Projection Measurement and Holography

Transcription

1 April 10, 2017 Projection Measurement and Holography Tokiro Numasawa Osaka University Particle Physics Theory Based on JHEP08(2016)077 arxiv: [hep-th] Collaboration with N.Shiba(Harvard) T.Takayanagi and K.Watanabe(YITP)

2 (1)Introduction

3 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

4 h0 (x) (y) 0i 6=0 S A = c 3 log l c l

5 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

6 S A S A

7

8 (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

9 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

10 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

11 i = e ph P 0i = ih = h i h O(x 1 )O(x 2 ) i = O(x 1 ) O(x 2 )

12 X( ) =2ip K( / p )+K( p ) 1 2 t 2q p-it p+it X 1 ζ K( ) = q p X 2k 2k 2k 1 2k k=1 q p 2 log e 2 ξ Identify 0 y -2π log(ρ) w Identify 0 x

13 P q 0 q x 0.5 x +0.5 =0.6 p =0.5 q =5.3 S A (x) S A (x) S ground A (x) x

14 (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)

15 q P q =0.6 p =0.5 q =5.3 S A (t) S A (t) S ground A (t) t

16 O 1 (w 1 ) O 2 (w 2 ) w O 2 ( 2 ) O 1 ( 1 ) P 2 = P = r q + w q w

17 2q l n w n(w 1 ) S A = c 6 log 2l(l +2q) q b + b

18 2q l 1 l 2 w n(w 2 ) n(w 1 ) S A = 1 6 log 2l 1(l 1 +2q) q log 2l 2(l 2 +2q) q +2 b 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

19 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

20

21 (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

22 CFT1 CFT2 CFT1 CFT2 P P

23 = 1 + i S ent = c 3 2

24 CFT1 CFT2 CFT1 CFT2 w -q+ip Cy q+ip P P Cx -q-ip q-ip

25 -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

26 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

27

28 (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

29 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 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

30

31 A A A A

32 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

33 2q l 1 l S A l 2 l 1 q Phase-1 Phase A A P Q P Q 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

34

35 ds 2 = R2 AdS z 2 h dz z 2 2 2dx z 2 2 2dy 2 i Disconnected Geodesics Connected Geodesic BH Horizon

36 =0.6 p =0.5 q =5.3 q 0 q x 0.5 x S A x 1.0

37 q 0 q S A (t)

38

39 (5)Conclusions and Discussion

40 CFT1 CFT2 CFT1 CFT2 P P

41 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