Derivation of holographic negativity in AdS 3 /CFT 2
DDerivation of holographic negativity in AdS /CFT Yuya Kusuki ∗ Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP),Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan.
Jonah Kudler-Flam † and Shinsei Ryu ‡ Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA (Dated: September 23, 2019)We present a derivation of the holographic dual of logarithmic negativity in AdS /CFT thatwas recently conjectured in [Phys. Rev. D 99, 106014 (2019)]. This is given by the area of anextremal cosmic brane that terminates on the boundary of the entanglement wedge. The derivationconsists of relating the recently introduced R´enyi reflected entropy to the logarithmic negativityin holographic conformal field theories. Furthermore, we clarify previously mysterious aspects ofnegativity at large central charge seen in conformal blocks and comment on generalizations to genericdimensions, dynamical settings, and quantum corrections. Introduction.—
The von Neumann entropy of the re-duced density matrix is an excellent measure of the en-tanglement between bipartite subsystems in a pure state.In particular, it has played a major role in the under-standing of how bulk geometry holographically emergesfrom microscopic degrees of freedom in the AdS/CFTcorrespondence. This is due to the fact that the von Neu-mann entropy of a boundary subregion A is equal to thearea of the extremal bulk surface γ A that is homologousto A , [1–3] S vN ( ρ A ) = area( γ A )4 G N , (1)where G N is the bulk Newton constant. For mixed states ρ AB , the von Neumann entropy fails to serve as a cor-relation measure because S vN ( ρ A ) (cid:54) = S vN ( ρ B ) and thecorresponding mutual information [ I AB ≡ S vN ( ρ A ) + S vN ( ρ B ) − S vN ( ρ AB )] fails to quantify the entanglementbetween A and B . Rather, the mutual information cap-tures classical correlations, such as thermodynamic en-tropy, on top of the quantum correlations.In general, one may be interested in characterizing theentanglement structure of many-body systems in mixedstates. In particular, it is an interesting question to askif and how mixed state entanglement manifests itself ge-ometrically in the bulk in AdS/CFT. For this purpose,we study the logarithmic negativity, a suitable measureof entanglement for mixed states [4], based on the posi-tive partial transpose criterion [5–10]. While most mixedstate entanglement measures are defined in terms of in-tractable optimization procedures, the logarithmic nega-tivity is operationally defined and generally computable.For a bipartite density matrix ρ AB , the partial transposeis an operation that transposes just one of the subsystems (cid:104) i A , j B | ρ T B AB | k A , l B (cid:105) = (cid:104) i A , l B | ρ AB | k A , j B (cid:105) , (2) ∗ [email protected] † jkudlerfl[email protected] ‡ [email protected] where i A , j B , k A , and l B are bases for subsystems A and B . The logarithmic negativity is then defined as E ( ρ AB ) ≡ log (cid:12)(cid:12)(cid:12) ρ T B AB (cid:12)(cid:12)(cid:12) , (3)where |O| = Tr √OO † is the trace norm.With motivations from quantum error-correcting codesand preliminary examples in 2D conformal field theory,two of the authors conjectured that logarithmic negativ-ity in holographic conformal field theories is dual to abackreacted entanglement wedge cross section in asymp-totically antide Sitter (AdS) space-times [11], giving aconcrete proposal for how mixed state entanglement isgeometrized in the bulk. In this Letter, we provide aproof of the conjecture. For symmetric configurations,we provide direct comparisons between conformal fieldtheory (CFT) computations of negativity and entangle-ment wedge cross sections. Holographic reflected entropy.—
Additional quantitieshave been shown to be related to the entanglement wedgecross section in holographic theories, including the entan-glement of purification, odd entropy, and reflected en-tropy [12–15]. While each quantity is intriguing in itsown right, we will use the reflected entropy for our pur-poses in deriving the holographic dual for logarithmicnegativity in AdS /CFT .We start by reviewing the construction of the reflectedentropy and then state the conjecture for the holographicdual of logarithmic negativity (13). Generically, mixeddensity matrices may be decomposed into a sum of purestates ρ AB = (cid:88) a p a ρ ( a ) AB . (4)We may then perform a Schmidt decomposition on eachpure state ρ ( a ) AB = (cid:88) i,j (cid:113) λ ( a ) i λ ( a ) j | i ( a ) (cid:105) (cid:104) j ( a ) | A ⊗ | i ( a ) (cid:105) (cid:104) j ( a ) | B . (5) a r X i v : . [ h e p - t h ] S e p A canonical purification of the original mixed state maythen be constructed in the doubled Hilbert space H A ⊗H A ∗ ⊗ H B ⊗ H B ∗ |√ ρ AB (cid:105) ≡ (cid:88) a,i,j (cid:113) p ( a ) λ ( a ) i λ ( a ) j | i ( a ) (cid:105) A | j ( a ) (cid:105) A ∗ | i ( a ) (cid:105) B | j ( a ) (cid:105) B ∗ . (6)The reflected entropy is then defined by [15] S R ( A : B ) ≡ S vN ( ρ AA ∗ ) , (7)where S vN is the von Neumann entropy and ρ AA ∗ is thereduced density matrix on H A ⊗ H A ∗ . In holographicconformal field theories, this was shown to be dual totwice the area of the entanglement wedge cross section[15] S R = 2 E W . (8)This was proven for time reflection symmetric states us-ing the Lewkowycz-Maldacena gravitational replica trick[16]. It is natural to think that such a correspondenceholds in generic time-dependent settings. In fact, manynontrivial checks of the time-dependent conjecture havebeen performed in Ref. [17].We consider the R´enyi reflected entropies S ( n ) R ( A : B ) ≡ S ( n ) ( ρ AA ∗ ) . (9)As shown by Dong [18], the R´enyi entropies are relatedto the modular entropies which are dual to cosmic branesin the bulk gravity theory˜ S ( n ) ≡ n ∂ n (cid:18) n − n S ( n ) (cid:19) = area(Cosmic Brane n )4 G N . (10)The cosmic branes are codimension-two objects with ten-sion T n = n − nG N . (11)The modular reflected entropies are then dual to twicethe area of cosmic branes that terminate on the entan-glement wedge [19]˜ S ( n ) R ≡ n ∂ n (cid:18) n − n S ( n ) R (cid:19) = 2 area(Cosmic Brane n )4 G N (cid:12)(cid:12)(cid:12) E W . (12)While this was not explicitly stated in Ref. [15], it is asimple corollary of their gravitational construction of thestate |√ ρ AB (cid:105) . Holographic negativity conjecture.—
In Ref. [11], it wasconjectured that the logarithmic negativity in holo-graphic conformal field theories is dual to a backreact-ing entanglement wedge cross section. The conjecture may be concisely stated in terms of the R´enyi reflectedentropy as E = S (1 / R E = X d E W , (14)where X d is a constant that depends on the dimension ofthe CFT X d = (cid:18) x d − d (cid:0) x d (cid:1) − (cid:19) , (15) x d = 2 d (cid:32) (cid:114) − d d (cid:33) . (16)We will consider 2D CFTs where X = 3 /
2. We willexplicitly compute the holographic negativity for disjointintervals in the vacuum state and a single interval atfinite temperature from conformal blocks, finding preciseagreement with (14).
Deriving holographic negativity.—
We now provide asimple derivation of (13) in 2D CFTs. We consider twoarbitrary subsystems A = n A (cid:91) i =1 [ u i , v i ] , B = n B (cid:91) i =1 [ w i , y i ] . (17)The logarithmic negativity in the vacuum state (andgeneric conformal transformations from the vacuum) maybe computed by a correlation function of twist fields E = lim n e → log (cid:42) n A (cid:89) i ( σ n e ( u i )¯ σ n e ( v i )) n B (cid:89) i (¯ σ n e ( w i ) σ n e ( y i )) (cid:43) . (18)See Ref. [21] for a thorough exposition of computing neg-ativity in conformal field theory. The twist fields haveconformal dimensions h n e = ¯ h n e = c (cid:18) n e − n e (cid:19) , (19) h (2) n e = ¯ h (2) n e = c (cid:18) n e − n e (cid:19) , (20)where h (2) n e are the double twist fields that arise whenfusing two twist fields of the same chirality. We compare(18) to the correlation function of generalized twist fieldsthat computes half of the R´enyi reflected entropy S (1 / R / m → lim n → / log (cid:42) n A (cid:89) i (cid:16) σ g A ( u i ) σ g − A ( v i ) (cid:17) × n B (cid:89) i (cid:16) σ g B ( w i ) σ g − B ( y i ) (cid:17) (cid:43) CFT ⊗ mn . (21)These generalized twist fields have the action of movingfields between sheets in two directions labeled by m and n . The g B g − A twist field does not appear in the correla-tion function, rather in the conformal block, as it is thelowest weight primary operator in the operator productexpansion (OPE) of g B and g − A . For more precise def-initions, see Ref. [15]. The generalized twist fields haveconformal dimensions h g B = h g − A = cn ( m − m , (22) h g B g − A = 2 c ( n − n . (23)The conformal dimensions, positions, and dominant in-termediate channels of the operators in (18) and (21) pre-cisely match. Thus, in the limit of large central charge,we confirm (13) for the class of states that may be ob-tained by conformal transformations from the vacuum.For completely generic states that include primary oper-ator insertions, we are also able to confirm (13), thoughwe leave the details to the Supplemental Material. Symmetric examples.—
In symmetric configurations,we may use the simplification of (14). First, we focus onthe negativity of disjoint intervals in the vacuum. Thenegativity is a conformally invariant quantity [22] andcomputed by E = lim n e → log (cid:104) σ n e ( ∞ )¯ σ n e (1)¯ σ n e ( x, ¯ x ) σ n e (0) (cid:105) , (24)where we set the two intervals to A = [0 , x ] and B =[1 , ∞ ] for simplicity. We restrict ourselves to the holo-graphic CFTs where the correlator can be approximatedby a single conformal block. In the x → x → σ n e , has a conformal dimension of the order of c ; there-fore, the explicit form is unknown. Nevertheless, we canuse the Zamolodchikov recursion relation [24, 25] to eval-uate the conformal block numerically to arbitrarily highprecision.On this background, the negativity should be com-pared to the entanglement wedge cross section, whichis given by [12] E W = c √ − x − √ − x , < x < / , , / < x < . (25)In Fig. 1, we show the entanglement wedge cross sec-tion and the negativity calculated by the Zamolodchikovrecursion relation. One can immediately find that thenegativity perfectly matches the minimal entanglementwedge cross section. This result resolves the mysteriesthat arose when computing (24) using less direct ap-proaches in Refs. [11, 23, 26]. FIG. 1. The blue line is the negativity that comes from theVirasoro block computed to order q using Zamolodchikov’srecursion relation where q is the elliptic nome. The yellow lineshows the minimal entanglement wedge cross section (25).Here we set c = 10 and (cid:15) = 10 − , and in this plot, we dividethese quantities by c to rescale. We progress to the negativity of a single interval oflength l at finite temperature β − , which can be calcu-lated by [27] [28] E = lim L →∞ lim n e → log (cid:16) (cid:104) σ n e ( − L )¯ σ n e ( − l ) σ n e (0)¯ σ n e ( L ) (cid:105) β (cid:17) × (2 (cid:15) ) h (2) ne (cid:16) C σ ne ¯ σ ne σ ne (cid:17) = c (cid:18) β π(cid:15) e πlβ (cid:19) + lim L →∞ lim n e → log (cid:104) σ n e ( ∞ )¯ σ n e (1) σ n e ( x, ¯ x )¯ σ n e (0) (cid:105) , (26)where (cid:15) is an UV regulator, and the cross ratio is givenby x −−−−→ L →∞ e − πlβ . (27)We may fix the OPE coefficient in the replica limit byconsidering the pure state limits of (18) and (21)lim n e → C σ ne ¯ σ ne σ ne = 2 c/ . (28)In the holographic CFT, this four-point function canbe approximated by a single conformal block. The dom-inant conformal block has two candidates.(i) s channel: In the semiclassical limit, the s channelblock is simplified because the only contribution to theintermediate state is the primary exchange [29]. Thus,the approximated correlator is given by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:3041) (cid:3280) (cid:2870) (cid:3041) (cid:3280) (cid:3041) (cid:3280) (cid:2870)(cid:3041) (cid:3280) (cid:3041) (cid:3280) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) C σ ne ¯ σ ne σ ne (cid:17) −−−−→ n e → (cid:12)(cid:12) x c (cid:12)(cid:12) − c . (29)Substituting this into (26), we obtain E = c βπ(cid:15) . (30)(ii) t channel: The t channel block is just the HHLL block [29, 30], whose explicit form is given by F HHLL (0 | − z ) = z h L (1 − z ) h H F LLHH (0 | − z )= z h L ( δ +1) (1 − z ) h H (cid:18) − z δ δ (cid:19) − h L , (31)where δ = (cid:113) − c h H . Using this, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:3041) (cid:3280) (cid:2870) (cid:3041) (cid:3280) (cid:3041) (cid:3280) (cid:2870)(cid:3041) (cid:3280) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −−−−→ n e → (1 − x ) c . (32)This leads to the negativity E = c (cid:18) βπ(cid:15) sinh πlβ (cid:19) . (33)Thus, we can conclude that the negativity at finitetemperature is E = min (cid:34) c βπ(cid:15) , c (cid:18) βπ(cid:15) sinh πlβ (cid:19)(cid:35) . (34)This result perfectly matches the minimal entanglementwedge cross section in the Ba˜nados Teitelboim Zanelliblack hole geometry [12] upon using (14). Discussion.—
In this Letter, we have derived the holo-graphic dual of logarithmic negativity in AdS /CFT ,confirming the conjecture from Ref. [11]. There are sev-eral directions for future understanding. In particular, it is important to prove the holographic conjecture in higherdimensions. 3D gravity and 2D conformal field theory arequite special, so we are far from a complete derivation.Such a derivation may require a clever implementation ofthe gravitational replica trick for the partially transposeddensity matrix or a comparison of correlation functionsof higher-dimensional twist operators for negativity andR´enyi reflected entropy. Furthermore, the reflected en-tropy is not proven to (though it is believed to) be dual tothe entanglement wedge cross section in generic dynami-cal settings. Because we have used the reflected entropyas a crutch in our derivation, the dynamical proposal forlogarithmic negativity in AdS /CFT is also still con-jectural. Finally, it would be interesting to investigatequantum corrections to the holographic formula in thesense of Ref. [31]. General expectations and hints fromerror-correcting codes lead us to guess that the leadingcorrection comes from the logarithmic negativity betweenthe bulk fields on either side of the entanglement wedgecross section. ACKNOWLEDGMENTS
We thank Souvik Dutta, Tom Faulkner, and TadashiTakayanagi for useful discussions. S.R. is supported bya Simons Investigator Grant from the Simons Founda-tion. Y.K. is supported by a JSPS fellowship. J.K.F. andS.R. thank the Yukawa Institute for Theoretical Physics(YITP-T-19-03) for hospitality during the completion ofthis work. [1] S. Ryu and T. Takayanagi, Phys. Rev. Lett. , 181602(2006), arXiv:hep-th/0603001 [hep-th].[2] S. Ryu and T. Takayanagi, Journal of High EnergyPhysics , 045 (2006), arXiv:hep-th/0605073 [hep-th].[3] V. E. Hubeny, M. Rangamani, and T. Takayanagi,Journal of High Energy Physics , 062 (2007),arXiv:0705.0016 [hep-th].[4] It is important to note that logarithmic negativity onlyprovides an upper bound on the distillable entanglement.[5] A. Peres, Phys. Rev. Lett. , 1413 (1996).[6] M. Horodecki, P. Horodecki, and R. Horodecki, PhysicsLetters A , 1 (1996), arXiv:quant-ph/9605038 [quant-ph].[7] J. Eisert and M. B. Plenio, Journal of Modern Optics ,145 (1999), arXiv:quant-ph/9807034 [quant-ph].[8] R. Simon, Phys. Rev. Lett. , 2726 (2000), arXiv:quant-ph/9909044 [quant-ph].[9] G. Vidal and R. F. Werner, Phys. Rev. A , 032314(2002), arXiv:quant-ph/0102117 [quant-ph]. [10] M. B. Plenio, Phys. Rev. Lett. , 090503 (2005),arXiv:quant-ph/0505071 [quant-ph].[11] J. Kudler-Flam and S. Ryu, Phys. Rev. D , 106014(2019).[12] K. Umemoto and T. Takayanagi, Nature Physics , 573(2018), arXiv:1708.09393 [hep-th].[13] P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel,and B. Swingle, Journal of High Energy Physics ,98 (2018), arXiv:1709.07424 [hep-th].[14] K. Tamaoka, Phys. Rev. Lett. , 141601 (2019).[15] S. Dutta and T. Faulkner, arXiv e-prints ,arXiv:1905.00577 (2019), arXiv:1905.00577 [hep-th].[16] A. Lewkowycz and J. Maldacena, Journal of High EnergyPhysics , 90 (2013), arXiv:1304.4926 [hep-th].[17] Y. Kusuki and K. Tamaoka, arXiv e-prints ,arXiv:1907.06646 (2019), arXiv:1907.06646 [hep-th].[18] X. Dong, Nature Communications , 12472 (2016),arXiv:1601.06788 [hep-th].[19] In higher dimensions, the Engelhardt-Wall gluing proce-dure may be complicated by shockwaves associated withthe conical singularities [32]. [20] Strictly speaking, this equality is only proven for thevacuum state [33, 34]. However, it is expected (and haspassed multiple tests) that it holds whenever both thesubsystem configuration and quantum state are spheri-cally symmetric. We stress that it does not hold in genericsituations.[21] P. Calabrese, J. Cardy, and E. Tonni, Journal of Sta-tistical Mechanics: Theory and Experiment , 02008(2013), arXiv:1210.5359 [cond-mat.stat-mech].[22] P. Calabrese, J. Cardy, and E. Tonni, Phys. Rev. Lett. , 130502 (2012), arXiv:1206.3092 [cond-mat.stat-mech].[23] M. Kulaxizi, A. Parnachev, and G. Policastro, Journalof High Energy Physics , 10 (2014), arXiv:1407.0324[hep-th].[24] A. B. Zamolodchikov, Theoretical and MathematicalPhysics , 1088 (1987).[25] A. B. Zamolodchikov, Commun. Math. Phys. , 419(1984).[26] J. Kudler-Flam, M. Nozaki, S. Ryu, and M. TianTan, arXiv e-prints , arXiv:1906.07639 (2019),arXiv:1906.07639 [hep-th].[27] The factor of (2 (cid:15) ) h (2) ne has been largely ignored in theliterature but is necessary. It comes from the OPE of the twist fields in the original six-point correlation functioninto double twist fields in the resulting four-point func-tion.[28] P. Calabrese, J. Cardy, and E. Tonni, Journal ofPhysics A Mathematical General , 015006 (2015),arXiv:1408.3043 [cond-mat.stat-mech].[29] A. L. Fitzpatrick, J. Kaplan, and M. T. Walters,Journal of High Energy Physics , 145 (2014),arXiv:1403.6829 [hep-th].[30] A. L. Fitzpatrick, J. Kaplan, and M. T. Walters,Journal of High Energy Physics , 200 (2015),arXiv:1501.05315 [hep-th].[31] T. Faulkner, A. Lewkowycz, and J. Maldacena, Journalof High Energy Physics , 74 (2013), arXiv:1307.2892[hep-th].[32] N. Engelhardt and A. C. Wall, Journal of High EnergyPhysics , 160 (2019), arXiv:1806.01281 [hep-th].[33] L.-Y. Hung, R. C. Myers, M. Smolkin, and A. Yale,Journal of High Energy Physics , 47 (2011),arXiv:1110.1084 [hep-th].[34] M. Rangamani and M. Rota, Journal of High EnergyPhysics , 60 (2014), arXiv:1406.6989 [hep-th].[35] Y. Kusuki and K. Tamaoka, arXiv e-prints ,arXiv:1909.06790 (2019), arXiv:1909.06790 [hep-th]. SUPPLEMENTAL MATERIAL
We consider normalized states created by generic operator insertions | ψ (cid:105) = (cid:81) n O i O i ( x i ) | (cid:105) (cid:68)(cid:81) n O i [ O i ( x i )] † O i ( x i ) (cid:69) / (35)The negativity is then computed as E = lim n e → log (cid:68)(cid:81) n A i ( σ n e ( u i )¯ σ n e ( v i )) (cid:81) n B i (¯ σ n e ( w i ) σ n e ( y i )) (cid:81) n O i (cid:2) O ⊗ n e i ( x i ) (cid:3) † O ⊗ n e i ( x i ) (cid:69) CF T ⊗ ne (cid:16)(cid:68)(cid:81) n O i [ O i ( x i )] † O i ( x i ) (cid:69)(cid:17) n e . (36)This should be compared to the R´enyi reflected entropy which has a different normalization due to the canonicalpurification procedure and computed by the following path integral (see Ref. [15] for further details) S ( n ) R = 11 − n lim m e → log Z n,m e ( Z ,m e ) n , (37)where the replica partition function is defined by Z n,m e ≡ (cid:42) n A (cid:89) i ( σ g A ( u i ) σ g − A ( v i )) n B (cid:89) i ( σ g B ( w i ) σ g − B ( y i )) n O (cid:89) i [ O ⊗ m e ni ( x i )] † O ⊗ m e ni ( x i ) (cid:43) CFT ⊗ men . (38)We use the notation m e to remind the reader that we are taking the limit from even integers to one. In the following,we will show the negativity matches a half of the R´enyi reflected entropy at index 1/2, S (1 / R m e → log Z / ,m e ( Z ,m e ) / . (39)These correlation functions appear, for example, when studying the reflected entropy following a local quantum quenchor in a heavy state [17]. There is a subtlety that we must discuss regarding the structure of the primary operators inthe replica theory. For the negativity, the operator O ⊗ n e has the tensor structure O ⊗ n e = O ⊗ n e / ⊗ O ⊗ n e / , (40)where the subscript 1(2) implies that the operator acts like the local primary operator O on the odd (even) numberedsheets and the identity on the even (odd) sheets. Analogously, this is how the double twist field (for n e ) decomposesin general σ n e = σ (1) n e / ⊗ σ (2) n e / . (41)For the reflected entropy, the decomposition is O ⊗ m e n = O ⊗ n (0) ⊗ · · · ⊗ O ⊗ n ( m e / ⊗ . . . (42)where the subscript labels the replica copy in the m e direction where the operator acts. The key point is that in the m e → O ⊗ m e n does not reduce to O ⊗ n as one would naively expect but rather the squarelim m e → O ⊗ m e n = O ⊗ n (0) ⊗ O ⊗ n (1 / . (43)Similarly, this is how the twist operators from Ref. [15] reduce in the limitlim m e → σ g − A g B = σ (0) n ⊗ σ (1 / n . (44)As a result from this squaring, we obtain the squared correlation function,lim m e → Z ,m e = (cid:42) n O (cid:89) i [ O i ( x i )] † O i ( x i ) (cid:43) (45)One can find that substituting this into the denominator of (39) completely reproduces the denominator of (36). Notethat this is crucial for the R´enyi reflected entropy to reduce to twice the R´enyi entropy for pure states, which is atrue statement for any quantum state. We also stress that these operators do not interact with one another, just asthe operators for negativity on opposite parity sheets did not interact. This leads to a decoupling of the componentsof the conformal blocks e.g. (cid:104) σ (1) n e / ⊗ σ (2) n e / | O ⊗ n e / ⊗ O ⊗ n e / | σ (1) n e / ⊗ σ (2) n e / (cid:105) = (cid:104) σ (1) n e / | O ⊗ n e / | σ (1) n e / (cid:105) (cid:104) σ (2) n e / | O ⊗ n e / | σ (2) n e / (cid:105) , (46) (cid:104) σ (0) n ⊗ σ ( m e / n | O ⊗ n (0) ⊗ O ⊗ n ( m e / | σ (0) n ⊗ σ ( m e / n (cid:105) = (cid:104) σ (0) n | O ⊗ n (0) | σ (0) n (cid:105) (cid:104) σ ( m e / n | O ⊗ n ( m e / | σ ( m e / n (cid:105) . (47)Then, in the appropriate n e → , n → / , m e → , c → ∞ limit, both the numerators of (36) and (39) are describedby precisely the same conformal blocks, hence confirming (13) of the main text for generic states in AdS /CFT2