Non-geometric states and Holevo information in AdS3/CFT2

Transcription

1 Non-geometric states and Holevo information in AdS3/CFT2 Feng-Li Lin workshop 11/2018 based on , & with Jiaju Zhang(Milano-Bicocca) & Wu-Zhong Guo(NCTS) 1

2 Motivations Recently, there are lots of discussions on quantum thermalization in the context of AdS/CFT duality, inspired by the holographic entanglement. One of the key issues is how we can characterize the local (dis-)similarity between a typical pure state and thermal state, and how general it should be? The general principle/criterion is formulated as the eigenstate thermalization hypothesis (Srednicki) or canonical typicality (Popescu et al). On the bulk side, the story becomes how we can characterize the microstates of a black hole, or can we tell them locally?!2

3 In this talk, I will discuss two issues on quantum thermalization in the context of AdS3/CFT2. We will examine 1/c behaviors through short-interval expansion of entanglement entropy. In CFT side, 1/c effects will fine-grain the microstates and lead to expected failure of ETH for integrable models. In the bulk side, 1/c ~ G effects encode quantum gravity corrections.

4 1. Are all CFT states geometric? A folklore in 3D gravity, all the geometries are described by the Banados metrics: The holomorphic function stress tensor: L (z) is related to the vev of boundary This seemingly suggests that every CFT state corresponds to a Banados metric. However, a simple example shows the opposite: the linear supposition of the CFT quantum states. This is because the bulk gravity is classical. Can we formulate a general criterion for (non-)geometric states?!4

5 <latexit sha1_base64="0usbjei3ea1oz7novgkt1zjwxg0=">aaacchicbva7t8mwghr4lvikmdjgusgvpupqjrgldlbroc+pislhdvqrdhzzdlivdwthr7awgbarp4gnf4p7gkdljevnu/tkfxcmjcrton/wwuls8spqbi2/vrg5tw3v7dauscumdsyykk0qkcjotoqaakzaissih4w0w/7lyg8+ekmoigt6kbcfo25mi4qrnljgh9wh2v1t6cnkyxbuczof5avj/aaijwo74jscmea8caekakaobvax1xe45stwmcgl2q6tad9dulpmyddvpyokcpdrl7qnjrenys/giwzhkve6mblsnfjdsfp7ikncqqeptzij3voz3kj8z2unojrzmxonqsyxnjwupqxqauetwa6vbgs2marhsc1fie4hiba23evnce7syvokcvjydb8tfyox0zpyyb8cgijwwsmoggtqbxwawsn4bq/gzxqyxqx362msxbcmm3vgd6zphwvomko=</latexit> <latexit sha1_base64="0usbjei3ea1oz7novgkt1zjwxg0=">aaacchicbva7t8mwghr4lvikmdjgusgvpupqjrgldlbroc+pislhdvqrdhzzdlivdwthr7awgbarp4gnf4p7gkdljevnu/tkfxcmjcrton/wwuls8spqbi2/vrg5tw3v7dauscumdsyykk0qkcjotoqaakzaissih4w0w/7lyg8+ekmoigt6kbcfo25mi4qrnljgh9wh2v1t6cnkyxbuczof5avj/aaijwo74jscmea8caekakaobvax1xe45stwmcgl2q6tad9dulpmyddvpyokcpdrl7qnjrenys/giwzhkve6mblsnfjdsfp7ikncqqeptzij3voz3kj8z2unojrzmxonqsyxnjwupqxqauetwa6vbgs2marhsc1fie4hiba23evnce7syvokcvjydb8tfyox0zpyyb8cgijwwsmoggtqbxwawsn4bq/gzxqyxqx362msxbcmm3vgd6zphwvomko=</latexit> <latexit sha1_base64="0usbjei3ea1oz7novgkt1zjwxg0=">aaacchicbva7t8mwghr4lvikmdjgusgvpupqjrgldlbroc+pislhdvqrdhzzdlivdwthr7awgbarp4gnf4p7gkdljevnu/tkfxcmjcrton/wwuls8spqbi2/vrg5tw3v7dauscumdsyykk0qkcjotoqaakzaissih4w0w/7lyg8+ekmoigt6kbcfo25mi4qrnljgh9wh2v1t6cnkyxbuczof5avj/aaijwo74jscmea8caekakaobvax1xe45stwmcgl2q6tad9dulpmyddvpyokcpdrl7qnjrenys/giwzhkve6mblsnfjdsfp7ikncqqeptzij3voz3kj8z2unojrzmxonqsyxnjwupqxqauetwa6vbgs2marhsc1fie4hiba23evnce7syvokcvjydb8tfyox0zpyyb8cgijwwsmoggtqbxwawsn4bq/gzxqyxqx362msxbcmm3vgd6zphwvomko=</latexit> <latexit sha1_base64="0usbjei3ea1oz7novgkt1zjwxg0=">aaacchicbva7t8mwghr4lvikmdjgusgvpupqjrgldlbroc+pislhdvqrdhzzdlivdwthr7awgbarp4gnf4p7gkdljevnu/tkfxcmjcrton/wwuls8spqbi2/vrg5tw3v7dauscumdsyykk0qkcjotoqaakzaissih4w0w/7lyg8+ekmoigt6kbcfo25mi4qrnljgh9wh2v1t6cnkyxbuczof5avj/aaijwo74jscmea8caekakaobvax1xe45stwmcgl2q6tad9dulpmyddvpyokcpdrl7qnjrenys/giwzhkve6mblsnfjdsfp7ikncqqeptzij3voz3kj8z2unojrzmxonqsyxnjwupqxqauetwa6vbgs2marhsc1fie4hiba23evnce7syvokcvjydb8tfyox0zpyyb8cgijwwsmoggtqbxwawsn4bq/gzxqyxqx362msxbcmm3vgd6zphwvomko=</latexit> Geometric state criterion By studying the (short-interval expansion of) entanglement entropy, we find some states have no bulk description a la Banados metric. This is based on following simple observation: If a CFT state of order c stress tensor can be described by the Banados metric (order one in 1/c expansion), then S RT A 4G O(c) The criterion for a CFT2 state to be bulk geometric, its entanglement/renyi entropy calculated from CFT should be also order c at most in the large c limit.!5

6 2. Can we locally distinguish the microstates of a black hole? By ETH or canonical typicality, a typical state locally looks like thermal state. In dual gravity, this means that black hole microstate geometry (aads metric) locally looks like BTZ black hole. Can we distinguish the black hole s microstates? In our previous works , & , we showed that the EE of heavy primary states are the same as the EE of thermal state at leading order of c. Can we have quantify this (in-)distinguishability?!6

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sha1_base64="9vlfw73+i9xppwq+zgigzrureqy=">aaab/3icbvdlssnafj3uv62vqodgzwarxjvebn0irtcuk9haaekytcft0hmemylqyhf+ihsxirj1n9z5n07allt1wiuz59zlvxpilfftpo/bqswtr6yuvddrg5tb2zvu7l5hy0xh0sassdwnksamcti21ddstrvbpgbkph5df/79a1gasnfnxikjoroimlcmjjui9ybqq3kz6ixhfka2indei7funbwp4clxs1ihjvqr+xx0jc44eqyzphxp91it5kgzihmz1ijmkxtherqqnquccaldfhr/bb5bpq8tqwwja6fq74kcca3hpladhjmhnvck8t+vl5nkisypsdndbj4tsjigjyrfglbpfcggjs1bwff7k8rdpba2nrkadcgf//ii6zw2fmtvz+rnqzkokjger+ae+oacnmenaie2woarpinx8oy8os/ou/mxa6045cw++apn8weka5xh</latexit> <latexit sha1_base64="9vlfw73+i9xppwq+zgigzrureqy=">aaab/3icbvdlssnafj3uv62vqodgzwarxjvebn0irtcuk9haaekytcft0hmemylqyhf+ihsxirj1n9z5n07allt1wiuz59zlvxpilfftpo/bqswtr6yuvddrg5tb2zvu7l5hy0xh0sassdwnksamcti21ddstrvbpgbkph5df/79a1gasnfnxikjoroimlcmjjui9ybqq3kz6ixhfka2indei7funbwp4clxs1ihjvqr+xx0jc44eqyzphxp91it5kgzihmz1ijmkxtherqqnquccaldfhr/bb5bpq8tqwwja6fq74kcca3hpladhjmhnvck8t+vl5nkisypsdndbj4tsjigjyrfglbpfcggjs1bwff7k8rdpba2nrkadcgf//ii6zw2fmtvz+rnqzkokjger+ae+oacnmenaie2woarpinx8oy8os/ou/mxa6045cw++apn8weka5xh</latexit> <latexit sha1_base64="9vlfw73+i9xppwq+zgigzrureqy=">aaab/3icbvdlssnafj3uv62vqodgzwarxjvebn0irtcuk9haaekytcft0hmemylqyhf+ihsxirj1n9z5n07allt1wiuz59zlvxpilfftpo/bqswtr6yuvddrg5tb2zvu7l5hy0xh0sassdwnksamcti21ddstrvbpgbkph5df/79a1gasnfnxikjoroimlcmjjui9ybqq3kz6ixhfka2indei7funbwp4clxs1ihjvqr+xx0jc44eqyzphxp91it5kgzihmz1ijmkxtherqqnquccaldfhr/bb5bpq8tqwwja6fq74kcca3hpladhjmhnvck8t+vl5nkisypsdndbj4tsjigjyrfglbpfcggjs1bwff7k8rdpba2nrkadcgf//ii6zw2fmtvz+rnqzkokjger+ae+oacnmenaie2woarpinx8oy8os/ou/mxa6045cw++apn8weka5xh</latexit> Holevo information If we encode a random variable X (with outcome probability denoted by ) into a quantum state p i = X i p i i By measuring the quantum state to obtain the outcome Y, then the maximum of the mutual information I(X;Y) is the Holevo information. Apply this to the local distinguishability of black hole X microstates, the Holevo information is A is monotonically increasing with A. A = S A i p i S A,i 0 apple A apple S BH = S thermal := X i p i log p i!7

8 <latexit sha1_base64="sgdi4o6qqhq0jlixnjs7ffvgq0m=">aaab7xicbzbnswmxeizn/az1q+rrs7ainsquchqsevfywx5au5rsmm1js8mszaql9d948aciv/+pn/+nabshbx0h8pdodjl5o1qki77/7a2srq1vbba2its7u3v7pypdhtwzybzotnsmfvhlpvc8jgilb6wg0yssvbknb6f15hm3vmj1gkouhwntkxelrtfzjq4bio51t1t2k/5mzbmchmqqq9ytfxv6mmujv8gktbyd+cmgy2pqmmknxu5meurzkpz526gicbfhelbthjw6p0dibdxtsgbu74kxtawdjzhrtcgo7gjtav5xa2cyx4vjodimuwlzj+jmetrkejrpccmzypedyoxwuxi2oiyydaevxqjb4snl0divbi7vl8rvmzyoahzdczxbajdqhtuoqr0ypmizvmkbp70x7937mleuepnmefyr9/kdp6co5g==</latexit> <latexit sha1_base64="sgdi4o6qqhq0jlixnjs7ffvgq0m=">aaab7xicbzbnswmxeizn/az1q+rrs7ainsquchqsevfywx5au5rsmm1js8mszaql9d948aciv/+pn/+nabshbx0h8pdodjl5o1qki77/7a2srq1vbba2its7u3v7pypdhtwzybzotnsmfvhlpvc8jgilb6wg0yssvbknb6f15hm3vmj1gkouhwntkxelrtfzjq4bio51t1t2k/5mzbmchmqqq9ytfxv6mmujv8gktbyd+cmgy2pqmmknxu5meurzkpz526gicbfhelbthjw6p0dibdxtsgbu74kxtawdjzhrtcgo7gjtav5xa2cyx4vjodimuwlzj+jmetrkejrpccmzypedyoxwuxi2oiyydaevxqjb4snl0divbi7vl8rvmzyoahzdczxbajdqhtuoqr0ypmizvmkbp70x7937mleuepnmefyr9/kdp6co5g==</latexit> <latexit sha1_base64="sgdi4o6qqhq0jlixnjs7ffvgq0m=">aaab7xicbzbnswmxeizn/az1q+rrs7ainsquchqsevfywx5au5rsmm1js8mszaql9d948aciv/+pn/+nabshbx0h8pdodjl5o1qki77/7a2srq1vbba2its7u3v7pypdhtwzybzotnsmfvhlpvc8jgilb6wg0yssvbknb6f15hm3vmj1gkouhwntkxelrtfzjq4bio51t1t2k/5mzbmchmqqq9ytfxv6mmujv8gktbyd+cmgy2pqmmknxu5meurzkpz526gicbfhelbthjw6p0dibdxtsgbu74kxtawdjzhrtcgo7gjtav5xa2cyx4vjodimuwlzj+jmetrkejrpccmzypedyoxwuxi2oiyydaevxqjb4snl0divbi7vl8rvmzyoahzdczxbajdqhtuoqr0ypmizvmkbp70x7937mleuepnmefyr9/kdp6co5g==</latexit> <latexit sha1_base64="sgdi4o6qqhq0jlixnjs7ffvgq0m=">aaab7xicbzbnswmxeizn/az1q+rrs7ainsquchqsevfywx5au5rsmm1js8mszaql9d948aciv/+pn/+nabshbx0h8pdodjl5o1qki77/7a2srq1vbba2its7u3v7pypdhtwzybzotnsmfvhlpvc8jgilb6wg0yssvbknb6f15hm3vmj1gkouhwntkxelrtfzjq4bio51t1t2k/5mzbmchmqqq9ytfxv6mmujv8gktbyd+cmgy2pqmmknxu5meurzkpz526gicbfhelbthjw6p0dibdxtsgbu74kxtawdjzhrtcgo7gjtav5xa2cyx4vjodimuwlzj+jmetrkejrpccmzypedyoxwuxi2oiyydaevxqjb4snl0divbi7vl8rvmzyoahzdczxbajdqhtuoqr0ypmizvmkbp70x7937mleuepnmefyr9/kdp6co5g==</latexit> To calculate A, the difficulty lies on how to take thermal average of EE for all CFT state. To bypass this, Bao & Ooguri take the facts in the large c limit: (i) light sates are sparse, (ii) heavy states locally look thermal (not true for non-geometric states) (iii) verified by holographic EE One should expect the plateaux will be lifted by lifted by 1/c effect as I will show below.

9 Entanglement Entropy in 2D CFT!9

10 !10

11 Renyi entropy c.f. Calabrese & Cardy (2004) Renyi entropy is defined as As n >1, we have entanglement entropy: For 2D CFT, it can be obtained by 2-point function of twist operators of n-fold CFT on Riemann surface : 1-fold CFT n-fold CFT n-fold CFT!11

12 OPE & Enumerating Quasi-primaries T (z) T (w) = c n (z w) 2h T X Assuming translational symmetry, we have Counting the quasiprimaries of n-fold CFT: K c p K := Cp h K +p 1 C p, d K = 2h K +p 1 d K 1 X p=0 1 K`hK K = normalization of quasiprimary operator [x +(1 x)tr x L 0 ] n =1+nx 2 + tr A n A = c n `2h T n(n + 1) 2 X c p K p! (z w)h p K(w), lim z!1 z2h K h K (z)i Rn,1, K K2CFT n d K`hK h K i R x 4 + n(n2 +3n + 8) 6 j 1 j k K = X j 1 1 Xj k k ) hx j 1 1 Xj k k i R = hx 1 i R hx k i R with b X1 X k := X X 1 X k with some constraints for 0 apple j 1,,j k apple n 1. j 1,,j k d j1 jk x 6 + n(n + 1)(n2 +5n + 30) 24 x 8 +!12

13 !13

14 EE in small interval expansion S (n) A = c(n + 1) 12n log ` =) S A = c 6 log ` + 1 X Explicitly, n 1 1 log 1+ X k=1 {X 1,,X k } S A = c 6 log ` + `2a T ht i + `4a TT ht i 2 + `6a TTT ht i 3 + `8 a AA hai 2 + a TTA ht i 2 hai + a TTTT ht i 4 nx X k=1 {X 1,,X k } + `10 a T AA ht i hai 2 + a TTTA ht i 3 hai + a TTTTT ht i 5 + `12 a BB hbi 2 + a DD hdi 2 + a T AB ht i hai hbi + a T AD ht i hai hdi + a AAA hai 3 + a TTTB ht i 3 hbi + a TTTD ht i 3 hdi + a TTAA ht i 2 hai 2 + a TTTTA ht i 4 hai + a TTTTTT ht i 6 + O(`14 ). `hx 1 + +h Xk bx1 X k hx 1 i R hx k i R `hx 1 + +h Xk ax1 X k hx 1 i R hx k i R with a X1 X k := lim n!1 b X1 X k n 1.!14 a T = 1 6, a TT = a AA = a T AA = a BB = a T AB = a TTTB = a TTTTA = 1 30c, a TTT = 4 315c 2, 1 126c(5c + 22), a TTA = 1 315c 2, a TTTT = c c 3, c 2 (5c + 22), a TTTA = 3 385c 3, a TTTTT = c(70c + 29), a DD = c 2 (70c + 29), a T AD = 5(14c + 43) 9009c 3 (70c + 29), a TTTD = 16(c + 5) 3465c 4, 70c c(2c 1)(5c + 22)(7c + 68), c 2 (5c + 22), a AAA = c 3, a TTAA = 2(33c + 784) 45045c 4, a TTTTTT = 2(11c c ) 45045c 5. 4(5c + 64) 3003c 2 (5c + 22) 2, 585c c 3 (5c + 22),

15 EEs For the ground state of a CFT on a circle of size L: ht i L = 2 c 6L 2, hai L = 4 c(5c + 22) 180L 4, hbi L = 62 6 c 525L 6, hdi L = 6 c(2c 1)(5c + 22)(7c + 68) 216(70c + 29)L 6 =) S A = c 6 log ` 2 c`2 36L 2 4 c`4 1080L 4 6 c` L 6 8 c` L 8 For a low temperature state on a circle: ht i L,q = 2 6L 2 [c 48q2 72q 3 144q 4 + O(q 5 )], hai L,q = hbi L,q = 4!15 10 c` L c` L 12 + O(`14 ) q := e 2 /L << 1 180L 4 [c(5c + 22) + 480(5c + 22)q (5c + 22)q (c + 6)q 4 + O(q 5 )], L 6 [31c 1008(120c + 1)q2 1512(720c + 161)q (1640c + 841)q 4 + O(q 5 )], hdi L,q = 6 (2c 1)(7c + 68) 216(70c + 29)L 6 [c(5c + 22) (5c + 22)q (5c + 22)q (215c 638)q 4 + O(q 5 S A = c 6 log ` h + c i q2 ` 2 h 3 +2q3 +4q 4 + O(q 5 c ) + L q q3 15 4(c i 8)q4 ` 4 h + + O(q 5 c ) + 15c L q q3 8(c i 16)q4 ` O(q 5 ) c L h c q q3 4(159c + 728)q 4 i ` 8 + O(q 5 ) c L h c q q3 104(295c )q 4 i ` 10 + O(q 5 ) c L h 691c q q3 8( c )q 4 i ` + O(q 5 ) c L +O(`14 ) 12

16 Geometric vs Non-geometric!16

17 Conditions on expectation values of quasiprimaries As shown, the expectation values of the quasiprimaries dictates the entanglement/renyi entropy. Requiring the Renyi entropy of A in a state order c yield the following constraints: ht i = c (w)+ (w)+ (w) c 1 + O c 2, 1 c hai = c 2 (w) 2 + c (w)+ (w)+o, hbi = c 2h 0 (w) 2 4 i 5 (w) 00 (w) + c (w)+o(c 0 ), hdi = c 3 (w) 3 +3c 2 (w)[ (w) (w) (w)] + c (w)+o(c 0 ), hei = c 2n 00 (w) o 63 [ (w) (4) (w) 7 0 (w) (3) (w)] + O(c), h hhi = c 3 (w) 0 (w) 2 4 i 5 (w) 00 (w) + c 2h 0 (w) 2 (w) 2 (w) 0 (w) 0 (w)+ 4 5 (w)2 00 (w) (w) 00 (w) (w)+ 0 (w) (w) 5 00 (w) (w) 9 (w) 00 (w)+ 13 i 9 (w) (w) + O(c), hii = c 4 (w) 4 +2c 3 (w) 2 [3 (w) 4 (w) (w)] + c 2 [12 (w) 2 (w) 2 +4 (w) 3 (w) 12 (w) (w) (w)+3 (w) 2 6 (w) 2 (w)+4 (w) (w)] + O(c) c.f. n >1 for EE: to be at most ht i = c (w)+o(c 0 ), hai = c 2 (w) 2 + O(c). with (w), (w), (w), (w), (w), (w), (w) being arbitrary order c 0 holomorphic functions.!17

18 <latexit sha1_base64="pyytqrrsrfjupwos6/vaovuzsdm=">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</latexit> <latexit sha1_base64="pyytqrrsrfjupwos6/vaovuzsdm=">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</latexit> <latexit sha1_base64="pyytqrrsrfjupwos6/vaovuzsdm=">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</latexit> <latexit sha1_base64="pyytqrrsrfjupwos6/vaovuzsdm=">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</latexit> Examples Heavy primary states are geometrical as can be checked straightforwardly. Thermal states (dual to BTZ) satisfy the constraints as expected. The above twos are consistent with the ETH. A more nontrivial example is the ``coherent CFT state, which is dual to a bulk moving particle: (z0 ) = 1 1 z0 z 0 z 0 2h (z 0 ) 0ih0 (1/ z 0 ). This coherent state example suggests existence of some kind of correspondence principle for the geometric states.!18

19 <latexit sha1_base64="ylultsqpwqel5qmmxsnk8xc7txi=">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</latexit> <latexit 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Quantum KdV equations: T = 1 c 6 T 000 3(TT) 0 = 5c + 22 T 000 3A J 2k (w) lim Q 2k 1 = Zc!1 L 0 6 k c k hj 2k(w)i dw L J 2k(w) Classical KdV equations: U = U UU 0. J 2 = U, J 4 = U 2, J 6 = U U 02, J 8 = U 4 + U 2 U UU(4).!19

20 non-examples As the classical gravity cannot reflect the quantum linear superposition principle, we expect the superpositions of the primary states are non-geometric. This is indeed the case. Besides, we find the following descendant states are nongeometric: (m) i with h + m O(c), i with h O(c), (m) i with h + m O(c), T (m) i with m O(c), A (m) i with m O(c). := (T ) 3 2(h + 1) 00!20

21 Implications In perturbative quantum gravity, the metric is corrected by G~1/c correction, and thus we will not expect these effect will turn the non-non-geometric states into the geometric ones. By Cardy s formula, we find that the fraction of primary states is negligible in the thermodynamical limit. If most of the descendant states are non-geometric, they will be also non-thermal. This will then implies the canonical typicality of quantum thermalization fails in the large c 2D CFTs.!21

22 Holevo information of microstates!22

23 <latexit sha1_base64="6arjkdpuwwijggwufbefiku3rai=">aaab8nicbzbns8naeiy39avwr6phl4tf8fqsefry9olbqwx7aukom+2kxbrzhn2juep/hhcpinj113jz37htc9dwfxye3plhz94ok8kg6347pbx1jc2t8nzlz3dv/6b6enq2aa45thgqu92nmaepflrqoirupoelkyronlqd1ttpoi1i1soomwgtnlaifpyhtfwgamq0kjle96o1t+7orvfbk6bgcjv71a+gn/i8ayvcmmn8z80wndcngkuyvolcqmb4ia3at6hyaiaczfee0jpr9gmcavsu0rn7e2lcempgswq7e4zds1ybmf/v/bzj63aivjyjkl74km4lxzto7qd9oygjhftgxau7k+vdphlhm1lfhuatn7wk7yu6z/nhsta4keiokxnyss6jr65ig9yrjmkrtllytf7jm4poi/pufcxas04xc0z+ypn8avjwkka=</latexit> <latexit sha1_base64="6arjkdpuwwijggwufbefiku3rai=">aaab8nicbzbns8naeiy39avwr6phl4tf8fqsefry9olbqwx7aukom+2kxbrzhn2juep/hhcpinj113jz37htc9dwfxye3plhz94ok8kg6347pbx1jc2t8nzlz3dv/6b6enq2aa45thgqu92nmaepflrqoirupoelkyronlqd1ttpoi1i1soomwgtnlaifpyhtfwgamq0kjle96o1t+7orvfbk6bgcjv71a+gn/i8ayvcmmn8z80wndcngkuyvolcqmb4ia3at6hyaiaczfee0jpr9gmcavsu0rn7e2lcempgswq7e4zds1ybmf/v/bzj63aivjyjkl74km4lxzto7qd9oygjhftgxau7k+vdphlhm1lfhuatn7wk7yu6z/nhsta4keiokxnyss6jr65ig9yrjmkrtllytf7jm4poi/pufcxas04xc0z+ypn8avjwkka=</latexit> <latexit sha1_base64="6arjkdpuwwijggwufbefiku3rai=">aaab8nicbzbns8naeiy39avwr6phl4tf8fqsefry9olbqwx7aukom+2kxbrzhn2juep/hhcpinj113jz37htc9dwfxye3plhz94ok8kg6347pbx1jc2t8nzlz3dv/6b6enq2aa45thgqu92nmaepflrqoirupoelkyronlqd1ttpoi1i1soomwgtnlaifpyhtfwgamq0kjle96o1t+7orvfbk6bgcjv71a+gn/i8ayvcmmn8z80wndcngkuyvolcqmb4ia3at6hyaiaczfee0jpr9gmcavsu0rn7e2lcempgswq7e4zds1ybmf/v/bzj63aivjyjkl74km4lxzto7qd9oygjhftgxau7k+vdphlhm1lfhuatn7wk7yu6z/nhsta4keiokxnyss6jr65ig9yrjmkrtllytf7jm4poi/pufcxas04xc0z+ypn8avjwkka=</latexit> <latexit sha1_base64="6arjkdpuwwijggwufbefiku3rai=">aaab8nicbzbns8naeiy39avwr6phl4tf8fqsefry9olbqwx7aukom+2kxbrzhn2juep/hhcpinj113jz37htc9dwfxye3plhz94ok8kg6347pbx1jc2t8nzlz3dv/6b6enq2aa45thgqu92nmaepflrqoirupoelkyronlqd1ttpoi1i1soomwgtnlaifpyhtfwgamq0kjle96o1t+7orvfbk6bgcjv71a+gn/i8ayvcmmn8z80wndcngkuyvolcqmb4ia3at6hyaiaczfee0jpr9gmcavsu0rn7e2lcempgswq7e4zds1ybmf/v/bzj63aivjyjkl74km4lxzto7qd9oygjhftgxau7k+vdphlhm1lfhuatn7wk7yu6z/nhsta4keiokxnyss6jr65ig9yrjmkrtllytf7jm4poi/pufcxas04xc0z+ypn8avjwkka=</latexit> Thermal average of EEs To find the 1/c correction to the Holevo information, we need to find the thermal average of EEs over all states (inclu. primaries and descendants). In the small interval expansion of EE, this can be done by finding the thermal average of one-point functions (and its powers) of quasiprimaries. This relies on the following: X p i hx i i i = hx i, X = T,A, X 2 r p i ht i r P (exp cl 12 i i X i = p ix i ), L i exp cl 12 X 2 i 2r p i ht i r (q@ q ) r Z(q) i =, with Z(q) =q L Z(q) i X p i hx i i hyi i i = 1 L Z L/2 L/2 L dxhx (x)y(0)i (= [T,A] =0. c 24 [1 + q 2 + q 3 +2q 4 + O(q 5 )], q := e 2 L 1 <latexit sha1_base64="irlurv1ny35+dklf1063cgiho2e=">aaaccnicbvc7sgnbfj31genr1djmnag2ht0gkiiqtlgwigaekf3d7orummt24cysejatbfwvgwtfbp0co//gsbkfjh4yojxzd3fu8wloplksb2nufmfxabmwulxdw9/ynle2gzjkbiu6jxgkwh6rwfkidcuuh1ysgaqeh6y3ubz5zqcqkkxhrrrg4aakfzkfuak01dh37s/o4s49sitozldjgslyixqcx2ezwzm2o2bjkltj4fli56sectq65pftjwgsqkgoj1k2bstwbkqeyprdvnqsctgha9kdtqyhcuc66fiudb9opyv9sogxkjxwfydsekg5ddw9grdvl9pespzpayfkp3vtfsajgpbofvkjxyrco15wlwmgig81ivqw/vdm+0qqqnr7rv2cpx3ylgluyrbmn8el6kverwhton10igx0gqroctvqhvh0ij7rk3oznowx4934mizogxlmb/2b8fkdwryzzq==</latexit> <latexit sha1_base64="irlurv1ny35+dklf1063cgiho2e=">aaaccnicbvc7sgnbfj31genr1djmnag2ht0gkiiqtlgwigaekf3d7orummt24cysejatbfwvgwtfbp0co//gsbkfjh4yojxzd3fu8wloplksb2nufmfxabmwulxdw9/ynle2gzjkbiu6jxgkwh6rwfkidcuuh1ysgaqeh6y3ubz5zqcqkkxhrrrg4aakfzkfuak01dh37s/o4s49sitozldjgslyixqcx2ezwzm2o2bjkltj4fli56sectq65pftjwgsqkgoj1k2bstwbkqeyprdvnqsctgha9kdtqyhcuc66fiudb9opyv9sogxkjxwfydsekg5ddw9grdvl9pespzpayfkp3vtfsajgpbofvkjxyrco15wlwmgig81ivqw/vdm+0qqqnr7rv2cpx3ylgluyrbmn8el6kverwhton10igx0gqroctvqhvh0ij7rk3oznowx4934mizogxlmb/2b8fkdwryzzq==</latexit> <latexit sha1_base64="irlurv1ny35+dklf1063cgiho2e=">aaaccnicbvc7sgnbfj31genr1djmnag2ht0gkiiqtlgwigaekf3d7orummt24cysejatbfwvgwtfbp0co//gsbkfjh4yojxzd3fu8wloplksb2nufmfxabmwulxdw9/ynle2gzjkbiu6jxgkwh6rwfkidcuuh1ysgaqeh6y3ubz5zqcqkkxhrrrg4aakfzkfuak01dh37s/o4s49sitozldjgslyixqcx2ezwzm2o2bjkltj4fli56sectq65pftjwgsqkgoj1k2bstwbkqeyprdvnqsctgha9kdtqyhcuc66fiudb9opyv9sogxkjxwfydsekg5ddw9grdvl9pespzpayfkp3vtfsajgpbofvkjxyrco15wlwmgig81ivqw/vdm+0qqqnr7rv2cpx3ylgluyrbmn8el6kverwhton10igx0gqroctvqhvh0ij7rk3oznowx4934mizogxlmb/2b8fkdwryzzq==</latexit> <latexit sha1_base64="irlurv1ny35+dklf1063cgiho2e=">aaaccnicbvc7sgnbfj31genr1djmnag2ht0gkiiqtlgwigaekf3d7orummt24cysejatbfwvgwtfbp0co//gsbkfjh4yojxzd3fu8wloplksb2nufmfxabmwulxdw9/ynle2gzjkbiu6jxgkwh6rwfkidcuuh1ysgaqeh6y3ubz5zqcqkkxhrrrg4aakfzkfuak01dh37s/o4s49sitozldjgslyixqcx2ezwzm2o2bjkltj4fli56sectq65pftjwgsqkgoj1k2bstwbkqeyprdvnqsctgha9kdtqyhcuc66fiudb9opyv9sogxkjxwfydsekg5ddw9grdvl9pespzpayfkp3vtfsajgpbofvkjxyrco15wlwmgig81ivqw/vdm+0qqqnr7rv2cpx3ylgluyrbmn8el6kverwhton10igx0gqroctvqhvh0ij7rk3oznowx4934mizogxlmb/2b8fkdwryzzq==</latexit>!23

24 High T case: A = 2 3` L 8 4`6( cl + 12 ) 945c 5 L O(`12 ). NB: order c 0!! To the order we considered, it vanishes in the thermodynamic limit, i.e., L! 1 but with, ` fixed!24

25 Low T case: A = h 32q 2 i 15c + 24q3 5c + 64q4 ` 5c + O(q5 ) L h 128(c 16)q 2 32(c 24)q c c 2 + +O(`8) 4 NB: order 1/c!! 256(c 40)q4 105c 2 + O(q 5 ) i ` L 6!25

26 <latexit sha1_base64="elbpmooo5mf8pzysaivqotoqipc=">aaacf3icdvdlsgmxfm3uv62vuzduuovgqkxfufyvunelfewd2loymds2ndmzkoxyhvkln/6kgxekunwdf2omd6iva4hdoedyc48bcqa043xamyxfpewv7gpubx1jc8ve3qkreukknsq4ke2xkoasgjpmmkmzleb8l0pdhz6nfumwpgiiunajedo+6qesxyjrruraxbagoy39+ljyhox0qoie0xshz3rmhzra03q+n3ttqqnojigdx2rmfdau1a790fyejxwinoveqvbjcxunjlizyihjtsmfiafd0oewoqhxqxxi8v0jpjckh3tcmhdopfbnj2likzxyxzp0ir6on14q/uw1it077cqsccmnaz0s6kuca4htkrdh0pp5ybbcjtn/xxrajkhavjmbl+f/uj8qlgy/oi6uk9m6smgp7andveinqiwuubxveex36be9oxfrwxqyxq23strjtwd20tdy719fzp9z</latexit> <latexit sha1_base64="elbpmooo5mf8pzysaivqotoqipc=">aaacf3icdvdlsgmxfm3uv62vuzduuovgqkxfufyvunelfewd2loymds2ndmzkoxyhvkln/6kgxekunwdf2omd6iva4hdoedyc48bcqa043xamyxfpewv7gpubx1jc8ve3qkreukknsq4ke2xkoasgjpmmkmzleb8l0pdhz6nfumwpgiiunajedo+6qesxyjrruraxbagoy39+ljyhox0qoie0xshz3rmhzra03q+n3ttqqnojigdx2rmfdau1a790fyejxwinoveqvbjcxunjlizyihjtsmfiafd0oewoqhxqxxi8v0jpjckh3tcmhdopfbnj2likzxyxzp0ir6on14q/uw1it077cqsccmnaz0s6kuca4htkrdh0pp5ybbcjtn/xxrajkhavjmbl+f/uj8qlgy/oi6uk9m6smgp7andveinqiwuubxveex36be9oxfrwxqyxq23strjtwd20tdy719fzp9z</latexit> <latexit sha1_base64="elbpmooo5mf8pzysaivqotoqipc=">aaacf3icdvdlsgmxfm3uv62vuzduuovgqkxfufyvunelfewd2loymds2ndmzkoxyhvkln/6kgxekunwdf2omd6iva4hdoedyc48bcqa043xamyxfpewv7gpubx1jc8ve3qkreukknsq4ke2xkoasgjpmmkmzleb8l0pdhz6nfumwpgiiunajedo+6qesxyjrruraxbagoy39+ljyhox0qoie0xshz3rmhzra03q+n3ttqqnojigdx2rmfdau1a790fyejxwinoveqvbjcxunjlizyihjtsmfiafd0oewoqhxqxxi8v0jpjckh3tcmhdopfbnj2likzxyxzp0ir6on14q/uw1it077cqsccmnaz0s6kuca4htkrdh0pp5ybbcjtn/xxrajkhavjmbl+f/uj8qlgy/oi6uk9m6smgp7andveinqiwuubxveex36be9oxfrwxqyxq23strjtwd20tdy719fzp9z</latexit> <latexit sha1_base64="elbpmooo5mf8pzysaivqotoqipc=">aaacf3icdvdlsgmxfm3uv62vuzduuovgqkxfufyvunelfewd2loymds2ndmzkoxyhvkln/6kgxekunwdf2omd6iva4hdoedyc48bcqa043xamyxfpewv7gpubx1jc8ve3qkreukknsq4ke2xkoasgjpmmkmzleb8l0pdhz6nfumwpgiiunajedo+6qesxyjrruraxbagoy39+ljyhox0qoie0xshz3rmhzra03q+n3ttqqnojigdx2rmfdau1a790fyejxwinoveqvbjcxunjlizyihjtsmfiafd0oewoqhxqxxi8v0jpjckh3tcmhdopfbnj2likzxyxzp0ir6on14q/uw1it077cqsccmnaz0s6kuca4htkrdh0pp5ybbcjtn/xxrajkhavjmbl+f/uj8qlgy/oi6uk9m6smgp7andveinqiwuubxveex36be9oxfrwxqyxq23strjtwd20tdy719fzp9z</latexit> Long-interval cases For the long-interval case, we use the fact S B,i = S A,i, and the know result for thermal state EE: S B = c 3 log sinh ` + cl 3 I(1 exp 2 ` ), c.f. Chen & Wu, , where I(x) is the mutual inforamtion of two intervals on a complex plane with cross ration x. High T case: NB: order c 0 correction!! B = cl (4 L 7 )` L ` (32 2 L L)` L O(`11, 1/c). Low T case: B = S(L) h 32 ( 2 + L 2 )(4 2 + L 2 ) i ` 15L 5 q 2 + O(q 3 ) L 4 + O(`5).!26

27 Micro-canonical ensemble (MCE) We can also consider the Holevo information in the MCE with fixed high energy E, i.e., p i = (E E i), with (E) = (E) r cl 6E I 1 We can obtain the multi-point functions/ees in MCE by inverse Laplace transform of the ones in Canonical ensemble (CE), similar to the trick deriving the Cardy s formula for (E). Short-interval: A = 3`4[ cl(i 3 I 1 ) + 24 I 2 ] LI O(`12 ) Long-interval: B = S(L)+O(`12 ) NB: order c!! r 2 cle 3, There are also non-universal contribution from non-identity families.!27

28 c=30 here!!28

29 Implications & Conclusions We see that the Holevo information got 1/c corrections but is consistently vanishing as interval size goes to zero. Recall that Holevo information=average of relative entropy over all states, including descendants. Since most of states are descendants (at least in the thermodynamic limit), many of which we shown are non-geometric and should contribute with higher order of c. Thus, our results implies there is a subtle cancellation of non-geometric state s contributions in the average of relative entropy to yield order c Holevo information. This implies that even the non-perturbative quantum gravity effect looks wild, it is miraculously to yield averaging classical geometry of BTZ.!29