Holographic Entanglement Negativity for Adjacent Subsystems in $\mathrm{AdS_{d+1}/CFT_{d}}$

Quantum entanglement has developed into an ubiquitous feature of modern fundamental physics in recent times connecting a spectrum of diverse areas from condensed matter physics to quantum gravity. In this regard entanglement entropy has evolved as the most significant and convenient measure to characterize the entanglement of a bipartite quantum system in a pure state. From quantum information theory this is defined as the von Neumann entropy of the reduced density matrix of the corresponding subsystem. For $(1+1)$-dimensional conformal field theories ($CFT_{1+1}$) the entanglement entropy may be computed through the replica technique developed by Calabrese et al in [1, 2].

In the context of the $AdS/CFT$ correspondence, Ryu and Takayanagi in [3, 4] proposed a prescription to compute the entanglement entropy of holographic $CFT$s. For a subsystem described by a spatial region $A$ on the boundary the entanglement entropy is given from this conjecture by the area of the co-dimension two bulk $AdS_{d+1}$ extremal surface anchored on the region $A$. In the recent past this has led to intense research activity into entanglement issues for diverse holographic $CFT$s both at zero and finite temperatures [5, 6, 7, 8, 9, 10, 11].

Although the entanglement entropy was crucial for the characterization of entanglement for bipartite systems in pure states, it was inadequate as an entanglement measure for mixed quantum states. In a seminal work Vidal and Werner [12] introduced a quantity termed entanglement negativity as a computable measure for the upper bound on the distillable entanglement in mixed states. The non convexity property of this entanglement measure was subsequently established in [13]. Interestingly, the authors in [14, 15, 16] computed this quantity in ${CFT_{1+1}}$ employing a variant of the usual replica technique involving a certain four point function of the twist/anti-twist fields. This technique has been extensively employed to compute the entanglement negativity of various mixed state configurations in $CFT_{1+1}$[17, 18, 19, 20, 21, 22].

Naturally, it was critical to establish a holographic description for the entanglement negativity of boundary $CFT$s in terms of the bulk dual geometry in the $AdS/CFT$ scenario. In spite of interesting insights in [23, 24], a clear holographic prescription for the entanglement negativity of $CFT$s remained an unresolved issue. Two of the present authors (VM and GS) in the articles [25, 26, 27] (CMS) proposed a holographic conjecture for the entanglement negativity of such boundary $CFT_{d}$s which exactly reproduced the $CFT_{1+1}$ [16] results in the large central charge limit.

It is important to emphasize that the CMS conjecture mentioned above refers to the entanglement negativity of a single subsystem within an infinite system described by the boundary $CFT_{d}$. In the articles [15, 16], the authors computed the entanglement negativity of a mixed state characterized by two finite intervals $A_{1}$ and $A_{2}$ in a $CFT_{1+1}$ both at zero and finite temperatures. In a recent communication [28], the present authors established an independent holographic conjecture for the entanglement negativity between the two intervals mentioned above in the context of $AdS_{3}/CFT_{2}$. It was shown there that the corresponding entanglement negativity was characterized by a certain algebraic sum of the geodesic lengths in the bulk $AdS_{3}$ space time anchored on the two adjacent intervals, which reduced to the holographic mutual information. Remarkably the holographic entanglement negativity computed from the above prescription exactly reproduced the $CFT_{1+1}$ results both for zero and finite temperatures in the large central charge limit [15, 29]. The holographic conjecture for the entanglement negativity [28] alluded above allowed a direct generalization to the $AdS_{d+1}/CFT_{d}$ scenario. In this case the entanglement negativity could be characterized in terms of an algebraic sum of the areas of bulk co-dimension two extremal surfaces anchored on the respective subsystems in the boundary $CFT_{d}$. As earlier this reduces to the holographic mutual information between the subsystems.

In this article we provide the first non trivial higher dimensional examples in the context of the $AdS_{d+1}/CFT_{d}$ correspondence to establish the efficacy of our conjecture. To this end we consider the mixed state of two adjacent subsystems $A_{1}$ and $A_{2}$ characterized by rectangular strip geometries and compute the corresponding holographic entanglement negativity in $CFT_{d}$s at both zero and finite temperatures. For zero temperature the bulk configuration is described by the $AdS_{d+1}$ vacuum whereas the finite temperature scenario is described by the $AdS_{d+1}$-Schwarzschild black hole. For the finite temperature case the computation of the holographic entanglement negativity requires both a low and a high temperature approximations for the areas of the corresponding bulk extremal surfaces. At low temperatures the leading contribution arises from the $AdS_{d+1}$ vacuum corrected by sub leading thermal contributions. Interestingly for the high temperature case on the other hand the thermal contribution are precisely subtracted out. Hence at the leading order the entanglement negativity at high temperature is characterized by the area of the entangling surface on the boundary.

This article is organized as follows, in section $2$ we briefly review the computation of the holographic entanglement negativity of two adjacent intervals in the $AdS_{3}/CFT_{2}$ scenario described in [28]. In section $3$ we establish the corresponding holographic conjecture for the entanglement negativity of two adjacent subsystems in the context of the $AdS_{d+1}/CFT_{d}$ correspondence. In the subsequent section $4$ we employ our conjecture to compute the holographic entanglement negativity for two adjacent subsystems of rectangular strip geometries at zero temperature from the bulk $AdS_{d+1}$ vacuum. In section $5$ we describe the corresponding computation for the finite temperature scenario from a bulk $AdS_{d+1}$-Schwarzschild black hole. In the final section $6$ we summarize our results and present our conclusions and future open issues.

In this section, we briefly recapitulate the essential elements for the entanglement negativity of mixed states in a $CFT_{1+1}$  [14, 15]. To this end we consider a tripartition in the $CFT_{1+1}$ described by the spatial intervals $A_{1}$, $A_{2}$ and $B$ with $A=A_{1}\cup A_{2}=[u_{1},v_{1}]\cup[u_{2},v_{2}]$, and $B=A^{c}$ represents the rest of the system¹¹1Note that the definition of entanglement negativity requires the concept of purification which involves embedding the given bipartite system ($A_{1}\cup A_{2}=A$) in a mixed state, inside a larger system $B$ such that the full system $A_{1}\cup A_{2}\cup B$ is in a pure quantum state. The larger system $B$ is then traced out to obtain the required mixed state $\rho_{A}$ of the bipartite quantum system. . The reduced density matrix of the subsystem $A$ is defined as $\rho_{A}=\mathrm{Tr}_{B}~\rho$ and $\rho_{A}^{T_{2}}$ is the partial transpose of the reduced density matrix with respect to the interval $A_{2}$. The entanglement negativity  $\mathcal{E}$ is defined as the logarithm of the trace norm of the partially transposed reduced density matrix [12], which is expressed as

$$\mathcal{E}=\ln\mathrm{Tr}|\rho_{A}^{T_{2}}|.$$

(2.1)

The entanglement negativity may now be obtained through a replica technique as discussed in [14, 15] to determine $\mathrm{Tr}~(\rho_{A}^{T_{2}})^{n_{e}}$ and the replica limit is given as the analytic continuation of $n_{e}$ through even sequences to $n_{e}\to 1$. This leads to the following expression for the entanglement negativity

$$\mathcal{E}=\lim_{n_{e}\rightarrow 1}\ln\mathrm{Tr}(\rho_{A}^{T_{2}})^{n_{e}}.$$

(2.2)

For the mixed state described by the two intervals as shown in Fig. (1), the quantity $\mathrm{Tr}(\rho_{A}^{T_{2}})^{n_{e}}$ is given by a four point function of the twist operators on the complex plane from the replica technique described in [14, 15], as follows

$$\mathrm{Tr}(\rho_{A}^{T_{2}})^{n_{e}}=\langle\mathcal{T}_{n_{e}}(u_{1})\overline{\mathcal{T}}_{n_{e}}(v_{1})\overline{\mathcal{T}}_{n_{e}}(u_{2})\mathcal{T}_{n_{e}}(v_{2})\rangle_{\mathbb{C}}.$$

(2.3)

In this section we establish the holographic entanglement negativity conjecture for a mixed state described by two adjacent subsystems in a boundary $CFT_{d}$ in the context of the $AdS_{d+1}/CFT_{d}$ scenario which was alluded to in the article [28]. As mentioned in the Introduction this would involve an algebraic sum of the areas of the bulk co-dimension two extremal surfaces anchored on the respective subsystems. From the conjecture described in [28] the holographic entanglement negativity may then be expressed as follows

$$\mathcal{E}=\frac{3}{16G^{(d+1)}_{N}}\big{(}\mathcal{A}_{1}+\mathcal{A}_{2}-\mathcal{A}_{12}\big{)},$$

(3.1)

where $\mathcal{A}_{i}$ is the extremal area of the co-dimension two surface anchored on the subsystem $A_{i}$. Using the Ryu-Takayanagi prescription [4, 3], it is possible to express the holographic entanglement negativity in the following form

$$\mathcal{E}=\frac{3}{4}(S_{A_{1}}+S_{A_{2}}-S_{A_{1}\cup A_{2}})=\frac{3}{4}[{\cal I}(A_{1},A_{2})],$$

(3.2)

where $S_{A_{i}}$ is the holographic entanglement entropy of the subsystem $A_{i}$. Interestingly the expression in eq. (3.2) is the holographic mutual information between the two adjacent subsystems modulo a constant factor. We are now ready to employ our holographic conjecture to evaluate the entanglement negativity for two adjacent subsystems described by $(d-1)$-dimensional spatial rectangular strip geometries in the boundary $CFT_{d}$ which we will describe in the subsequent sections.

As mentioned above in this section we now proceed to the computation of the holographic entanglement negativity for two adjacent subsystems described by rectangular strip geometries in the boundary $CFT_{d}$ at zero temperature. The corresponding bulk dual geometry in this case is the $AdS_{d+1}$ vacuum space time whose metric in the Poincare coordinates is given as

$$ds^{2}=\frac{1}{z^{2}}\Big{(}-dt^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}+dz^{2}\Big{)},$$

(4.1)

where the $AdS$ radius has been set to $R=1$. The respective rectangular strip geometries of the two subsystems $A_{1}$ and $A_{2}$ depicted in Fig. (3) are then specified as follows

$$x=x^{1}\equiv[-\frac{l_{j}}{2},\frac{l_{j}}{2}]~~x^{i}=[-\frac{L}{2},\frac{L}{2}],~~i=2,...,(d-1),~~j=1,2.$$

(4.2)

We now briefly describe the computation for the areas of bulk co-dimension two extremal surfaces anchored on rectangular strip geometries in the boundary $CFT_{d}$ [4]. The corresponding area functional is expressed in the following way

$${\cal A}=L^{d-2}\int_{-l/2}^{l/2}dx\frac{\sqrt{1+(\frac{dz}{dx})^{2}}}{z^{d-1}}.$$

(4.3)

The Euler-Lagrange equation for the extremization problem is then given as

$$\frac{dz}{dx}=\frac{\sqrt{z_{*}^{2(d-1)}-z^{2(d-1)}}}{z^{d-1}},$$

(4.4)

where $z=z_{*}$ is the turning point of the extremal surface. The extremal area may then be described as

$$\mathcal{A}=\frac{2}{d-2}\Big{(}\frac{L}{a}\Big{)}^{d-2}-2I\Big{(}\frac{L}{z_{*}}\Big{)}^{d-2},$$

(4.5)

where $a$ is the UV cutoff and the constant $I$ is given as

$$I=\frac{1}{d-2}-\int_{0}^{1}\frac{dy}{y^{d-1}}\Big{(}\frac{1}{\sqrt{1-y^{2(d-1)}}}-1\Big{)}=-\frac{\sqrt{\pi}~\Gamma\Big{(}\frac{2-d}{2(d-1)}\Big{)}}{\Gamma\Big{(}\frac{1}{2(d-1)}\Big{)}}.$$

(4.6)

Using the eq.(4.4), eq. (4.5) and eq. (4.6) it is now possible to express the area of the extremal co-dimension two surface as

$${\cal A}={\cal A}_{div}-s_{0}~\Big{(}\frac{L}{l}\Big{)}^{d-2},$$

(4.7)

where the divergent part of the area ${\cal A}_{div}$ and the constant $s_{0}$ are given as

	$\displaystyle s_{0}$	$\displaystyle=\frac{2^{d-1}\pi^{(d-1)/2}}{d-2}\Bigg{(}\frac{\Gamma(\frac{d}{2(d-1)})}{\Gamma(\frac{1}{2(d-1)})}\Bigg{)}^{d-1},$		(4.8)
	$\displaystyle{\cal A}_{div}$	$\displaystyle=\frac{2}{d-2}\Big{(}\frac{L}{a}\Big{)}^{d-2}.$

One may now determine the holographic entanglement negativity $\mathcal{E}$ for the mixed state at zero temperature in the boundary $CFT_{d}$ described by the two strip geometries in Fig. (3) to be as follows

$\displaystyle\mathcal{E}=$

$\displaystyle\frac{3}{16G_{N}^{(d+1)}}\Bigg{[}\frac{2}{d-2}\Big{(}\frac{L}{a}\Big{)}^{d-2}-s_{0}\bigg{\{}\Big{(}\frac{L}{l_{1}}\Big{)}^{d-2}+\Big{(}\frac{L}{l_{2}}\Big{)}^{d-2}-\Big{(}\frac{L}{l_{1}+l_{2}}\Big{)}^{d-2}\bigg{\}}\Bigg{]}.$

(4.9)

The first term in the above expression is the divergent term which is proportional to the area of the entangling surface between the two spatial strips on the $d$ dimensional boundary and the second term describes the finite part of the negativity.

At finite temperatures the boundary $CFT_{d}$ is dual to the $AdS_{d+1}$-Schwarzschild black hole with the following metric where the $AdS$-radius has been set to $R=1$

$$ds^{2}=-r^{2}\Big{(}1-\frac{r_{h}^{d}}{r^{d}}\Big{)}dt^{2}+\frac{dr^{2}}{r^{2}\Big{(}1-\frac{r_{h}^{d}}{r^{d}}\Big{)}}+r^{2}d\vec{x}^{2}.$$

(5.1)

The horizon radius $r_{h}$ is related to the Hawking temperature as $T=r_{h}d/4\pi$ and $\vec{x}\equiv(x,x^{i})$ are the coordinates on the boundary. We first briefly review the computation for the area ${\cal A}$ of the bulk $AdS_{d+1}$ co-dimension two extremal surface anchored on a single rectangular strip on the boundary as described in [9]. This will be subsequently employed to compute the holographic entanglement negativity for the configuration Fig. (3) in question. The extremal area functional anchored on a single rectangular strip is given as

$${\cal A}=L^{d-2}\int drr^{d-2}\sqrt{r^{2}x^{\prime 2}+\frac{1}{r^{2}(1-\frac{r_{h}^{d}}{r^{d}})}}.$$

(5.2)

The corresponding Euler-Lagrange equation for the extremization problem leads to the following

$$\frac{l}{2}=\frac{1}{r_{c}}\int^{1}_{0}\frac{u^{d-1}du}{\sqrt{(1-u^{2d-2})}}(1-\frac{r_{h}^{d}}{r_{c}^{d}}u^{d})^{-\frac{1}{2}},~~~u=\frac{r_{c}}{r},$$

(5.3)

where $r_{c}$ as earlier describes the turning point. The area functional in terms of the variable $u$ may now be expressed as follows

$${\cal A}=2L^{d-2}r_{c}^{d-2}\int^{1}_{0}\frac{du}{u^{d-1}\sqrt{(1-u^{2d-2})}}(1-\frac{r_{h}^{d}}{r_{c}^{d}}u^{d})^{-\frac{1}{2}}.$$

(5.4)

This leads us to the final expression for the area functional as

$${\cal A}=\big{(}\mathcal{A}_{div}+\mathcal{A}_{finite}\big{)},$$

(5.5)

where $\mathcal{A}_{div}$ is the temperature independent divergent part and $\mathcal{A}_{finite}$ is the finite part. These may be expressed as follows

$\displaystyle\begin{split}\mathcal{A}_{div}&=\frac{2}{d-2}\Big{(}\frac{L}{a}\Big{)}^{d-2},\\ \mathcal{A}_{finite}&=2L^{d-2}r_{c}^{d-2}\Bigg{[}\frac{\sqrt{\pi}\Gamma\Big{(}-\frac{d-2}{2(d-1)}\Big{)}}{2(d-1)\Gamma\Big{(}\frac{1}{2(d-1)}\Big{)}}+\sum_{n=1}^{\infty}\Big{(}\frac{1}{2(d-1)}\Big{)}\frac{\Gamma\Big{(}n+\frac{1}{2}\Big{)}\Gamma\Big{(}\frac{d(n-1)+2}{2(d-1)}\Big{)}}{\Gamma\big{(}1+n\big{)}\Gamma\Big{(}\frac{dn+1}{2(d-1)}\Big{)}}\Big{(}\frac{r_{h}}{r_{c}}\Big{)}^{nd}\Bigg{]}.\end{split}$

(5.6)

Note that $r_{c}>r_{h}$ from [37] ensuring the convergence of the series in $\mathcal{A}_{finite}$. The holographic entanglement negativity for the mixed state described by the two intervals in the boundary $CFT_{d}$ ( Fig. (3) ) may then be obtained from our conjecture eq. (3.1) as

$$\begin{split}{\cal E}&=\frac{3}{16G_{N}^{(d+1)}}\Bigg{[}\frac{2}{d-2}(\frac{L}{a})^{d-2}\\ &+2L^{d-2}r_{c1}^{d-2}\bigg{\{}\frac{\sqrt{\pi}\Gamma\Big{(}-\frac{d-2}{2(d-1)}\Big{)}}{2(d-1)\Gamma\Big{(}\frac{1}{2(d-1)}\Big{)}}+\sum_{n=1}^{\infty}\Big{(}\frac{1}{2(d-1)}\Big{)}\frac{\Gamma\Big{(}n+\frac{1}{2}\Big{)}\Gamma\Big{(}\frac{d(n-1)+2}{2(d-1)}\Big{)}}{\Gamma\big{(}1+n\big{)}\Gamma\Big{(}\frac{dn+1}{2(d-1)}\Big{)}}\Big{(}\frac{r_{h}}{r_{c1}}\Big{)}^{nd}\bigg{\}}\\ &+2L^{d-2}r_{c2}^{d-2}\bigg{\{}\frac{\sqrt{\pi}\Gamma\Big{(}-\frac{d-2}{2(d-1)}\Big{)}}{2(d-1)\Gamma\Big{(}\frac{1}{2(d-1)}\Big{)}}+\sum_{n=1}^{\infty}\Big{(}\frac{1}{2(d-1)}\Big{)}\frac{\Gamma\Big{(}n+\frac{1}{2}\Big{)}\Gamma\Big{(}\frac{d(n-1)+2}{2(d-1)}\Big{)}}{\Gamma\big{(}1+n\big{)}\Gamma\Big{(}\frac{dn+1}{2(d-1)}\Big{)}}\Big{(}\frac{r_{h}}{r_{c2}}\Big{)}^{nd}\bigg{\}}\\ &-2L^{d-2}r_{c3}^{d-2}\bigg{\{}\frac{\sqrt{\pi}\Gamma\Big{(}-\frac{d-2}{2(d-1)}\Big{)}}{2(d-1)\Gamma\Big{(}\frac{1}{2(d-1)}\Big{)}}+\sum_{n=1}^{\infty}\Big{(}\frac{1}{2(d-1)}\Big{)}\frac{\Gamma\Big{(}n+\frac{1}{2}\Big{)}\Gamma\Big{(}\frac{d(n-1)+2}{2(d-1)}\Big{)}}{\Gamma\big{(}1+n\big{)}\Gamma\Big{(}\frac{dn+1}{2(d-1)}\Big{)}}\Big{(}\frac{r_{h}}{r_{c3}}\Big{)}^{nd}\bigg{\}}\Bigg{]}.\end{split}$$

(5.7)

Here $r_{c1},r_{c2},r_{c3}$ are the turning points of the extremal surfaces in the bulk anchored on the strips $A_{1},A_{2}$ and $A_{1}\cup A_{2}$ on the boundary respectively. It is required to evaluate the quantity $r_{ci}$ from the eq. (5.3) in terms of $l_{i}$ and $r_{h}$. The corresponding integral is not analytically solvable but may be determined perturbatively for low and high temperature approximations described in the following subsections.

To summarize we have established a holographic conjecture for the entanglement negativity for mixed states of adjacent subsystems in zero and finite temperature boundary $CFT_{d}$s. The relevant subsystems are described by $(d-1)$-dimensional spatial rectangular strip geometries at the boundary in a $AdS_{d+1}/CFT_{d}$ scenario. Our conjecture involves a certain algebraic sum of the areas of bulk co dimension two extremal surfaces anchored on the corresponding subsystems on the $AdS_{d+1}$ boundary and was motivated by the corresponding analysis for the $AdS_{3}/CFT_{2}$ scenario in [28]. It is interesting that the algebraic sum described above actually characterizes the holographic mutual information between the two adjacent subsystems. Note that these two measures are completely distinct quantities in quantum information theory. Our conjecture states that only their universal parts (which are dominant in the holographic limit) match upto a numerical factor for the particular mixed state configuration of adjacent subsystems. We emphasize that such a matching between the universal parts of negativity and mutual information of adjacent intervals in a $CFT_{1+1}$ has also been demonstrated for both local and global quench problems in [30, 20].

The holographic entanglement negativity for the boundary $CFT_{d}$ at zero temperature could then be computed from the bulk dual geometry described by the $AdS_{d+1}$ vacuum from our conjecture. The corresponding holographic entanglement negativity for the boundary $CFT_{d}$ at finite temperature however involved a bulk dual geometry described by the $AdS_{d+1}$-Schwarzschild black hole with a planar horizon. In the latter case the area integrals are not analytically solvable and were evaluated in a perturbative expansion for low and high temperatures. It was observed from our computation that the leading contribution to the holographic entanglement negativity at low temperature arises from the $AdS_{d+1}$ vacuum with subleading thermal corrections. Interestingly on the other hand at high temperatures the finite part of the holographic entanglement negativity is proportional to the area of the entangling surface on the boundary whereas the volume dependent thermal parts cancel out. It has been demonstrated that entanglement negativity does obey such area laws in various condensed matter systems confirming the expectation from quantum information theory (See [38, 39]).

Through these examples we demonstrated that our conjecture provides a direct and elegant holographic prescription to compute the entanglement negativity for mixed states described by the specific configuration in boundary $CFT_{d}$ both at zero and finite temperatures. However for the higher dimensional $AdS_{d+1}/CFT_{d}$ scenario this remains a conjecture. So in higher dimensions our conjecture requires further analysis towards a possible proof from the bulk side which remains a non trivial open issue. In this context our examples serve as a first consistency check in higher dimensions and lead to interesting results described above.

It is well known from quantum information theory that the entanglement negativity characterizes the upper bound on the distillable entanglement for mixed states. It is expected that our conjecture will lead to a deeper understanding of entanglement issues for diverse applications in higher dimensional conformal field theories from condensed matter physics to quantum gravity. It would be interesting to compute the holographic entanglement negativity for subsystems described by more general geometries other than the rectangular strip geometries considered by us. This would possibly lead to deeper insights into the nature of holographic quantum entanglement and its relation to issues of quantum gravity. We expect to return to these exciting issues in the near future.

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.