Holographic entanglement negativity conjecture for adjacent intervals in $AdS_3/CFT_2$
Parul Jain Vinay Malvimat Sayid Mondal Gautam Sengupta
hep-th


 





parul.jain@ca.infn.it
Dipartimento di Fisica, Università di Cagliari
Cittadella Universitaria, 09042 Monserrato, Italy
 INFN, Sezione di Cagliari, Italy



 vinaymm@iitk.ac.in 
Department of Physics,

Indian Institute of Technology Kanpur,

Kanpur 208016, INDIA

 sayidphy@iitk.ac.in
Department of Physics,

Indian Institute of Technology Kanpur,

Kanpur 208016, INDIA

sengupta@iitk.ac.in
Department of Physics,

Indian Institute of Technology Kanpur,

Kanpur 208016, INDIA











We propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in (1+1)-dimensional dual conformal field theories through the AdS/CFT correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture
we obtain the entanglement negativity of adjacent intervals in zero and finite temperature (1+1)-dimensional holographic conformal field theories dual to the bulk AdS_3 vacuum and the Euclidean BTZ black hole respectively. Our holographic conjecture exactly reproduces the conformal field theory results obtained through the replica technique, in the large central charge limit.  We briefly elucidate the corresponding issue for the AdS_d+1/CFT_d scenario.




Holographic entanglement negativity conjecture for adjacent intervals in 
AdS_3/CFT_2 
    Gautam Sengupta
    December 30, 2023
======================================================================================



§ INTRODUCTION




Quantum entanglement in recent times has impacted an expansive list of theoretical issues from condensed matter physics to quantum gravity through the holographic AdS/CFT correspondence <cit.>. This geometric connection has allowed the characterization of quantum entanglement in extended systems like holographic conformal field theories. For bipartite quantum systems in a pure state this involves the entanglement entropy which is defined as the von Neumann entropy of the reduced density matrix. In a series of interesting communications Calabrese et al advanced a comprehensive procedure to compute the entanglement entropy of (1+1)-dimensional conformal field theories (CFT_1+1) <cit.> utilizing the replica technique. 

Following <cit.>, in a seminal work Ryu and Takayanagi proposed a holographic characterization of the entanglement entropy in d-dimensional conformal field theories (CFT_d), involving bulk dual AdS_d+1 geometries through the AdS/CFT correspondence <cit.> ( for an extensive review see <cit.>). According to the Ryu and Takayanagi ( RT) conjecture the universal part of the entanglement entropy of a subsystem in a dual CFT_d was described by the area of a co-dimension two bulk AdS_d+1 static minimal surface homologous to the subsystem. For the
AdS_3/CFT_2 scenario the static minimal surface reduces to a space like geodesic in the bulk AdS_3 geometry anchored on the appropriate spatial interval in the dual CFT_1+1.
The holographic entanglement entropy obtained from the RT conjecture exactly reproduces the corresponding CFT_1+1 results obtained through the replica technique in the large central 
charge limit.




It is well known however in quantum information theory that the entanglement entropy ceases to be a valid measure for the characterization of mixed state entanglement where it receives contributions from irrelevant correlations. This is a complex issue in quantum information theory and necessitates the introduction of suitable entanglement measures for the distillable entanglement. In a seminal communication Vidal and Werner <cit.> addressed this issue and proposed a computable measure termed entanglement negativity which provides an upper bound on the distillable entanglement whose non convexity was subsequently demonstrated by Plenio in <cit.>. In the recent past Calabrese et al in <cit.> utilized an alternative replica technique to compute the entanglement negativity for mixed states in both zero and finite temperature CFT_1+1. 

Naturally the above developments lead to the significant issue of a holographic description for the entanglement negativity involving the bulk geometry for dual conformal field theories in the AdS/CFT frame work <cit.>. In a recent interesting communication two of the present authors (VM and GS) in the collaborations <cit.>(CMS),  proposed a holographic conjecture for the entanglement negativity of mixed states in holographic CFT_ds. The conjecture involves a specific  algebraic sum of the areas of co-dimension two bulk extremal surfaces ( geodesic lengths in AdS_3/CFT_2 ) anchored on the corresponding subsystems. Note that the proposed conjecture was in relation to the entanglement negativity for a singly connected subsystem in an infinite system. Remarkably, this conjecture precisely characterizes the upper bound on  the distillable entanglement through the elimination of the thermal contributions for finite temperature mixed states. Furthermore it could be shown that the holographic entanglement negativity was proportional  to the sum of the holographic mutual information for the relevant partitioning of the system.

In this article motivated by the CMS conjecture, <cit.> we advance an independent holographic entanglement negativity conjecture for the mixed states of adjacent intervals in both 
zero and finite temperature CFT_1+1. In the context of the AdS_3/CFT_2 scenario we establish that the holographic entanglement negativity for this configuration is described by a specific algebraic sum of the bulk geodesic lengths which is proportional to the holographic mutual information between the intervals. Note that our case is distinct from <cit.> where the entanglement entropy characterizing the entanglement between the intervals and the rest of the system was considered. On the other hand here we consider the entanglement negativity which characterizes the entanglement between these adjacent intervals. Our computation for the entanglement negativity is expected to have significant applications to various entanglement issues in diverse fields.

This article is organized as follows. In section <ref> we briefly review the 
characterization of entanglement negativity in quantum  information theory and describe the computation of this quantity for mixed states of adjacent intervals in a CFT_1+1. In the subsequent section <ref> we establish our holographic conjecture for the entanglement negativity which exactly reproduces the replica technique results for the CFT_1+1 in the large central charge limit. We then summarize our results in the last section <ref>.

 





§ ENTANGLEMENT NEGATIVITY



In this section we review the characterization of entanglement negativity in quantum information theory and the computation of this entanglement measure for mixed state configuration of adjacent intervals in a CFT_1+1 for both zero and finite temperatures
as well as for a finite size system.



 §.§ Entanglement negativity in quantum information theory

We begin by briefly  reviewing the definition of entanglement negativity in quantum information theory as described in <cit.>. In this context it is required to consider a tri-partition of a quantum system into the subsystems A_1, A_2 and B, with A=A_1∪ A_2 and B=A^c representing the rest of the system. The Hilbert space ℋ for the bipartite system A may be written as a direct product  ℋ=ℋ_1 ⊗ℋ_2 of the corresponding Hilbert spaces of the individual subsystems A_1 and A_2 respectively. The reduced density matrix of the subsystem A is given as ρ_A=Tr_A^c(ρ). The partial transpose of this reduced density matrix with respect to A_2 may be defined as follows

    ⟨ e^(1)_ie^(2)_j|ρ_A^T_2|e^(1)_ke^(2)_l⟩ = 
    ⟨ e^(1)_ie^(2)_l|ρ_A|e^(1)_ke^(2)_j⟩,

where |e^(1)_i⟩ and |e^(2)_j⟩ are the bases for the respective Hilbert spaces
ℋ_1 and  ℋ_2.
The entanglement negativity between the subsystems A_1 and A_2 is then described as
follows

    ℰ = lnTr|ρ_A^T_2|,

where the trace norm Tr|ρ_A^T_2| is given by the sum of absolute eigenvalues of the partially transposed reduced density matrix ρ_A^T_2. 




 §.§ Entanglement negativity in CFT_1+1


Here we briefly review the computation of the entanglement negativity for mixed states of two disjoint intervals in a CFT_1+1 as described in <cit.>. To this end the authors considered the tri-partition of the system as depicted in Fig. (<ref>), where
A_1 describes the interval (A_1∈[u_1,v_1]) of length l_1 and A_2 is the interval (A_2∈[u_2,v_2]) of length l_2 while B=A^c represents the rest of the system.

The entanglement negativity between the subsystems A_1 and A_2 is obtained through a suitable replica technique which involves the construction of the quantity Tr(ρ_A^T_2)^n_e for even n_e and considering the analytic continuation of the even sequence to n_e→1 as [Note that the explicit construction of this analytic continuation is complex and remains an open issue except for some simple conformal field theories <cit.> (See also <cit.>).] 

    ℰ = lim_n_e → 1 lnTr(ρ_A^T_2)^n_e.

It was shown that the quantity Tr(ρ_A^T_2)^n_e in the above expression may be expressed as a four point function of the twist fields on the complex plane as follows

    Tr(ρ_A^T_2)^n_e = 
    ⟨𝒯_n_e(u_1)𝒯_n_e(v_1)𝒯_n_e(u_2)𝒯_n_e(v_2)⟩_ℂ.





 §.§ Adjacent intervals in vacuum

The formulation described above was utilized to characterize the 
entanglement negativity for mixed states of adjacent intervals in <cit.>. This configuration is obtained through the limit v_1→ u_2 for Fig. (<ref>) which reduces to the configuration depicted in Fig. (<ref>). 




The four point function eq. (<ref>) in this limit effectively reduces to a three point function of the twist fields as follows

    Tr(ρ_A^T_2)^n_e = 
    ⟨𝒯_n_e(-l_1)𝒯^2_n_e(0)𝒯_n_e(l_2)⟩,

The entanglement negativity for the adjacent intervals at zero temperature is obtained by employing eqs. (<ref>) and (<ref>), which may be expressed as

    ℰ = c/4ln[l_1l_2/(l_1+l_2)a]+ const,

where a and c are the UV cut off and the central charge of the corresponding CFT. Note that the first term in the above expression is universal whereas the
second `const' term depends on the detailed operator content of the CFT.




 §.§ Adjacent intervals in vacuum for finite size



For a finite sized system of length L with a periodic boundary condition, the entanglement negativity for the adjacent intervals may be computed from the corresponding 
three point function  eq. (<ref>) on a cylinder<cit.>. This is obtained through the conformal map 
z → w = iL/2πln z, from the complex plane to the cylinder with the circumference 
L. The entanglement negativity may then be expressed as

    ℰ =
    c/4ln[(L/π a)sin(π l_1/L)sin(π l_2/L)/sinπ (l_1+l_2)/L] +const.

As earlier the `const' term in the above expression is non universal.





 §.§ Adjacent intervals at finite temperature

An analysis as above may be performed to obtain the entanglement negativity for the mixed state of adjacent intervals in a finite temperature CFT_1+1 which is defined on
a cylinder of circumference β=1/T which is the inverse temperature. This is related to the CFT_1+1 on the complex plane through the conformal 
map z → w = β/2πln z.
Using the above transformation and  eq. (<ref>), the entanglement negativity is given as

    ℰ =
    c/4ln[(β/π a)sinh(π l_1/β)sinh(π l_2/β)/sinhπ (l_1+l_2)/β] +const,

where the `const' as earlier is non universal.



 §.§ Large central charge limit

Here we review the large central charge analysis for the entanglement negativity of the mixed state configuration of adjacent intervals at zero temperature as described in <cit.>. According to the AdS_3/CFT_2 correspondence, the holographic limit in which the bulk spacetime is semiclassical ( Newton's constant G_N^(3)→0) is equivalent to the large central charge limit c→∞ in the dual CFT_1+1 as these quantities are related through the Brown-Henneaux formula c=3 R/2 G_N^(3). In <cit.>, such a large central charge analysis for the four point correlator in eq. (<ref>) was obtained by employing the monodromy technique to determine the conformal blocks providing the dominant contribution. The authors demonstrated that in the limit v_1→ u_2, which corresponds to the case of adjacent intervals depicted in Fig. (<ref>),  the dominant contribution to the four point function in eq. (<ref>) arises from the conformal block corresponding to the intermediate operator 𝒯^2. This led  to the following expression for the large central charge limit of the entanglement negativity as

    E=c/4ln[l_1l_2/(l_1+l_2)a].

Note that this matches exactly with the universal part of the entanglement negativity of the adjacent intervals given in eq. (<ref>), implying that the non-universal constant term in eq. (<ref>) is sub leading in the large central charge limit[ Note that the monodromy technique was also utilized earlier to determine the large central charge limit of the entanglement entropy for multiple disjoint intervals in <cit.> (and through a different technique in <cit.>). The authors demonstrated that in this limit the entanglement entropy is universal and matched exactly with the predictions of the holographic Ryu-Takayanagi conjecture.]. The authors also demonstrated numerically that the entanglement negativity vanishes E→ 0 in the limit of disjoint intervals placed far apart. In the subsequent sections we will establish a holographic conjecture which exactly reproduces the large-c results for the case of the adjacent intervals from the corresponding dual bulk geometry.




§ HOLOGRAPHIC ENTANGLEMENT NEGATIVITY FOR ADJACENT INTERVALS



In this section we establish our holographic entanglement negativity conjecture involving the bulk AdS_3 geometry for the mixed state configuration of adjacent intervals in a dual CFT_1+1. As described in (<ref>) the large central charge analysis for the entanglement negativity clearly indicates the plausibility of a holographic characterization for the entanglement negativity in a dual CFT_1+1 through the AdS_3/CFT_2 correspondence. To this end we consider the adjacent intervals A_1 and A_2 of lengths l_1 and l_2 respectively as depicted in Fig. (<ref>), where the subsystem A_1∪ A_2 is in a mixed state. As described in (<ref>) for this mixed state configuration in the CFT_1+1, the entanglement negativity involves the three point twist correlator in eq. (<ref>) whose form is fixed from conformal invariance as follows

    ⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ
            =  
    c_n_e^2 C_𝒯_n_e𝒯_n_e^2𝒯_n_e/|z_12|^Δ_𝒯^2_n_e
    |z_23|^Δ_𝒯^2_n_e|z_13|^2Δ_𝒯_n_e-Δ_𝒯^2_n_e,

where c_n are normalization constants and C_𝒯_n_e𝒯_n_e^2𝒯_n_e is the structure constant for the relevant twist field OPEs. The scaling dimensions Δ_𝒯_n_e and 
Δ_𝒯_n_e^2 of the twist fields 𝒯_n_e and 𝒯_n_e^2 are respectively given as <cit.>

    Δ_𝒯_n_e =  c/12(n_e - 1/n_e),
    Δ_𝒯_n_e^2 =2Δ_𝒯_n_e/2= c/6(n_e/2- 2/n_e).


In this connection we note here that the two point twist correlators in the CFT_1+1 are given as
as follows <cit.>

    < T_n_e(z_k) T_n_e(z_l)>_ℂ=C_^(1)/|z_kl|^Δ_𝒯_n_e,
    < T^2_n_e(z_k) T^2_n_e(z_l)>_ℂ=C_^(2)/|z_kl|^Δ_𝒯^2_n_e,

where C^(1),  C^(2) are normalization constants.
A close examination of the form of the above two point twist correlator now clearly indicates that the three point twist correlator in eq. (<ref>) may be factorized as follows 

    ⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ=C < T_n_e(z_1) T_n_e(z_3)>_ℂ
       (< T^2_n_e(z_1) T^2_n_e(z_2)>_ℂ< T^2_n_e(z_2) T^2_n_e(z_3)>_ℂ/< T^2_n_e(z_1) T^2_n_e(z_3)>_ℂ)^1/2,

where the constant C is given as follows

    C=c_n_e^2 C_𝒯_n_e𝒯_n_e^2𝒯_n_eC_^(1)( C_^(2))^1/2.

Interestingly, the leading universal part of the three point function eq. (<ref>), which is dominant in the large central charge limit may be expressed as follows

    ⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ=C < T_n_e(z_1) T_n_e(z_3)>_ℂ
             < T_n_e/2(z_1) T_n_e/2(z_2)>_ℂ< T_n_e/2(z_2) T_n_e/2(z_3)>_ℂ/< T_n_e/2(z_1) T_n_e/2(z_3)>_ℂ.

Note that in the above expression in eq. (<ref>) we have used the following factorization in a CFT_1+1 as described in <cit.>

    ⟨ T^2_n_e(z_i) T^2_n_e(z_j)⟩_ℂ=< T_n_e/2(z_i) T_n_e/2(z_j)>^2_ℂ .

We have ignored the non-universal constants c_n and C_𝒯_n_e𝒯_n_e^2𝒯_n_e  in eq. (<ref>), in the large central charge limit as mentioned in (<ref>).

It is well known from the AdS_3/CFT_2 correspondence that the two point function of the twist fields located at z_i and z_j in a CFT_1+1 is related to the length of the dual bulk space like geodesic ℒ_ij  anchored on the corresponding interval as follows <cit.>

    < T_n_e(z_k) T_n_e(z_l)>_ℂ∼ e^-Δ_n_e L_kl/R,
       < T_n_e/2(z_i) T_n_e/2(z_j)>_ℂ∼ e^-Δ_n_e/2 L_ij/R,

where R is the AdS_3 radius. The three point twist correlator in eq. (<ref>), upon employing eqs. (<ref>) and (<ref>) may be expressed as 

    ⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ
          
    =exp[-Δ_𝒯_n_eℒ_13-Δ_𝒯_n_e/2(ℒ_12+ℒ_23-ℒ_13)/R].

Here the corresponding points defining the adjacent intervals
are (z_1=-l_1, z_2=0,z_3=l_2). In the replica limit[Note that the large central charge limit has to be taken prior to the replica limit to arrive at eq.(16). This order of limits is critical and is also true for the case of the entanglement entropy where the scaling dimension of the twist field 
τ_n vanishes in the replica limit and has to be understood in the sense of an analytic continuation (see <cit.>). In a similar fashion the negative scaling dimension of the twist field τ_n_e/2 in the replica limit is in the sense of the non-trivial analytic continuation involved in the replica definition for the entanglement negativity.](n_e→ 1) the scaling dimensions Δ_𝒯_n_e→ 0 and Δ_𝒯_n_e/2→-c/8. Hence by utilizing eqs. (<ref>) and (<ref>) in eq. (<ref>), the holographic entanglement negativity for the mixed state of the adjacent intervals in a dual CFT_1+1 may be expressed in terms of a specific algebraic sum of the lengths of the bulk space like geodesics ℒ_ij (i,j∈ 1,2,3) anchored on appropriate intervals as follows

    ℰ = 3/16G^3_N(ℒ_12+ℒ_23-ℒ_13),

where we have used the Brown-Henneaux formula c=3R/2G^3_N  <cit.>. Upon employing the Ryu-Takayanagi conjecture <cit.> for the holographic entanglement entropy  the eq. (<ref>) above may be expressed   in terms of a specific algebraic sum of the holographic entanglement entropies as follows
 
    ℰ =  3/4(S_A_1+S_A_2-S_A_1∪ A_2)=3/4 I(A_1, A_2).

This algebraic sum of the holographic entanglement entropies is precisely the
holographic mutual information I(A_1,A_2)  between the two intervals for the mixed state configuration under consideration.



It is important to note that the mutual information and the entanglement negativity are entirely distinct entanglement measures in quantum information theory. Negativity is an upper bound on the distillable entanglement and the mutual information is the upper bound on the total correlation in the bipartite system. However the universal parts of these two quantities which are dominant in the holographic (large-c) limit are proportional for the particular mixed state configuration,
although this is not the case for other configurations where a more intricate relation between these quantities are valid in the holographic limit [ Interestingly this proportionality has also been observed in the context of global and local quench in a CFT_1+1 <cit.> for the same configuration utilizing the replica technique. However note that our results are distinct and pertain to a holographic description ( distinct from the replica technique) of entanglement negativity
for the mixed state of a dual CFT_1+1 without quench.].

As explained in the introduction the entanglement negativity in our case characterizes the entanglement between the adjacent intervals for the mixed state configuration of a CFT_1+1 in question. It is important to mention here that eq. (<ref>) provide an elegant holographic construction involving the dual bulk AdS_3 geometry for the entanglement negativity of this mixed state in terms of a specific algebraic sum of the lengths of space like geodesics anchored on the intervals and their union. Interestingly as shown in eq. (<ref>) this may be expressed as an algebraic sum of the corresponding entanglement entropies which reduces to the holographic mutual information for the mixed state configuration in question. Our construction characterizes a significant progress for the critical issue of a holographic description of entanglement negativity in a generic AdS_d+1/CFT_d scenario started by two of the present authors in <cit.>. The present work is an important sequel towards further elucidation of this crucial issue and provides critical insight into the structure of holographic entanglement negativity for generic mixed states in CFT_ds. Our construction for the mixed state configuration under consideration constitutes a significant example 
for the holographic description of entanglement negativity and indicates 
an elegant geometrical technique which may generalized for application to other mixed state configurations in a CFT_1+1.

The construction described here suggests a possible higher dimensional generalization of the holographic negativity conjecture for mixed states of adjacent subsystems in a generic AdS_d+1/CFT_d scenario. The entanglement negativity in this case would involve a similar algebraic sum of the areas of bulk co-dimension two static minimal surfaces anchored on the appropriate subsystems, which may be expressed as follows

    ℰ = 3/16G^(d+1)_N(𝒜_1+𝒜_2-𝒜_12)=3/4 I(A_1, A_2),

where 𝒜_i denotes the area of the bulk co dimension two static minimal surface
anchored on the subsystem A_i. Note that the above expression is also proportional to the holographic mutual information between the two subsystems. However the higher dimensional generalization requires substantiation through specific examples and a possible proof from a bulk perspective which constitute an open issue [ Note that the higher dimensional holographic entanglement negativity construction suggested above has been implemented for mixed state configurations of adjacent subsystems described by rectangular strip geometries in CFT_ds dual to bulk pure AdS_d+1 space time, AdS_d+1-Schwarzschild and RN-AdS_d+1 black holes in <cit.>. The results reproduce certain universal features of entanglement negativity for CFT_1+1s which are consistent with quantum information theory expectations and constitute significant consistency checks for the higher dimensional conjecture.].




 §.§ Adjacent intervals in the vacuum


The holographic conjecture established above in eq. (<ref>) which describes the entanglement negativity for the mixed state configuration of adjacent intervals in terms of the lengths of the bulk space like geodesics anchored on appropriate intervals may now be applied to specific examples. In this context we first obtain the holographic entanglement negativity for the mixed state configuration of adjacent intervals at zero temperature in a CFT_1+1 dual to a bulk pure AdS_3 geometry. The bulk metric in the Poincaré coordinates is given as

    ds^2 = -(r^2/R^2)dt^2 +(r^2/R^2)^ -1 dr^2 + r^2/R^2 dx^2,

where R is the AdS_3 radius and the coordinate x ∈ℝ. The bulk spacelike geodesic length L_γ anchored to the interval γ of length l_γ  in the CFT_1+1 is given as follows <cit.>

    ℒ_γ = 2R lnl_γ/a,

where a is the UV cut off. The holographic entanglement negativity for the adjacent intervals of lengths l_1 and l_2 may now be expressed
by employing  eq. (<ref>) as follows

    ℰ = c/4ln[l_1l_2/(l_1+l_2)a],

where we have used the Brown-Henneaux formula.
Remarkably, the holographic entanglement negativity eq. (<ref>) matches  exactly with the corresponding replica technique result in eq. (<ref>) at the large central charge limit.




 §.§ Adjacent intervals for finite sized systems in vacuum


We now proceed to compute the holographic entanglement negativity for mixed states of adjacent intervals in a finite sized system of length L at zero temperature in a dual CFT_1+1 from our conjecture described in eq. <ref>. In this case as explained in (<ref>) the corresponding CFT_1+1 is defined on an infinite cylinder 
with the spatial direction compactified on a circle of circumference L. The bulk dual configuration for this case is the AdS_3 vacuum in global coordinates with the following metric <cit.> 

    ds^2= R^2(-cosh^2 ρ dt^2 + dρ^2 + sinh^2  ρ d ϕ^2 ),

where ϕ is periodic with a period of 2π. The length of a bulk space like geodesic anchored on the subsystem 
γ of length l_γ in the 
boundary CFT_1+1 is given as <cit.>

    ℒ_γ = 2R ln(L/π asinπ l_γ/L).

The holographic entanglement negativity for the adjacent intervals of lengths l_1 and l_2 may now be computed by substituting the respective geodesic lengths given by eq. (<ref>) in eq. (<ref>), which leads to the following expression

    ℰ = c/4ln[(L/π a)sin(π l_1/L)sin(π l_2/L)/sinπ (l_1+l_2)/L],

where we have utilized the the Brown-Henneaux formula. Interestingly, the holographic entanglement negativity obtained in eq. (<ref>) exactly reproduces the replica technique results eq. (<ref>)  in the large central charge limit. 



 §.§ Adjacent intervals at finite temperature

We now proceed to compute the holographic entanglement negativity for the mixed state of 
adjacent intervals in a dual CFT_1+1 at a finite temperature utilizing our conjecture described in eq. (<ref>). The corresponding CFT_1+1 is defined on a spatially infinite cylinder with the Euclidean time direction compactified on a circle of circumference β.
The appropriate bulk dual configuration is then described by a Euclidean BTZ black hole <cit.> 
at a Hawking temperature T=1/β with the ϕ direction uncompactified[Note that this configuration is actually a BTZ black string.]. The metric for this is given as

    ds^2 = (r^2 - r_h^2)/R^2dτ^2 +R^2/(r^2 - r_h^2)dr^2 + r^2/R^2dϕ^2 ,

where r=r_h denotes the event horizon and τ is the Euclidean time.
The length of a bulk spacelike geodesic anchored on a subsystem γ is given as <cit.>

    ℒ_γ = 2R ln(β/π asinhπ l_γ/β),

where β=2π R^2/r_h is the inverse Hawking temperature. 

It is now possible to compute the holographic entanglement negativity for the mixed state of adjacent intervals of lengths l_1 and l_2 in the finite temperature CFT_1+1 from our conjecture by utilizing the eqs. (<ref>) and (<ref>), which leads to the following expression

    ℰ = c/4ln[(β/π a)sinh(π l_1/β)sinh(π l_2/β)/sinhπ (l_1+l_2)/β],

where once again we have employed the Brown-Henneaux formula. As earlier the holographic entanglement negativity eq. (<ref>) exactly reproduces the replica technique result eq. (<ref>) for the finite temperature  CFT_1+1 in the large central charge limit. This clearly constitutes compelling evidence for the validity of our holographic entanglement negativity conjecture.




§ SUMMARY AND DISCUSSION



To summarize we have established a holographic   entanglement negativity conjecture for mixed states of adjacent intervals in zero and finite temperature dual CFT_1+1 
in the AdS_3/CFT_2 scenario. Note that this configuration is a mixed state as the degrees of freedom of the rest of the system are traced over and hence the corresponding entanglement negativity characterizes the entanglement between the adjacent intervals. Our conjecture involves a specific algebraic sum of the lengths of bulk space like geodesics anchored on respective intervals in the dual holographic CFT_1+1 which reduces to the holographic mutual information between the intervals. 

In the AdS_3/CFT_2 scenario, the bulk dual configuration for the zero temperature CFT_1+1 is the pure AdS_3 vacuum expressed in terms of the Poincare coordinates for the infinite sized system and in the global coordinates for the finite sized system respectively. The finite temperature case on the other hand involves a bulk Euclidean BTZ black hole (black string). In both the cases we have clearly demonstrated that our conjecture exactly reproduces the corresponding replica technique results for both zero and finite temperature dual CFT_1+1s in the large central charge limit. Note however that our holographic conjecture in this article is specific to the mixed state configuration of adjacent intervals. From this perspective a more general holographic entanglement negativity conjecture for the mixed state of disjoint intervals is desirable. This constitutes an interesting open issue for a future investigation.

Naturally the holographic conjecture proposed here in the context of the AdS_3/CFT_2 scenario suggests a possible generalization in a AdS_d+1/CFT_d scenario. The mixed state in this case would be defined by the configuration of adjacent subsystems in the dual holographic CFT_d. The holographic entanglement negativity would involve a specific algebraic sum of the areas of bulk co dimension two extremal surfaces anchored on the respective subsystems, which would be proportional to the holographic mutual information between them. However as such a higher dimensional generalization would require substantiation through specific examples and a possible proof from  a bulk perspective.

We emphasize here that our holographic conjecture provides a simple and elegant method to compute the entanglement negativity for such mixed states in holographic CFTs both at zero and finite temperatures. Naturally this is expected to have significant applications in diverse areas like strongly coupled theories in condensed matter physics, topological phases, quantum phase transitions and critical issues in quantum gravity. These are interesting open questions for the future.



§ ACKNOWLEDGEMENT

Parul Jain would like to thank Prof. Mariano Cadoni for his guidance and 
the Department of Physics, Indian Institute of Technology Kanpur, India for their warm hospitality. Parul Jain's work is financially supported by Università di Cagliari, Italy and INFN, Sezione di Cagliari, Italy.

utphys 

