# Holographic entanglement negativity for disjoint intervals in \(AdS_3/CFT_2\)

## Abstract

We advance a holographic construction for the entanglement negativity of bipartite mixed state configurations of two disjoint intervals in \((1+1)\) dimensional conformal field theories (\(CFT_{1+1}\)) through the \(AdS_3/CFT_2\) correspondence. Our construction constitutes the large central charge analysis of the entanglement negativity for mixed states under consideration and involves a specific algebraic sum of bulk space like geodesics anchored on appropriate intervals in the dual \(CFT_{1+1}\). The construction is utilized to compute the holographic entanglement negativity for such mixed states in \(CFT_{1+1}\)s dual to bulk pure \(AdS_3\) geometries and BTZ black holes respectively. Our analysis exactly reproduces the universal features of corresponding replica technique results in the large central charge limit which serves as a consistency check.

## 1 Introduction

Quantum entanglement has attracted intense focus recently in diverse disciplines from condensed matter physics to issues of quantum gravity [1, 2, 3, 4, 5, 6]. The entanglement for bipartite pure states may be characterized by the entanglement entropy which is defined as the von Neumann entropy of the reduced density matrix for the subsystem under consideration. However entanglement entropy fails to be a viable measure for the characterization of mixed state entanglement as it incorporates correlations irrelevant to the specific bipartite system in question. This significant issue in quantum information theory was addressed by Vidal and Werner in [7], where they introduced a computable measure termed entanglement negativity which characterized the upper bound on the distillable entanglement for the bipartite mixed state.^{1} The entanglement negativity was defined as the logarithm of the trace norm of the partially transposed density matrix with respect to one of the subsystems of a bipartite system. It was shown by Plenio in [8] that the entanglement negativity was not convex but was an entanglement monotone under local operations and classical communication (LOCC).

In [9, 10, 11, 12] the authors advanced a comprehensive procedure to compute the entanglement entropy in \((1+1)\) dimensional conformal field theories (\(CFT_{1+1}\)) employing a replica technique. For configurations involving multiple disjoint intervals the entanglement entropy computed through the replica technique receives non universal contributions which depend on the full operator content of the \(CFT_{1+1}\). It was later shown in [13, 14] that these non universal contributions were sub leading in the large central charge limit. Subsequently a variant of the above replica technique could be utilized to compute the entanglement negativity of various bipartite pure and mixed state configurations in a \(CFT_{1+1}\) [15, 16, 17]. Interestingly the entanglement negativity for a bipartite pure state was given by the Rényi entropy of order half in conformity with quantum information theory. Following this, in [18] the large central charge limit of the entanglement negativity for a mixed state configuration of two disjoint intervals was investigated. Interestingly in this case the entanglement negativity is non universal in general except when the two intervals are in proximity where a universal contribution may be extracted in the large central charge limit [18]. Remarkably through a monodromy analysis it could be numerically demonstrated that the entanglement negativity exhibited a phase transition [18, 19].

In the context of the *AdS* / *CFT* correspondence Ryu and Takayanagi (RT) [20, 21] advanced a holographic conjecture to describe the universal part of the entanglement entropy of a subsystem in a dual \(CFT_d\). This was given by the area of the co dimension two static minimal surface in the bulk \(AdS_{d+1}\) geometry, homologous to the subsystem. This development attracted intense interest in obtaining the holographic entanglement entropy of bipartite systems described by dual \(CFT_d\)s (for a detailed review see [21, 22, 23, 24] and references therein). A covariant generalization of the Ryu–Takayanagi conjecture was subsequently advanced in [25] by Hubeny, Rangamani and Takayanagi (HRT). A proof of the RT conjecture was established from the bulk perspective initially in the context of \(AdS_3/CFT_2\) framework and later generalized to the \(AdS_{d+1}/CFT_d\) scenario in [26, 27, 28, 29]. Subsequently the covariant HRT conjecture was proved in [30]. The developments described above naturally led to the interesting issue of a corresponding holographic characterization for the universal part of the entanglement negativity of \(CFT_d\)s in the \(AdS_{d+1}/CFT_d\) scenario. A holographic computation of the entanglement negativity for the pure vacuum state of a \(CFT_d\) dual to a bulk pure \(AdS_{d+1}\) geometry was given in [31]. Despite this progress a clear holographic construction for the entanglement negativity of bipartite states in \(CFT_d\)s remained an outstanding issue.

In [32, 33] two of the present authors (VM and GS) proposed a holographic entanglement negativity conjecture and its covariant generalization for bipartite states in the \(AdS_3/CFT_2\) scenario. This was substantiated by a large central charge analysis of the entanglement negativity of the \(CFT_{1+1}\) utilizing the monodromy technique in [34]. This proposal was subsequently extended in [35] to higher dimensions in the context of the \(AdS_{d+1}/CFT_d\). However a bulk proof of this conjecture along the lines of [28, 29] remains an outstanding issue. Following [32, 33] in [36, 37] a holographic entanglement negativity conjecture and its covariant extension was proposed for bipartite mixed state configurations of adjacent intervals in dual \(CFT_{1+1}\)s. Subsequently through the \(AdS_{d+1}/CFT_d\) framework a higher dimensional generalization of the above construction was proposed in [38]. This could be applied to investigate such mixed states in \(CFT_d\)s dual to the bulk pure \(AdS_{d+1}\) geometry, \(AdS_{d+1}\)-Schwarzschild black hole and the \(AdS_{d+1}\)-Reissner Nordstrom black hole in [38, 39].

As mentioned earlier the entanglement negativity for the mixed state of two disjoint intervals which is in general non universal exhibits an interesting behavior in the large central charge limit where a universal contribution may be isolated. A holographic description from a bulk perspective for this intriguing behavior of the entanglement negativity is a fascinating open issue. In this article we address this interesting issue and propose a holographic entanglement negativity conjecture for such mixed state configuration of two disjoint intervals in the \(AdS_3/CFT_2\) scenario. To this end we utilize the large central charge analysis presented in [18] to extract the universal part of the entanglement negativity for the mixed state in question both at zero and finite temperatures and also for a finite size system in \(CFT_{1+1}\). Interestingly we observe that the entanglement negativity for the mixed states in question are cut off independent. Following this analysis it is possible to establish a holographic conjecture characterizing the universal part of the entanglement negativity of the mixed state in question. Our construction involves a specific algebraic sum of the lengths of bulk space like geodesics anchored on intervals appropriate to the configuration of the mixed state in question and reduces to an algebraic sum of the holographic mutual informations between particular combinations of the intervals.^{2} Application of our conjecture to the examples of such mixed state configurations in \(CFT_{1+1}\) dual to bulk pure \(AdS_3\) geometries and the Euclidean BTZ black hole substantiates our conjecture and constitute significant consistency checks. Interestingly in the limit of the intervals being adjacent we are able to exactly reproduce the universal features of results described in [16, 17, 36] from our holographic construction for the disjoint case.

This article is organized as follows. In Sect. 2 we briefly review the computation of entanglement negativity for bipartite mixed state configuration of two disjoint intervals in a \(CFT_{1+1}\). In Sect. 3 we describe the large central charge analysis for the entanglement negativity utilizing the monodromy technique. Subsequently in Sect. 4 we advance a holographic entanglement negativity conjecture for the mixed state of disjoint intervals using the large central charge results and describe its application to various scenarios. Finally, we summarize our results in Sect. 5 and present our conclusions.

## 2 Entanglement negativity

*B*. The bipartite system \( A \equiv A_1 \cup A_2 \) in a mixed state, described by the reduced density matrix \( \rho _A = {\text {Tr}}_B (\rho ) \), may then be obtained by tracing over the subsystem \( B \equiv A^c \). It is assumed that the Hilbert space for the bipartite system

*A*may be expressed as a direct product \( {{{\mathcal {H}}}} = {{{\mathcal {H}}}}_1 \otimes {{{\mathcal {H}}}}_2 \) where \( {{{\mathcal {H}}}}_1 \) and \( {{{\mathcal {H}}}}_2 \) respectively describe the Hilbert spaces for the subsystems \( A_1 \) and \( A_2 \). The partial transpose of the reduced density matrix \( \rho _A \) with respect to \( A_2 \), is defined as

### 2.1 Entanglement negativity in a \(CFT_{1+1}\)

## 3 Entanglement negativity at large *c*

In this section we briefly review the large central charge analysis for the four point twist correlator in Eq. (4) above through the monodromy technique [13, 18, 42, 43, 44, 45, 46, 47, 48, 49, 50]. Our discussion will be focused on the explicit form of the four point twist correlator when the disjoint intervals depicted in Fig. 1 are in proximity as described in [18]. In this instance the entanglement negativity for the bipartite zero temperature mixed state configuration of disjoint intervals may be obtained explicitly in the large central charge limit.

*f*in the above expression may then be determined through the monodromy properties of the solutions to the second order differential equation given as

*T*(

*z*) , to satisfy

*f*in Eq. (7) may then be determined through the integration of the expression \( \partial f / \partial x = c_2 \).

### 3.1 Entanglement negativity of disjoint intervals in the \( x \rightarrow 1 \) channel

*c*limit, using Eq. (7) we then arrive at the following

^{3}In this case

*T*(

*z*) in Eq. (9) is given by

*f*in Eq. (13) as follows

^{4}

*c*limit as

Note that in [13], it was demonstrated that in the large central charge limit, the entanglement entropy of the configuration described by two disjoint intervals exhibits a phase transition from its value in the *s*-channel (\( x \rightarrow 0 \)) to its value in the *t*-channel (\( x \rightarrow 1 \)) at \( x = \frac{ 1 }{ 2 } \) . These correspond to different geodesic combinations in the dual bulk \(AdS_3\) geometry as predicted by the holographic proposal of Ryu and Takayanagi. Interestingly, in [18], the authors showed that a similar phase transition occurs for the entanglement negativity of the mixed state of two disjoint intervals as well. It was numerically demonstrated that this phase transition occurs for the negativity in the large *c* limit from its value in the *s*-channel (\( x \rightarrow 0 \)) where it vanishes, to its value in the *t*-channel which is given by Eq. (19). However, it was not possible to determine the exact value of the cross ratio *x* at which the phase transition occurs. Recently, in [19] it was shown that there exists a correspondence between the classical geometries dual to the Rényi entanglement entropy and the Rényi entanglement negativity which suggests that the phase transition once again occurs at \( x = \frac{ 1 }{ 2 } \).

### 3.2 Entanglement negativity for disjoint intervals in vacuum at large *c*

Interestingly the above result is cut off independent unlike the case for the mixed state of adjacent intervals as described in [16]. Furthermore it is to be noted that the above expression in Eq. (20) exactly reproduces the universal part of the entanglement negativity in the adjacent interval limit provided the separation length \( l_s \) is identified with the cut off as \( l_s \rightarrow a \) [16].

### 3.3 Entanglement negativity for disjoint intervals in vacuum for a finite size system at large *c*

*L*with a periodic boundary condition, the entanglement negativity for the mixed state in question, may be obtained from Eq. (19) through the conformal transformation \( z \rightarrow w = \left( i L / 2 \pi \right) \ln z \), from the complex plane to the cylinder of circumference

*L*[16]. Under this conformal map the four point twist correlator in Eq. (4) transforms as

*c*for the zero temperature mixed state configuration of disjoint intervals in proximity for this case is then obtained from Eq. (18) as follows

### 3.4 Entanglement negativity for disjoint intervals at a finite temperature at large *c*

*T*, the entanglement negativity at large

*c*may be obtained as above through the conformal map \( z \rightarrow w = \left( \beta / 2 \pi \right) \ln z \) from the complex plane to the cylinder where the Euclidean time direction has now been compactified to a circle with circumference \( \beta \equiv 1 / T \) [17]. As before, employing Eqs. (3), (4) and (21), with the transformation described above, the entanglement negativity at large

*c*, for the mixed state configuration of disjoint intervals in proximity at a finite temperature may be computed from Eq. (18) as follows

In Fig. 2a we graphically describe the behavior of the entanglement negativity as a function of the separation \( l_s \) between the disjoint intervals for the three scenarios described above. It is observed in all the cases that the entanglement negativity decreases as we increase separation length \( l_s \) between the intervals, which is in conformity with quantum information results. In Fig. 2b the entanglement negativity has been plotted against the length of the first interval \(l_1\). In this plot we observe that the entanglement negativity increases with the interval size and eventually saturates for large \( l_1 \) in all the cases.

Having presented the entanglement negativity of the mixed state under consideration in a \(CFT_{1+1}\) for the three different scenarios we now proceed to establish a holographic conjecture involving the dual bulk \(AdS_3\) geometry which correctly reproduces the above large central charge results.

## 4 Holographic entanglement negativity for disjoint intervals

*R*is the \(AdS_3\) length scale.

Note that as explained in Sect. 3.1, the result given in Eq. (19) is valid for the values of the cross ratio \( \frac{ 1 }{ 2 }< x < 1 \) which implies that the geodesic combination given above is also valid only in this regime. On the other hand in the regime described by \( 0< x < \frac{ 1 }{ 2 } \), the entanglement negativity is zero characterizing the phase transition at \(x=\frac{1}{2}\).

^{5}

### 4.1 Holographic entanglement negativity for two disjoint intervals in vacuum

*R*is the \(AdS_3\) radius. The length of the bulk space like geodesic anchored on an interval \( \gamma \) ( of length \( l_{\gamma } \)), in this geometry described by Eq. (30), may then be expressed as [20, 21, 54, 55]

*a*being the UV cut off. Using the expression in Eq. (31), the holographic entanglement negativity for the mixed state under consideration may now be obtained from Eq. (27) as follows

### 4.2 Holographic entanglement negativity of disjoint intervals in vacuum for a finite size system

*L*with a periodic boundary condition. For this purpose it is required to consider the \(CFT_{1+1}\) on an infinite cylinder with the spatial direction compactified on a circle of circumference

*L*, as discussed earlier in Sect. 3.3. The corresponding dual bulk configuration in this case is the pure \(AdS_3\) space time expressed in global coordinates as follows [20, 21, 54, 55]

*a*is once again the UV cut off. Utilizing the above expression given in Eq. (34) it is now possible to obtain the holographic entanglement negativity for the mixed state in question from Eq. (27) as follows

### 4.3 Holographic entanglement negativity for two disjoint intervals at a finite temperature

*T*is the temperature. The corresponding dual bulk \(AdS_3\) configuration is now the Euclidean BTZ black hole (black string) at a Hawking temperature

*T*[20, 21, 54, 55]. The metric for the Euclidean BTZ black hole is given as

*a*is the UV cut off. As earlier we now utilize the above expression in Eq. (37) to obtain the holographic entanglement negativity for the finite temperature mixed state under consideration from Eq. (27) as

## 5 Summary and conclusions

To summarize we have established a holographic entanglement negativity conjecture involving the bulk geometry for bipartite mixed states of disjoint intervals in a dual \(CFT_{1+1}\) through the \(AdS_3/CFT_2\) correspondence. In this context we have utilized the large central charge analysis involving the monodromy technique for the entanglement negativity of such mixed states in a holographic \(CFT_{1+1}\). Using the large central charge result we have established a holographic construction for the entanglement negativity of the above mixed state configurations, which involves a specific algebraic sum of the lengths of bulk space like geodesics anchored on appropriate intervals. Interestingly the holographic entanglement negativity reduces to an algebraic sum of the holographic mutual informations relevant to a certain combination of the intervals confirming other similar results in the literature.

Application of our conjecture exactly reproduces the entanglement negativity for bipartite mixed states of disjoint intervals in proximity for a holographic \(CFT_{1+1}\) obtained through the replica technique, in the large central charge limit and serves as a strong consistency check. In this context we have computed the holographic entanglement negativity for such bipartite mixed states in a \(CFT_{1+1}\) for various scenarios. These involve the zero temperature mixed state of disjoint intervals in proximity for both infinite and finite size systems described by a holographic \(CFT_{1+1}\). The corresponding bulk dual configurations are described by the pure \(AdS_3\) geometry in the Poincaré and global coordinates respectively. Furthermore we have extended our analysis to obtain the holographic entanglement negativity for the corresponding finite temperature mixed state of such disjoint intervals in a \(CFT_{1+1}\) dual to a bulk Euclidean BTZ black hole (black string). Interestingly in each of the scenarios described above we have been able to exactly reproduce the corresponding results for adjacent intervals in a \(CFT_{1+1}\) through the adjacent limit which provides further consistency check for our construction.

We would like to mention here that although our holographic entanglement negativity conjecture has been substantiated through applications to specific examples of zero and finite temperature mixed states under consideration, a bulk proof for our conjecture along the lines of [28] is a non trivial open issue that needs attention. Furthermore our analysis suggests a higher dimensional generalization of the holographic entanglement negativity conjecture for such mixed states of disjoint intervals in proximity through the \(AdS_{d+1}/CFT_d\) framework. Such an extension would involve a similar algebraic sum of bulk codimension two static minimal surfaces anchored on appropriate subsystems to describe the holographic entanglement negativity for such mixed states under consideration. Naturally such a higher dimensional generalization needs to be substantiated through consistency checks involving applications to specific examples and also a bulk proof along the lines of [29]. Our holographic entanglement negativity conjecture is expected to provide interesting insights into diverse physical phenomena such as topological phases, quantum phase transitions, strongly coupled theories in condensed matter physics and critical issues in quantum gravity, which involve such mixed state entanglement. These constitute fascinating open issues for future investigations.

## Footnotes

- 1.
Distillable entanglement characterizes the amount of pure entanglement that can be extracted from the state in question using only LOCC.

- 2.
- 3.
Note that the negative value of scaling dimension of the twist field in the replica limit has to be understood as an analytic continuation.

- 4.
- 5.
There seems to be an intriguing connection between the holographic entanglement negativity and the holographic mutual information although they are distinct quantities in quantum information theory. For the adjacent intervals they are identical and this is also reported in the literature in [52, 53].

## References

- 1.M. Van Raamsdonk, Comments on quantum gravity and entanglement. arXiv:0907.2939 [hep-th]
- 2.M. Van Raamsdonk, Building up spacetime with quantum entanglement. Gen. Relativ. Grav.
**42**, 2323–2329 (2010). arXiv:1005.3035 [hep-th]. https://doi.org/10.1142/S0218271810018529 - 3.M. Van Raamsdonk, Building up spacetime with quantum entanglement Int. J. Mod. Phys.
**D19**, 2429 (2010)ADSzbMATHGoogle Scholar - 4.B. Swingle, Entanglement renormalization and holography. Phys. Rev. D
**86**, 065007 (2012). arXiv:0905.1317 [cond-mat.str-el]ADSCrossRefGoogle Scholar - 5.J. Maldacena, L. Susskind, Cool horizons for entangled black holes. Fortsch. Phys.
**61**, 781–811 (2013). arXiv:1306.0533 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 6.T. Hartman, J. Maldacena, Time evolution of entanglement entropy from black hole interiors. JHEP
**05**, 014 (2013). arXiv:1303.1080 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 7.G. Vidal, R.F. Werner, Computable measure of entanglement. Phys. Rev. A
**65**, 032314 (2002). arXiv:quant-ph/0102117 ADSCrossRefGoogle Scholar - 8.M.B. Plenio, Logarithmic negativity: a full entanglement monotone that is not convex. Phys. Rev. Lett.
**95**(9), 090503 (2005). arXiv:quant-ph/0505071 ADSCrossRefGoogle Scholar - 9.P. Calabrese, J.L. Cardy, Entanglement entropy and quantum field theory. J. Stat. Mech.
**0406**, P06002 (2004). arXiv:hep-th/0405152 zbMATHGoogle Scholar - 10.P. Calabrese, J. Cardy, Entanglement entropy and conformal field theory. J. Phys. A
**42**, 504005 (2009). arXiv:0905.4013 [cond-mat.stat-mech]MathSciNetCrossRefGoogle Scholar - 11.P. Calabrese, J. Cardy, E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory. J. Stat. Mech.
**0911**, P11001 (2009). arXiv:0905.2069 [hep-th]MathSciNetCrossRefGoogle Scholar - 12.P. Calabrese, J. Cardy, E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II. J. Stat. Mech.
**1101**, P01021 (2011). arXiv:1011.5482 [hep-th]MathSciNetGoogle Scholar - 13.T. Hartman, Entanglement entropy at large central charge. arXiv:1303.6955 [hep-th]
- 14.M. Headrick, Entanglement Renyi entropies in holographic theories. Phys. Rev. D
**82**, 126010 (2010). arXiv:1006.0047 [hep-th]ADSCrossRefGoogle Scholar - 15.P. Calabrese, J. Cardy, E. Tonni, Entanglement negativity in quantum field theory. Phys. Rev. Lett.
**109**, 130502 (2012). arXiv:1206.3092 [cond-mat.stat-mech]ADSCrossRefGoogle Scholar - 16.P. Calabrese, J. Cardy, E. Tonni, Entanglement negativity in extended systems: a field theoretical approach. J. Stat. Mech.
**1302**, P02008 (2013). arXiv:1210.5359 [cond-mat.stat-mech]MathSciNetCrossRefGoogle Scholar - 17.P. Calabrese, J. Cardy, E. Tonni, Finite temperature entanglement negativity in conformal field theory. J. Phys. A
**48**(1), 015006 (2015). arXiv:1408.3043 [cond-mat.stat-mech]ADSMathSciNetCrossRefGoogle Scholar - 18.M. Kulaxizi, A. Parnachev, G. Policastro, Conformal blocks and negativity at large central charge. JHEP
**09**, 010 (2014). arXiv:1407.0324 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 19.X. Dong, S. Maguire, A. Maloney, H. Maxfield, Phase transitions in 3D gravity and fractal dimension. JHEP
**05**, 080 (2018). arXiv:1802.07275 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 20.S. Ryu, T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett.
**96**, 181602 (2006). arXiv:hep-th/0603001 ADSMathSciNetCrossRefGoogle Scholar - 21.S. Ryu, T. Takayanagi, Aspects of holographic entanglement entropy. JHEP
**08**, 045 (2006). arXiv:hep-th/0605073 ADSMathSciNetCrossRefGoogle Scholar - 22.T. Nishioka, S. Ryu, T. Takayanagi, Holographic entanglement entropy: an overview. J. Phys. A
**42**, 504008 (2009). arXiv:0905.0932 [hep-th]MathSciNetCrossRefGoogle Scholar - 23.M. Rangamani, T. Takayanagi, Holographic entanglement entropy. Lect. Notes Phys.
**931**, 1–246 (2017). arXiv:1609.01287 [hep-th]MathSciNetCrossRefGoogle Scholar - 24.T. Nishioka, Entanglement entropy: holography and renormalization group. arXiv:1801.10352 [hep-th]
- 25.V.E. Hubeny, M. Rangamani, T. Takayanagi, A Covariant holographic entanglement entropy proposal. JHEP
**07**, 062 (2007). arXiv:0705.0016 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 26.D.V. Fursaev, Proof of the holographic formula for entanglement entropy. JHEP
**09**, 018 (2006). arXiv:hep-th/0606184 ADSMathSciNetCrossRefGoogle Scholar - 27.H. Casini, M. Huerta, R.C. Myers, Towards a derivation of holographic entanglement entropy. JHEP
**05**, 036 (2011). arXiv:1102.0440 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 28.T. Faulkner, The entanglement Renyi entropies of disjoint intervals in AdS/CFT. arXiv:1303.7221 [hep-th]
- 29.A. Lewkowycz, J. Maldacena, Generalized gravitational entropy. JHEP
**08**, 090 (2013). arXiv:1304.4926 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 30.X. Dong, A. Lewkowycz, M. Rangamani, Deriving covariant holographic entanglement. JHEP
**11**, 028 (2016). arXiv:1607.07506 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 31.M. Rangamani, M. Rota, Comments on entanglement negativity in holographic field theories. JHEP
**10**, 060 (2014). arXiv:1406.6989 [hep-th]ADSCrossRefGoogle Scholar - 32.P. Chaturvedi, V. Malvimat, G. Sengupta, Holographic quantum entanglement negativity. JHEP
**05**, 172 (2018). arXiv:1609.06609 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 33.P. Chaturvedi, V. Malvimat, G. Sengupta, Covariant holographic entanglement negativity. Eur. Phys. J. C
**78**(9), 776 (2018). arXiv:1611.00593 [hep-th]ADSCrossRefGoogle Scholar - 34.V. Malvimat, G. Sengupta, Entanglement negativity at large central charge. arXiv:1712.02288 [hep-th]
- 35.P. Chaturvedi, V. Malvimat, G. Sengupta, Entanglement negativity, holography and black holes. Eur. Phys. J. C
**78**(6), 499 (2018). arXiv:1602.01147 [hep-th]ADSCrossRefGoogle Scholar - 36.P. Jain, V. Malvimat, S. Mondal, G. Sengupta, Holographic entanglement negativity conjecture for adjacent intervals in AdS\(_3\)/CFT\(_2\). arXiv:1707.08293 [hep-th]
- 37.P. Jain, V. Malvimat, S. Mondal, G. Sengupta, Covariant holographic entanglement negativity conjecture for adjacent subsystems in \({{\rm AdS}}_{3}/{{\rm CFT}}_{2}\). arXiv:1710.06138 [hep-th]
- 38.P. Jain, V. Malvimat, S. Mondal, G. Sengupta, Holographic entanglement negativity for adjacent subsystems in AdS\(_{d+1}\)/CFT\(_{d}\). Eur. Phys. J. Plus
**133**(8), 300 (2018). arXiv:1708.00612 [hep-th]Google Scholar - 39.P. Jain, V. Malvimat, S. Mondal, G. Sengupta, Holographic entanglement negativity for conformal field theories with a conserved charge. Eur. Phys. J. C
**78**(11), 908 (2018). arXiv:1804.09078 [hep-th]ADSCrossRefGoogle Scholar - 40.H. Shapourian, S. Ryu, Finite-temperature entanglement negativity of Fermi surface. arXiv:1807.09808 [cond-mat.stat-mech]
- 41.J. Kudler-Flam, S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories. arXiv:1808.00446 [hep-th]
- 42.A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B
**241**, 333–380 (1984)ADSMathSciNetCrossRefGoogle Scholar - 43.A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block. Theor. Math. Phys.
**73**, 1088–1093 (1987)CrossRefGoogle Scholar - 44.A.B. Zamolodchikov, A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory. Nucl. Phys. B
**477**, 577–605 (1996). arXiv:hep-th/9506136 ADSMathSciNetCrossRefGoogle Scholar - 45.D. Harlow, J. Maltz, E. Witten, Analytic continuation of Liouville theory. JHEP
**12**, 071 (2011). arXiv:1108.4417 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 46.A.L. Fitzpatrick, J. Kaplan, M.T. Walters, Universality of long-distance AdS physics from the CFT bootstrap. JHEP
**08**, 145 (2014). arXiv:1403.6829 [hep-th]ADSCrossRefGoogle Scholar - 47.K.B. Alkalaev, V.A. Belavin, Monodromic vs geodesic computation of Virasoro classical conformal blocks. Nucl. Phys. B
**904**, 367–385 (2016). arXiv:1510.06685 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 48.A.L. Fitzpatrick, J. Kaplan, M.T. Walters, Virasoro conformal blocks and thermality from classical background fields. JHEP
**11**, 200 (2015). arXiv:1501.05315 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 49.E. Perlmutter, Virasoro conformal blocks in closed form. JHEP
**08**, 088 (2015). arXiv:1502.07742 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 50.P. Ruggiero, E. Tonni, P. Calabrese, Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks. J. Stat. Mech.
**1811**(11), 113101 (2018). arXiv:1805.05975 [cond-mat.stat-mech]MathSciNetCrossRefGoogle Scholar - 51.J.D. Brown, M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity. Commun. Math. Phys.
**104**, 207–226 (1986)ADSMathSciNetCrossRefGoogle Scholar - 52.A. Coser, E. Tonni, P. Calabrese, Entanglement negativity after a global quantum quench. J. Stat. Mech.
**1412**(12), P12017 (2014). arXiv:1410.0900 [cond-mat.stat-mech]MathSciNetCrossRefGoogle Scholar - 53.X. Wen, P.-Y. Chang, S. Ryu, Entanglement negativity after a local quantum quench in conformal field theories. Phys. Rev. B
**92**(7), 075109 (2015). arXiv:1501.00568 [cond-mat.stat-mech]ADSCrossRefGoogle Scholar - 54.M. Cadoni, M. Melis, Holographic entanglement entropy of the BTZ black hole. Found. Phys.
**40**, 638–657 (2010). arXiv:0907.1559 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 55.M. Cadoni, M. Melis, Entanglement entropy of AdS black holes. Entropy
**12**(11), 2244–2267 (2010)ADSMathSciNetCrossRefGoogle Scholar

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