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In this paper we establish a connection between segmented strings propagating in $AdS_{d+1}$ and $CFT_d$ subsystems in Minkowski spacetime characterized by quantum information theoretic quantities calculated for the vacuum state. We show that the area of the world sheet of a string segment on the AdS side can be connected to fidelity susceptibility (the real part of the quantum geometric tensor) on the CFT side. This quantity has another interpretation as the computational complexity for infinitesimally separated states corresponding to causal diamonds that are displaced in a spacelike manner according to the metric of kinematic space. These displaced causal diamonds encode information for a unique reconstruction of the string world sheet segments in a holographic manner. Dually the bulk segments are representing causally ordered sets of consecutive boundary events in boosted inertial frames or in noninertial ones proceeding with constant acceleration. For the special case of $AdS_3$ one can also see the segmented stringy area in units of $4GL$ ($G$ is Newton's constant and $L$ is the AdS length) as the conditional mutual information $I(A,C\vert B)$ calculated for a trapezoid configuration arising from boosted spacelike intervals $A$,$B$ and $C$. In this special case the variation of the discretized Nambu-Goto action leads to an equation for entanglement entropies in the boundary theory of the form of a Toda equation. For arbitrary $d$ the string world sheet patches are living in the modular slices of the entanglement wedge. They seem to provide some sort of tomography of the entanglement wedge where the patches are linked together by the interpolation ansatz, i.e. the discretized version of the equations of motion for the Nambu-Goto action., Comment: 57 pages, 11 figures Title changed and new material added

We study the thermal state of a two dimensional conformal field theory which is dual to the static BTZ black hole in the high temperature limit. After partitioning the boundary of the static BTZ slice into $N$ subsystems we show that there is an underlying $C_{N-1}$ cluster algebra encoding entanglement patterns of the thermal state. We also demonstrate that the polytope encapsulating such patterns in a geometric manner for a fixed $N$ is the cyclohedron ${\mathcal C}_{N-1}$. Alternatively these patterns of entanglement can be represented in the space of geodesics (kinematic space) in terms of a Zamolodchikov $Y$-system of $C_{N-1}$ type. The boundary condition for such an $Y$-system is featuring the entropy of the BTZ black hole., Comment: 15 pages, 9 figures, LaTex

By employing the ${\rm AdS}_3/{\rm CFT}_2$ correspondence in this note we observe an analogy between the structures found in connection with the Arkani-Hamed-Bai-He-Yan (ABHY) associahedron used for understanding scattering amplitudes, and the one used for understanding space-time emerging from patterns of entanglement. The analogy suggests the natural interpretation for the associahedron as a holographic entanglement polytope associated to the ${\rm CFT}_2$ vacuum. Our observations hint at the possibility that the factorization properties of scattering amplitudes are connected to the notion of separability of space-time as used in the theory of holographic quantum entanglement., Comment: 15 pages , 5 figures LaTeX

We study the thermal state of a two dimensional conformal field theory which is dual to the static BTZ black hole in the high temperature limit. After partitioning the boundary of the static BTZ slice into $N$ subsystems we show that there is an underlying $C_{N-1}$ cluster algebra encoding entanglement patterns of the thermal state. We also demonstrate that the polytope encapsulating such patterns in a geometric manner for a fixed $N$ is the cyclohedron ${\mathcal C}_{N-1}$. Alternatively these patterns of entanglement can be represented in the space of geodesics (kinematic space) in terms of a Zamolodchikov $Y$-system of $C_{N-1}$ type. The boundary condition for such an $Y$-system is featuring the entropy of the BTZ black hole., Comment: 15 pages, 9 figures, LaTex

In the ${\rm AdS}_3/{\rm CFT}_2$ setup we elucidate how gauge invariant boundary patterns of entanglement of the CFT vacuum are encoded into the bulk via the coefficient dynamics of an $A_{N-3}$, $N\geq 4$ cluster algebra. In the static case this dynamics of encoding manifests itself in kinematic space, which is a copy of de Sitter space ${\rm dS}_2$, in a particularly instructive manner. For a choice of partition of the boundary into $N$ regions the patterns of entanglement, associated with conditional mutual informations of overlapping regions, are related to triangulations of geodesic $N$-gons. Such triangulations are then mapped to causal patterns in kinematic space. For a fixed $N$ the space of all causal patterns is related to the associahedron ${\mathcal K}^{N-3}$ an object well-known from previous studies on scattering amplitudes. On this space of causal patterns cluster dynamics acts by a recursion provided by a Zamolodchikov's $Y$-system of type $(A_{N-3},A_1)$. We observe that the space of causal patterns is equipped with a partial order, and is isomorphic to the Tamari lattice. The mutation of causal patterns can be encapsulated by a walk of $N-3$ particles interacting in a peculiar manner in the past light cone of a point of ${\rm dS}_2$., Comment: 18 pages, 15 figures

In the ${\rm AdS}_3/{\rm CFT}_2$ setup we elucidate how gauge invariant boundary patterns of entanglement of the CFT vacuum are encoded into the bulk via the coefficient dynamics of an $A_{N-3}$, $N\geq 4$ cluster algebra. In the static case this dynamics of encoding manifests itself in kinematic space, which is a copy of de Sitter space ${\rm dS}_2$, in a particularly instructive manner. For a choice of partition of the boundary into $N$ regions the patterns of entanglement, associated with conditional mutual informations of overlapping regions, are related to triangulations of geodesic $N$-gons. Such triangulations are then mapped to causal patterns in kinematic space. For a fixed $N$ the space of all causal patterns is related to the associahedron ${\mathcal K}^{N-3}$ an object well-known from previous studies on scattering amplitudes. On this space of causal patterns cluster dynamics acts by a recursion provided by a Zamolodchikov's $Y$-system of type $(A_{N-3},A_1)$. We observe that the space of causal patterns is equipped with a partial order, and is isomorphic to the Tamari lattice. The mutation of causal patterns can be encapsulated by a walk of $N-3$ particles interacting in a peculiar manner in the past light cone of a point of ${\rm dS}_2$., Comment: 18 pages, 15 figures

By studying the space of geodesics in $ADS_3/CFT_2$ and quantizing the geodesic motion, we relate scattering data to boundary entanglement of the CFT vacuum. The basic idea is to use a family of plane waves parametrized by coordinates of the space of geodesics i.e. kinematic space. This idea enables a simple calculation of the Berry curvature living on kinematic space. As a result we recover the Crofton form with a coefficient depending on the scattering energy. In arriving at these results the space of horocycles is used. We show that this new space used in concert with kinematic space incorporates naturally the gauge degrees of freedom responsible for an analogue of Berry's Phase. Horocycles also give a new geometric look to the strong subadditivity relation in terms of lambda lengths giving rise to shear coordinates of geodesic quadrangles. A generalization for geodesic polygons then reveals an interesting connection with $A_{n}$ cluster algebras. Here the cluster variables are the lambda lengths related to the regularized entropies of the boundary via the Ryu-Takayanagi relation. An elaboration of this idea indicates that cluster algebras might provide a natural algebraic means for encoding the gauge invariant entanglement patterns of certain boundary entangled states in the geometry of bulk geodesics. Finally using the language of integral geometry we show how certain propagators connected to the bulk, boundary and kinematic spaces are related to data of elementary scattering problems. We also present some hints how these ideas might be generalized for more general holographic scenarios., Comment: 21 pages, 6 figures, LaTEX

By studying the space of geodesics in $ADS_3/CFT_2$ and quantizing the geodesic motion, we relate scattering data to boundary entanglement of the CFT vacuum. The basic idea is to use a family of plane waves parametrized by coordinates of the space of geodesics i.e. kinematic space. This idea enables a simple calculation of the Berry curvature living on kinematic space. As a result we recover the Crofton form with a coefficient depending on the scattering energy. In arriving at these results the space of horocycles is used. We show that this new space used in concert with kinematic space incorporates naturally the gauge degrees of freedom responsible for an analogue of Berry's Phase. Horocycles also give a new geometric look to the strong subadditivity relation in terms of lambda lengths giving rise to shear coordinates of geodesic quadrangles. A generalization for geodesic polygons then reveals an interesting connection with $A_{n}$ cluster algebras. Here the cluster variables are the lambda lengths related to the regularized entropies of the boundary via the Ryu-Takayanagi relation. An elaboration of this idea indicates that cluster algebras might provide a natural algebraic means for encoding the gauge invariant entanglement patterns of certain boundary entangled states in the geometry of bulk geodesics. Finally using the language of integral geometry we show how certain propagators connected to the bulk, boundary and kinematic spaces are related to data of elementary scattering problems. We also present some hints how these ideas might be generalized for more general holographic scenarios., Comment: 21 pages, 6 figures, LaTEX

A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for $n$-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Pl\"ucker map well-known form twistor theory. The $n=2$ case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For $n\geq 3$ the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side., Comment: 64 pages, 9 figures, 4 tables, LaTex

In this paper we study the Hamming-like fermionic code encoding three-qubits into sixteen Majorana modes recently introduced by Hastings. We show that although this fermionic code cannot be obtained from a single qubit stabilizer code via the usual procedure however, it can be obtained from two, qubit stabilizer ones via a glueing procedure combining both single and double occupancy embeddings of qubits into fermionic Fock space. This technique identifies the code subspace as a Cartan subspace of the largest exceptional group $E_8$. In arriving at these results we develop a general formalism for implementing this glueing procedure via the use of intertwiners between different subsectors of the Fock space realizing embedded qubits., Comment: 14 pages

A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for $n$-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Pl\"ucker map well-known form twistor theory. The $n=2$ case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For $n\geq 3$ the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side., Comment: 64 pages, 9 figures, 4 tables, LaTex

In this paper we study the Hamming-like fermionic code encoding three-qubits into sixteen Majorana modes recently introduced by Hastings. We show that although this fermionic code cannot be obtained from a single qubit stabilizer code via the usual procedure however, it can be obtained from two, qubit stabilizer ones via a glueing procedure combining both single and double occupancy embeddings of qubits into fermionic Fock space. This technique identifies the code subspace as a Cartan subspace of the largest exceptional group $E_8$. In arriving at these results we develop a general formalism for implementing this glueing procedure via the use of intertwiners between different subsectors of the Fock space realizing embedded qubits., Comment: 14 pages

We investigate the structure of the three-qubit magic Veldkamp line (MVL). This mathematical notion has recently shown up as a tool for understanding the structures of the set of Mermin pentagrams, objects that are used to rule out certain classes of hidden variable theories. Here we show that this object also provides a unifying finite geometric underpinning for understanding the structure of functionals used in form theories of gravity and black hole entropy. We clarify the representation theoretic, finite geometric and physical meaning of the different parts of our MVL. The upshot of our considerations is that the basic finite geometric objects enabling such a diversity of physical applications of the MVL are the unique generalized quadrangles with lines of size three, their one point extensions as well as their other extensions isomorphic to affine polar spaces of rank three and order two. In a previous work we have already connected generalized quadrangles to the structure of cubic Jordan algebras related to entropy fomulas of black holes and strings in five dimensions. In some respect the present paper can be regarded as a generalization of that analysis for also providing a finite geometric understanding of four-dimensional black hole entropy formulas. However, we find many more structures whose physical meaning is yet to be explored. As a familiar special case our work provides a finite geometric representation of the algebraic extension from cubic Jordan algebras to Freudenthal systems based on such algebras., Comment: 52 pages, 15 figures

We investigate the structure of the three-qubit magic Veldkamp line (MVL). This mathematical notion has recently shown up as a tool for understanding the structures of the set of Mermin pentagrams, objects that are used to rule out certain classes of hidden variable theories. Here we show that this object also provides a unifying finite geometric underpinning for understanding the structure of functionals used in form theories of gravity and black hole entropy. We clarify the representation theoretic, finite geometric and physical meaning of the different parts of our MVL. The upshot of our considerations is that the basic finite geometric objects enabling such a diversity of physical applications of the MVL are the unique generalized quadrangles with lines of size three, their one point extensions as well as their other extensions isomorphic to affine polar spaces of rank three and order two. In a previous work we have already connected generalized quadrangles to the structure of cubic Jordan algebras related to entropy fomulas of black holes and strings in five dimensions. In some respect the present paper can be regarded as a generalization of that analysis for also providing a finite geometric understanding of four-dimensional black hole entropy formulas. However, we find many more structures whose physical meaning is yet to be explored. As a familiar special case our work provides a finite geometric representation of the algebraic extension from cubic Jordan algebras to Freudenthal systems based on such algebras., Comment: 52 pages, 15 figures

We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physically sound meaning of some of the direct sum structures showing up in the context of the so-called Black-Hole/Qubit Correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of $Spin(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with 8 modes. Here we find an algebraically independent generating set of the generalized SLOCC invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group $E_8$ is playing in these considerations., Comment: 35 pages, 5 figures

We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physically sound meaning of some of the direct sum structures showing up in the context of the so-called Black-Hole/Qubit Correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of $Spin(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with 8 modes. Here we find an algebraically independent generating set of the generalized SLOCC invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group $E_8$ is playing in these considerations., Comment: 35 pages, 5 figures

We propose a generalization of the usual SLOCC and LU classification of entangled pure state fermionic systems based on the Spin group. Our generalization uses the fact that there is a representation of this group acting on the fermionic Fock space which when restricted to fixed particle number subspaces recovers naturally the usual SLOCC transformations. The new ingredient is the occurrence of Bogoliubov transformations of the whole Fock space changing the particle number. The classification scheme built on the Spin group prohibits naturally entanglement between states containing even and odd number of fermions. In our scheme the problem of classification of entanglement types boils down to the classification of spinors where totally separable states are represented by so called pure spinors. We construct the basic invariants of the Spin group and show how some of the known SLOCC invariants are just their special cases. As an example we present the classification of fermionic systems with a Fock space based on six single particle states. An intriguing duality between two different possibilities for embedding three-qubit systems inside the fermionic ones is revealed. This duality is elucidated via an interesting connection to configurations of wrapped membranes reinterpreted as qubits., Comment: 28 pages, 1 figure

We propose a generalization of the usual SLOCC and LU classification of entangled pure state fermionic systems based on the Spin group. Our generalization uses the fact that there is a representation of this group acting on the fermionic Fock space which when restricted to fixed particle number subspaces recovers naturally the usual SLOCC transformations. The new ingredient is the occurrence of Bogoliubov transformations of the whole Fock space changing the particle number. The classification scheme built on the Spin group prohibits naturally entanglement between states containing even and odd number of fermions. In our scheme the problem of classification of entanglement types boils down to the classification of spinors where totally separable states are represented by so called pure spinors. We construct the basic invariants of the Spin group and show how some of the known SLOCC invariants are just their special cases. As an example we present the classification of fermionic systems with a Fock space based on six single particle states. An intriguing duality between two different possibilities for embedding three-qubit systems inside the fermionic ones is revealed. This duality is elucidated via an interesting connection to configurations of wrapped membranes reinterpreted as qubits., Comment: 28 pages, 1 figure

According to the Black Hole/Qubit Correspondence (BHQC) certain black hole entropy formulas in supergravity can be related to multipartite entanglement measures of quantum information. Here we show that the origin of this correspondence is a connection between Hitchin functionals used as action functionals for form theories of gravity related to topological strings, and entanglement measures for systems with a small number of constituents. The basic idea acting as a unifying agent in these seemingly unrelated fields is stability connected to the mathematical notion of special prehomogeneous vector spaces associated to Freudenthal systems coming from simple Jordan algebras. It is shown that the nonlinear function featuring these functionals and defining Calabi-Yau and generalized Calabi-Yau structures is the Freudenthal dual a concept introduced recently in connection with the BHQC. We propose to use the Hitchin invariant for three-forms in 7 dimensions as an entanglement measure playing a basic role in classifying three-fermion systems with seven modes. The representative of the class of maximal tripartite entanglement is the three-form used as a calibration for compactification on manifolds with $G_2$ holonomy. The idea that entanglement measures are related to action functionals from which the usual correspondence of the BHQC follows at the tree level suggests that one can use the BHQC in a more general context., Comment: 55 pages, Latex

According to the Black Hole/Qubit Correspondence (BHQC) certain black hole entropy formulas in supergravity can be related to multipartite entanglement measures of quantum information. Here we show that the origin of this correspondence is a connection between Hitchin functionals used as action functionals for form theories of gravity related to topological strings, and entanglement measures for systems with a small number of constituents. The basic idea acting as a unifying agent in these seemingly unrelated fields is stability connected to the mathematical notion of special prehomogeneous vector spaces associated to Freudenthal systems coming from simple Jordan algebras. It is shown that the nonlinear function featuring these functionals and defining Calabi-Yau and generalized Calabi-Yau structures is the Freudenthal dual a concept introduced recently in connection with the BHQC. We propose to use the Hitchin invariant for three-forms in 7 dimensions as an entanglement measure playing a basic role in classifying three-fermion systems with seven modes. The representative of the class of maximal tripartite entanglement is the three-form used as a calibration for compactification on manifolds with $G_2$ holonomy. The idea that entanglement measures are related to action functionals from which the usual correspondence of the BHQC follows at the tree level suggests that one can use the BHQC in a more general context., Comment: 55 pages, Latex

We link the recently discovered black hole-qubit correspondence to the structure of extra dimensions. In particular we show that for toroidal compactifications of type IIB string theory simple qubit systems arise naturally from the geometrical data of the tori parametrized by the moduli. We also generalize the recently suggested idea of the attractor mechanism as a distillation procedure of GHZ-like entangled states on the event horizon, to moduli stabilization for flux attractors in F-theory compactifications on elliptically fibered Calabi-Yau four-folds. Finally using a simple example we show that the natural arena for qubits to show up is an embedded one within the realm of fermionic entanglement of quantum systems with indistinguishable constituents., Comment: 32 pages Latex

We relate the U-duality invariants characterizing two-center extremal black hole solutions in the stu, st^2 and t^3 models of N=2, d=4 supergravity to the basic invariants used to characterize entanglement classes of four-qubit systems. For the elementary example of a D0D4-D2D6 composite in the t^3 model we illustrate how these entanglement invariants are related to some of the physical properties of the two-center solution. Next we show that it is possible to associate elliptic curves to charge configurations of two-center composites. The hyperdeterminant of the hypercube, a four-qubit polynomial invariant of order 24 with 2894276 terms, is featuring the j invariant of the elliptic curve. We present some evidence that this quantity and its straightforward generalization should play an important role in the physics of two-center solutions., Comment: 51 pages LaTex

We link the recently discovered black hole-qubit correspondence to the structure of extra dimensions. In particular we show that for toroidal compactifications of type IIB string theory simple qubit systems arise naturally from the geometrical data of the tori parametrized by the moduli. We also generalize the recently suggested idea of the attractor mechanism as a distillation procedure of GHZ-like entangled states on the event horizon, to moduli stabilization for flux attractors in F-theory compactifications on elliptically fibered Calabi-Yau four-folds. Finally using a simple example we show that the natural arena for qubits to show up is an embedded one within the realm of fermionic entanglement of quantum systems with indistinguishable constituents., Comment: 32 pages Latex

We relate the U-duality invariants characterizing two-center extremal black hole solutions in the stu, st^2 and t^3 models of N=2, d=4 supergravity to the basic invariants used to characterize entanglement classes of four-qubit systems. For the elementary example of a D0D4-D2D6 composite in the t^3 model we illustrate how these entanglement invariants are related to some of the physical properties of the two-center solution. Next we show that it is possible to associate elliptic curves to charge configurations of two-center composites. The hyperdeterminant of the hypercube, a four-qubit polynomial invariant of order 24 with 2894276 terms, is featuring the j invariant of the elliptic curve. We present some evidence that this quantity and its straightforward generalization should play an important role in the physics of two-center solutions., Comment: 51 pages LaTex

Concurrence is an entanglement measure characterizing the {\it mixed} state bipartite correlations inside of a pure state of an $n$-qubit system. We show that after organizing the charges and the moduli in the STU model of $N=2$, $d=4$ supergravity to a three-qubit state, for static extremal spherically symmetric BPS black hole solutions the vanishing condition for all of the bipartite concurrences on the horizon is equivalent to the attractor equations. As a result of this the macroscopic black hole entropy given by the three-tangle can be reinterpreted as a linear entropy characterizing the {\it pure} state entanglement for an arbitrary bipartite split. Both for the BPS and non-BPS cases explicit expressions for the concurrences are obtained, with their vanishing on the horizon is demonstrated., Comment: 15 pages, 2 figures, Latex

In a recent paper it has been shown that for double extremal static spherically symmetric BPS black hole solutions in the STU model the well-known process of moduli stabilization at the horizon can be recast in a form of a distillation procedure of a three-qubit entangled state of GHZ-type. By studying the full flow in moduli space in this paper we investigate this distillation procedure in more detail. We introduce a three-qubit state with amplitudes depending on the conserved charges the warp factor, and the moduli. We show that for the recently discovered non-BPS solutions it is possible to see how the distillation procedure unfolds itself as we approach the horizon. For the non-BPS seed solutions at the asymptotically Minkowski region we are starting with a three-qubit state having seven nonequal nonvanishing amplitudes and finally at the horizon we get a GHZ state with merely four nonvanishing ones with equal magnitudes. The magnitude of the surviving nonvanishing amplitudes is proportional to the macroscopic black hole entropy. A systematic study of such attractor states shows that their properties reflect the structure of the fake superpotential. We also demonstrate that when starting with the very special values for the moduli corresponding to flat directions the uniform structure at the horizon deteriorates due to errors generalizing the usual bit flips acting on the qubits of the attractor states., Comment: 38 pages LaTex

In this paper we describe the structure of extremal stationary spherically symmetric black hole solutions in the STU model of D=4, N=2 supergravity in terms of four-qubit systems. Our analysis extends the results of previous investigations based on three qubits. The basic idea facilitating this four-qubit interpretation is the fact that stationary solutions in D=4 supergravity can be described by dimensional reduction along the time direction. In this D=3 picture the global symmetry group $SL(2,R)^{\times 3}$ of the model is extended by the Ehlers SL(2,R) accounting for the fourth qubit. We introduce a four qubit state depending on the charges (electric, magnetic and NUT) the moduli and the warp factor. We relate the entanglement properties of this state to different classes of black hole solutions in the STU model. In the terminology of four qubit entanglement extremal black hole solutions correspond to nilpotent, and nonextremal ones to semisimple states. In arriving at this entanglement based scenario the role of the four algebraically independent four qubit SL(2,C) invariants is emphasized., Comment: 47 pages LATEX

Concurrence is an entanglement measure characterizing the {\it mixed} state bipartite correlations inside of a pure state of an $n$-qubit system. We show that after organizing the charges and the moduli in the STU model of $N=2$, $d=4$ supergravity to a three-qubit state, for static extremal spherically symmetric BPS black hole solutions the vanishing condition for all of the bipartite concurrences on the horizon is equivalent to the attractor equations. As a result of this the macroscopic black hole entropy given by the three-tangle can be reinterpreted as a linear entropy characterizing the {\it pure} state entanglement for an arbitrary bipartite split. Both for the BPS and non-BPS cases explicit expressions for the concurrences are obtained, with their vanishing on the horizon is demonstrated., Comment: 15 pages, 2 figures, Latex

It is shown that the $E_{6(6)}$ symmetric entropy formula describing black holes and black strings in D=5 is intimately tied to the geometry of the generalized quadrangle GQ$(2,4)$ with automorphism group the Weyl group $W(E_6)$. The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ$(2,4)$. Different truncations with $15, 11$ and 9 charges are represented by three distinguished subconfigurations of GQ$(2,4)$, well-known to finite geometers; these are the "doily" (i. e. GQ$(2,2)$) with 15, the "perp-set" of a point with 11, and the "grid" (i. e. GQ$(2,1)$) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non- commutative labelling for the points of GQ$(2,4)$. For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares -- objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the $E_{7(7)}$ symmetric entropy formula in D=4 by observing that the structure of GQ$(2,4)$ is linked to a particular kind of geometric hyperplane of the split Cayley hexagon of order two, featuring 27 points located on 9 pairwise disjoint lines (a distance-3-spread). We conjecture that the different possibilities of describing the D=5 entropy formula using Jordan algebras, qubits and/or qutrits correspond to employing different coordinates for an underlying non-commutative geometric structure based on GQ$(2,4)$., Comment: 17 pages, 3 figures, v2 a new paragraph added, typos corrected

Following the recent trend we develop further the black hole analogy between quantum information theory and the theory of extremal stringy black hole solutions. We show that the three-qubit interpretation of supersymmetric black hole solutions in the STU model can be extended also to include non-supersymmetric ones. First we show that the black hole potential can be expressed as one half the norm of a suitably chosen three-qubit entangled state containing the quantized charges and the moduli. The extremization of the black hole potential in terms of this entangled state amounts to either supressing bit flip errors (BPS-case) or allowing very special types of flips transforming the states between different classes of non-BPS solutions. We are illustrating our results for the example of the D2-D6 system. In this case the bit flip errors are corresponding to sign flip errors of the charges originating from the number of D2 branes. After moduli stabilization the states depending entirely on the charges are maximally entangled graph states (of the triangle graph) well-known from quantum information theory. An N=8 interpretation of the STU-model in terms of a mixed state with fermionic purifications is also given., Comment: 35 pages

Following the recent trend we develop further the black hole analogy between quantum information theory and the theory of extremal stringy black hole solutions. We show that the three-qubit interpretation of supersymmetric black hole solutions in the STU model can be extended also to include non-supersymmetric ones. First we show that the black hole potential can be expressed as one half the norm of a suitably chosen three-qubit entangled state containing the quantized charges and the moduli. The extremization of the black hole potential in terms of this entangled state amounts to either supressing bit flip errors (BPS-case) or allowing very special types of flips transforming the states between different classes of non-BPS solutions. We are illustrating our results for the example of the D2-D6 system. In this case the bit flip errors are corresponding to sign flip errors of the charges originating from the number of D2 branes. After moduli stabilization the states depending entirely on the charges are maximally entangled graph states (of the triangle graph) well-known from quantum information theory. An N=8 interpretation of the STU-model in terms of a mixed state with fermionic purifications is also given., Comment: 35 pages

Following the recent trend we develop further the black hole analogy between quantum information theory and the theory of extremal stringy black hole solutions. We show that the three-qubit interpretation of supersymmetric black hole solutions in the STU model can be extended also to include non-supersymmetric ones. First we show that the black hole potential can be expressed as one half the norm of a suitably chosen three-qubit entangled state containing the quantized charges and the moduli. The extremization of the black hole potential in terms of this entangled state amounts to either supressing bit flip errors (BPS-case) or allowing very special types of flips transforming the states between different classes of non-BPS solutions. We are illustrating our results for the example of the D2-D6 system. In this case the bit flip errors are corresponding to sign flip errors of the charges originating from the number of D2 branes. After moduli stabilization the states depending entirely on the charges are maximally entangled graph states (of the triangle graph) well-known from quantum information theory. An N=8 interpretation of the STU-model in terms of a mixed state with fermionic purifications is also given., Comment: 35 pages

Recently it has been observed that the group $E_7$ can be used to describe a special type of quantum entanglement of seven qubits partitioned into seven tripartite systems. Here we show that this curious type of entanglement is entirely encoded into the discrete geometry of the Fano plane. We explicitly work out the details concerning a qubit interpretation of the $E_7$ generators as representatives of tripartite protocols acting on the 56 dimensional representation space. Using these results we extend further the recently studied analogy between quantum information theory and supersymmetric black holes in four-dimensional string theory. We point out there is a dual relationship between entangled subsystems containing three and four tripartite systems. This relationship is reflected in the structure of the expressions for the black hole entropy in the N=4 and N=2 truncations of the $E_{7(7)}$ symmetric area form of N=8 supergravity. We conjecture that a similar picture based on other qubit systems might hold for black hole solutions in magic supergravities., Comment: 42 pages, 5 figures, LATEX

Recently striking multiple relations have been found between pure state 2 and 3-qubit entanglement and extremal black holes in string theory. Here we add further mathematical similarities which can be both useful in string and quantum information theory. In particular we show that finding the frozen values of the moduli in the calculation of the macroscopic entropy in the STU model, is related to finding the canonical form for a pure three-qubit entangled state defined by the dyonic charges. In this picture the extremization of the BPS mass with respect to moduli is connected to the problem of finding the optimal local distillation protocol of a GHZ state from an arbitrary pure three-qubit state. These results and a geometric classification of STU black holes BPS and non-BPS can be described in the elegant language of twistors. Finally an interesting connection between the black hole entropy and the average real entanglement of formation is established., Comment: 34 pages, 6 figures