15 1 0.10 0.05 0 2 4 6 8 10 12 0.05 5 10 15 0.10 1 0.15 2
(5)Conclusions and Discussion
CFT1 CFT2 CFT1 CFT2 P P
I E = 1 16 G N Z N p g(r 2 Lmatter ) 1 8 G N Z Q p h(k L Q matter) K ab Kh ab =8 G N T Q ab T ab = Th ab