Holographic entanglement negativity for a single subsystem in conformal field theories with a conserved charge
 

The recent past has witnessed the emergence of quantum entanglement as a central theme in the study of many body systems relevant to diverse fields from condensed matter physics to quantum gravity and black holes. For bipartite pure states the entanglement entropy serves as an appropriate entanglement measure. Remarkably for $(1+1)$-dimensional conformal field theories ($CFT_{1+1}$s) an explicit computation for the entanglement entropy utilizing the replica technique was established in [1, 2, 3, 4, 5]. For bipartite mixed states however the entanglement entropy incorporates irrelevant correlations and is hence unsuitable for characterization of the entanglement. In quantum information theory, several measures have been proposed for the characterization of mixed state entanglement which are in general not easily computable. Vidal and Werner in a significant work [6] proposed a computable measure termed entanglement negativity (logarithmic negativity) for bipartite mixed state entanglement which characterizes an upper bound on the distillable entanglement for the mixed state.¹¹1Distillable entanglement is the amount of pure entanglement that can be extracted from a given state by means of local operations and classical communication (LOCC). This measure was defined as the logarithm of the trace norm for the partially transposed reduced density matrix with respect to one of the subsystems and shown to be an entanglement monotone [7].²²2For an excellent review see [8]. Interestingly in a series of articles [9, 10, 11] the entanglement negativity for various bipartite pure and mixed states in a $CFT_{1+1}$ could be obtained through an appropriate replica technique.

In a different context a holographic characterization of the entanglement entropy for $d$-dimensional conformal field theories ($CFT_{d}$s) dual to bulk $AdS_{d+1}$ geometries was conjectured in a seminal work by Ryu and Takayanagi (RT) [12, 13]. Their proposal described the holographic entanglement entropy of a subsystem in the dual $CFT_{d}$ in terms of the area of a bulk codimension two static minimal surface homologous to the subsystem (RT Surface). The RT conjecture and its covariant generalization by Hubeny, Rangamani and Takayanagi (HRT) [14] were proved in a series of subsequent works [15, 16, 17, 18, 19, 20]. The RT/HRT conjectures inspired the remarkable development of the field of holographic quantum entanglement in dual $CFT_{d}$s summarized in [21, 22, 23, 24, 25, 26] and references therein.³³3Note that the field is extremely diverse and the list of references is not meant to be exhaustive.

The above developments naturally brought the critical issue of a holographic characterization for the entanglement negativity in dual $CFT_{d}$s into sharp focus. This significant issue was initially partially addressed for pure states in [8] although a clear holographic prescription for generic bipartite mixed states in dual $CFT_{d}$s remained an outstanding issue. In an interesting communication the authors in [27] proposed a holographic entanglement negativity conjecture for the bipartite states of a single interval in $CFT_{1+1}$s dual to bulk pure $AdS_{3}$ geometries and Euclidean BTZ black holes. This proposal involved a certain algebraic sum of the lengths of bulk geodesics homologous to appropriate combinations of the single interval and other auxiliary intervals. Remarkably the results for the holographic entanglement negativity computed from their proposal were in exact agreement with the corresponding $CFT_{1+1}$ replica technique results in the large central charge limit described in [9, 10, 11]. This construction was further substantiated through a rigorous large central charge analysis utilizing the monodromy technique in [28] for the $AdS_{3}/CFT_{2}$ scenario. Subsequently in a series of articles this holographic construction and its covariant generalization were utilized to obtain the entanglement negativity for diverse mixed state configurations in the $AdS_{3}/CFT_{2}$ scenario [29, 27, 30, 31, 32, 33].

A higher dimensional generalization of the above holographic conjecture for generic $AdS_{d+1}/CFT_{d}$ scenarios involving specific algebraic sums of the areas of the RT surfaces for certain combinations of subsystems, was subsequently proposed and utilized to compute the entanglement negativity of bipartite states of subsystems with long rectangular strip geometries in dual $CFT_{d}$s [34, 35, 36, 37]. The results conformed to quantum information theory expectations and reproduced certain significant features for the entanglement negativity also observed for the $AdS_{3}/CFT_{2}$ scenario which served as crucial consistency checks. Very recently a plausible bulk proof of these holographic entanglement negativity conjectures from a gravitational path integral perspective, along the lines of [18, 19] for spherical subsystem geometries based on [38] was described in [39].⁴⁴4A review of this plausible proof for single subsystems with spherical geometries has been reviewed in appendix A. However a generalization of this for generic subsystem geometries is still a non trivial outstanding issue.

We should mention here that subsequent to the developments described above, the authors in [40, 41] proposed an alternate holographic entanglement negativity conjecture for bipartite states in dual $CFT_{d}$s involving the minimal entanglement wedge cross section (EWCS) [42] backreacted by the cosmic brane for the conical defect of the bulk replicated geometry. For spherical entangling surfaces this backreaction could be computed and resulted in a dimension dependent constant numerical factor for the entanglement negativity. For the $AdS_{3}/CFT_{2}$ scenario the holographic results from their proposal matched with the corresponding field theory replica technique results in the large central charge limit modulo certain constants arising from the Markov gap [43]. The above holographic construction was further refined in [44] to resolve an outstanding issue and demonstrated the equivalence of the two proposals modulo the constants mentioned above.⁵⁵5Note that the constant backreaction factor $\chi_{d}=\frac{3}{2}$ for the $AdS_{3}/CFT_{2}$ scenario was incorporated in the proposal based on the algebraic sum of the lengths of bulk geodesics homologous to appropriate combinations of intervals described in [27]. However this alternative proposal requires further substantiation for the higher dimensional $AdS_{d+1}/CFT_{d}$ scenario.⁶⁶6See also the discussions in [45].

In this article we extend the higher dimensional holographic construction to compute the entanglement negativity for bipartite states of single subsystems described in [27] to $CFT_{d}$s with a conserved charge dual to bulk non extremal and extremal $AdS_{d+1}$-RN black holes. Our results provide further substantiation for the holographic conjecture for the entanglement negativity of bipartite states in higher dimensional $CFT_{d}$s. In this context following [27] we also consider subsystems with long rectangular strip geometries and utilize a perturbative technique described in [24, 23, 46, 47], to compute the areas of RT surfaces involving non trivial limits of certain parameters relevant for the bulk $AdS_{d+1}$-RN geometry.⁷⁷7 Note that an exact closed form analytic evaluation of the area has been performed through the use of Meijer G-functions in [48]. However the leading order behavior necessary for our purpose is correctly captured through the perturbative techniques utilized here. Interestingly as described in the literature, through our construction we reproduce certain significant features (such as elimination of the thermal contributions) of the holographic entanglement negativity for bipartite states in $CFT_{2}$s and $CFT_{d}$s described in [27, 34]. This conforms to quantum information theory expectations of the negativity being an upper bound on the distillable entanglement.

Note that in the context of the developments described in [40, 41, 44] our results also involve backreaction effects due to the cosmic brane arising from the conical defect of the replicated bulk geometry in the replica limit. However the explicit determination of these effects remains a non trivial outstanding issue for arbitrary subsystem geometries, in particular the long rectangular strip geometries investigated in this article.

In this connection we state here that in [49] the authors provided another construction to characterize the modification of the RT surfaces due to the presence of a charged bulk geometry. In this article they considered a charged topological black hole with a hyperbolic horizon coupled to a dual Maxwell gauge field to compute charged holographic Rényi entropies and the holographic entanglement entropy for spherical entangling surfaces in the dual $CFT_{d}$. Interestingly the corresponding charged Rényi entropies and the entanglement entropy for such spherical domains in the $CFT_{d}$ could be computed using a generalized replica technique for a thermal path integral via a background gauge field with a non trivial Wilson line on the Euclidean time circle following an appropriate conformal transformation.

This article is organized as follows. In section 2 we describe the holographic entanglement negativity construction for the bipartite configuration of a single subsystem characterized by long rectangular strip geometry in $CFT_{d}$s dual to bulk $AdS_{d+1}$ geometries. Subsequently in section 3 the construction is employed to compute the holographic entanglement negativity for mixed state configuration of a single subsystem in $CFT_{d}$s with a conserved charge dual to bulk non extremal and extremal $AdS_{d+1}$-RN black holes. Finally in section 4 we summarize our results and present our conclusions. We also provide some involved and lengthy results in the appendices.

In this section we present a brief review of the holographic entanglement negativity conjecture for the bipartite state of a single subsystem in a $CFT_{d}$ at a finite temperature dual to bulk $AdS_{d+1}$ geometries as described in [9, 10, 11, 34, 27]. In this context we begin with recapitulating the corresponding issue in the $AdS_{3}/CFT_{2}$ scenario [27]. For this purpose we consider a bipartite system ($A\cup A^{c}$) which is described by an interval $A$ of length $l$ and its complement $A^{c}$ in the dual $CFT_{1+1}$. To describe this construction it is also required to consider two auxiliary large but finite intervals $B_{1}$ and $B_{2}$ each of length $b$ on either side of $A$ and adjacent to it such that $B=(B_{1}\cup B_{2})\subset A^{c}$. The intervals may be represented as $B_{1}=(-b-l/2,-l/2),A=(-l/2,l/2),B_{2}=(l/2,b+l/2)$. The entanglement negativity for the mixed state of the single interval at a finite temperature in the dual $CFT_{1+1}$ may then be defined through a replica technique as follows [9]

$$\mathcal{E}=\lim_{B\to A^{c}}\lim_{n_{e}\to 1}\ln\mathrm{Tr}\left(\rho_{AB}^{T_{B}}\right)^{n_{e}},$$

(1)

where $\rho_{AB}^{T_{B}}$ is the partial transpose of the reduced density matrix $\rho_{AB}$ with respect to the subsystem $B$, and the replica limit $n_{e}\to 1$ is an analytic continuation of even sequences of $n_{e}$ to $n_{e}=1$. For two arbitrary subsystems $X$ and $Y$, the partial transpose $\rho_{XY}^{T_{Y}}$ of the reduced density matrix $\rho_{XY}$, with respect to the subsystem $Y$, may be defined as

$$\matrixelement{e^{(X)}_{i}e^{(Y)}_{j}}{\rho_{XY}^{T_{Y}}}{e^{(X)}_{k}e^{(Y)}_{l}}=\matrixelement{e^{(X)}_{i}e^{(Y)}_{l}}{\rho_{XY}}{e^{(X)}_{k}e^{(Y)}_{j}},$$

(2)

where ${|e^{(X)}_{i}\rangle}$ and $|e^{(Y)}_{j}\rangle$ describe orthonormal bases for the Hilbert spaces $\mathcal{H}_{X}$ and $\mathcal{H}_{Y}$ respectively. Note that the quantity $\mathrm{Tr}(\rho_{AB}^{T_{B}})^{n_{e}}$ may be expressed as a twist correlator in the $CFT_{1+1}$ appropriate to the bipartite configuration. For the case of the bipartite state of a single interval in the dual $CFT_{1+1}$ this is given by a four point twist field correlator on an infinitely long cylinder and the corresponding entanglement negativity is given as follows [11]

$$\mathcal{E}=\lim_{l/b\to 0}~{}\lim_{n_{e}\to 1}\ln\left[\left\langle\mathcal{T}_{n_{e}}(-b-l/2)\overline{\mathcal{T}}^{2}_{n_{e}}(-l/2)\mathcal{T}^{2}_{n_{e}}(l/2)\overline{\mathcal{T}}_{n_{e}}(b+l/2)\right\rangle_{cyl}\right],$$

(3)

where $l/b\to 0$ is the bipartite limit $B=(B_{1}\cup B_{2})\to A^{c}$. It is important to note that the order of the limits in eq. (3) is crucial where the bipartite limit ($l/b\to 0$) should be taken only after the replica limit $(n_{e}\to 1)$.

The twist fields $\mathcal{T}$s, $\overline{\mathcal{T}}$s above provide the boundary conditions between the $n_{e}$ replicas and reduce to primary fields on the complex plane in the replica limit $n_{e}\to 1$. As described in [28] the universal part of the four point twist field correlator described above may be obtained in the large central charge limit through a rigorous monodromy analysis as follows

$$\left\langle\mathcal{T}_{n_{e}}(z_{1})\overline{\mathcal{T}}^{2}_{n_{e}}(z_{2})\mathcal{T}^{2}_{n_{e}}(z_{3})\overline{\mathcal{T}}_{n_{e}}(z_{4})\right\rangle_{\mathbb{C}}\approx\frac{1}{x^{\Delta^{(2)}_{n_{e}}}z_{23}^{2\Delta^{(2)}_{n_{e}}}}\,,$$

(4)

where the cross ratio $x\equiv\frac{z_{12}z_{34}}{z_{13}z_{24}}$ and $z_{ij}=z_{i}-z_{j}$ and the scaling dimensions $\Delta_{n_{e}}$ and $\Delta_{n_{e}}^{(2)}$ of the operator $\mathcal{T}_{n_{e}}$ and $\mathcal{T}_{n_{e}}^{2}$ are related to each other as follows

$$\Delta_{n_{e}}=\frac{c}{12}\left(n_{e}-\frac{1}{n_{e}}\right),\qquad\qquad\Delta_{n_{e}}^{(2)}=2\Delta_{\frac{n_{e}}{2}}=\frac{c}{6}\left(\frac{n_{e}}{2}-\frac{2}{n_{e}}\right).$$

(5)

From the two point correlator of the twist fields in a $CFT_{1+1}$, given as

$$\left\langle\mathcal{T}_{n_{e}}(z_{i})\overline{\mathcal{T}}_{n_{e}}(z_{j})\right\rangle_{\mathbb{C}}~{}\sim~{}\left\lvert z_{ij}\right\rvert^{-2\Delta_{\mathcal{T}_{n_{e}}}},$$

(6)

it is observed that the large central charge form of the four point twist correlator in eq. (4) may be expressed in terms of the products of certain two point twist field correlators. Using the $AdS_{3}/CFT_{2}$ dictionary these two point twist correlators may in turn be expressed in terms of the lengths of space like geodesics in the bulk $AdS_{3}$ geometry (with length scale $R$) homologous to certain combinations of the intervals as follows

$$\left\langle\mathcal{T}_{n_{e}}(z_{i})\overline{\mathcal{T}}_{n_{e}}(z_{j})\right\rangle_{\mathbb{C}}~{}\sim~{}e^{-\frac{\Delta_{n_{e}}\mathcal{L}_{ij}}{R}}\,,$$

$$\left\langle\mathcal{T}_{\frac{n_{e}}{2}}(z_{k})\overline{\mathcal{T}}_{\frac{n_{e}}{2}}(z_{l})\right\rangle_{\mathbb{C}}~{}\sim~{}e^{-\frac{\Delta_{\frac{n_{e}}{2}}\mathcal{L}_{kl}}{R}}\,.$$

(7)

In the above expressions $\mathcal{L}_{ij}$ is the length of the geodesic homologous to the interval described by the endpoints $z_{i}$ and $z_{j}$. From eqs. (3), (4) and (7), the holographic entanglement negativity may then be expressed as

$$\mathcal{E}=\lim_{B\to A^{c}}~{}\left.\frac{3}{16G_{N}^{(3)}}\middle(2\mathcal{L}_{A}+\mathcal{L}_{B_{1}}+\mathcal{L}_{B_{2}}-\mathcal{L}_{A\cup B_{1}}-\mathcal{L}_{A\cup B_{2}}\right),$$

(8)

where we have utilized the Brown-Henneaux formula $c=\frac{3R}{2G_{N}^{(3)}}$ [50]. As discussed in the Introduction, the expression for the holographic entanglement negativity above contains an overall constant numerical factor due to the backreaction of the cosmic brane for the bulk conical defect in the replica limit.⁸⁸8For the $AdS_{3}/CFT_{2}$ scenario where the entangling surfaces are just points this overall dimension dependent backreaction factor $\chi_{d}$ may be explicitly determined as $\chi_{2}=\frac{3}{2}$ as described in [51, 40]. Note that using the RT prescription for the $AdS_{3}/CFT_{2}$ scenario the above equation may be expressed as follows

$$\mathcal{E}=\lim_{B\to A^{c}}\left.\frac{3}{4}\middle(2S_{A}+S_{B_{1}}+S_{B_{2}}-S_{A\cup B_{1}}-S_{A\cup B_{2}}\right).$$

(9)

Given the $AdS_{3}/CFT_{2}$ scenario discussed above, it was conjectured in [34] that in the context of the $AdS_{d+1}/CFT_{d}$ framework the corresponding holographic entanglement negativity for the mixed state of a single subsystem $A$ may be expressed in the bipartite limit $(B\to A^{c})$ as

$$\mathcal{E}_{A}=\lim_{B\to A^{c}}\left.\frac{3}{16G_{N}^{(d+1)}}\middle(2\mathcal{A}_{A}+\mathcal{A}_{B_{1}}+\mathcal{A}_{B_{2}}-\mathcal{A}_{A\cup B_{1}}-\mathcal{A}_{A\cup B_{2}}\right),$$

(10)

where $\mathcal{A}_{\gamma}$ denotes the area of the RT surface homologous to the subsystem $\gamma$, and $G_{N}^{(d+1)}$ is the gravitational constant in $(d+1)$ dimensions.⁹⁹9As discussed in the Introduction a plausible proof of the holographic conjectures expressed in eqs. (8) and (10) for spherical entangling surfaces discussed in [39] from replica symmetry breaking saddles for the bulk gravitational path integral [38] has been described in appendix A.

For the case when the subsystems $B_{1}$ and $B_{2}$ are symmetric with respect to the subsystem $A$, eq. (10) simplifies to

$$\mathcal{E}_{A}=\lim_{B\to A^{c}}\left.\frac{3}{8G_{N}^{(d+1)}}\middle(\mathcal{A}_{A}+\mathcal{A}_{B_{1}}-\mathcal{A}_{A\cup B_{1}}\right).$$

(11)

Using the standard Ryu-Takayanagi (RT) formula given as

$$\mathcal{S}_{A}=\frac{\mathcal{A}_{A}}{4G_{N}^{(d+1)}}\;,$$

(12)

the above conjecture for the holographic entanglement negativity for the bipartite state of a single subsystem $A$ may be expressed as¹⁰¹⁰10Note that this is a purely holographic result valid for the corresponding large $c$ limit of the dual $CFT$. This is explicit for the $AdS_{3}/CFT_{2}$ scenario where exact computations are possible for both the bulk and the boundary. To check its validity for the full quantum information theory measure, it would be required to compute the non universal contributions to the $CFT_{2}$ correlation functions, which is extremely non trivial.

$$\mathcal{E}_{A}=\lim_{B\to A^{c}}\left.\frac{3}{2}\middle(S_{A}+S_{B_{1}}-S_{A\cup B_{1}}\right).$$

(13)

Note that as discussed earlier for the $AdS_{3}/CFT_{2}$ scenario, the expressions in eqs. (9) and (10) for the $AdS_{d+1}/CFT_{d}$ framework will also involve effects due to the backreaction of the cosmic brane for the conical defect of the replicated bulk geometry. For arbitrary subsystem geometries the determination of the explicit form of these effects is still a non trivial open issue in particular for the long rectangular strip geometries considered in this article. However for spherical entangling surfaces the explicit form of the backreaction factor is described in [51, 40].

We now utilize the construction reviewed above to compute the holographic entanglement negativity for bipartite states of single subsystems in $CFT_{d}$s with a conserved charge, dual to bulk $AdS_{d+1}$-RN geometries. We employ a perturbative expansion of the areas of the corresponding bulk RT surfaces, in terms of the temperature $T$ and the chemical potential $\mu$ conjugate to the charge $Q$ of the $AdS_{d+1}$-RN black hole. It is convenient to express the holographic entanglement negativity in terms of the total energy $\varepsilon$ of the dual $CFT_{d}$, and an effective temperature $T_{\textit{ef{}f}}$, both of which are functions of the temperature and the chemical potential. Note that the parameter $\varepsilon$ is related to the expectation value of the component $T_{00}$ of the energy momentum tensor as described in [46]. To this end we first review the perturbative calculation of the area of the RT surfaces required for the computation of the entanglement negativity in subsection 3.1 before proceeding to the non extremal bulk $AdS_{d+1}$-RN black holes in subsection 3.2 and finally describing the corresponding bulk extremal $AdS_{d+1}$-RN black holes in subsection 3.3.