This thesis explores the numerous relationships between the entropy of black hole solutions in supergravity and the entanglement of multipartite systems in quantum information theory: the so-called black hole/qubit correspondence. We examine how, through the correspondence, the dyonic charges in the entropy of supersymmetric black hole solutions are directly matched to the state vector coefficients in the entanglement measures of their quantum information analogues. Moreover the Uduality invariance of the black hole entropy translates to the stochastic local operations and classical communication (SLOCC) invariance of the entanglement measures. Several examples are discussed, with the correspondence broadening when the supersymmetric classification of black holes is shown to match the entanglement classification of the qubit/qutrit analogues. On the microscopic front, we study the interpretation of D-brane wrapping configurations as real qubits/qutrits, including the matching of generating solutions on black hole and qubit sides. Tentative generalisations to other dimensions and qubit systems are considered. This is almost eclipsed by more recent developments linking the nilpotent U-duality orbit classi cation of black holes to the nilpotent classi cation of complex qubits. We provide preliminary results on the corresponding covariant classi cation. We explore the interesting parallel development of supersymmetric generalisations of qubits and entanglement, complete with two- and three-superqubit entanglement measures. Lastly, we briefly mention the supergravity technology of cubic Jordan algebras and Freudenthal triple systems (FTS), which are used to: 1) Relate FTS ranks to threequbit entanglement and compute SLOCC orbits. 2) Define new black hole dualities distinct from U-duality and related by a 4D/5D lift. 3) Clarify the state of knowledge of integral U-duality orbits in maximally extended supergravity in four, five, and six dimensions.
This thesis explores the numerous relationships between the entropy of black hole solutions in supergravity and the entanglement of multipartite systems in quantum information theory: the so-called black hole/qubit correspondence. We examine how, through the correspondence, the dyonic charges in the entropy of supersymmetric black hole solutions are directly matched to the state vector coefficients in the entanglement measures of their quantum information analogues. Moreover the Uduality invariance of the black hole entropy translates to the stochastic local operations and classical communication (SLOCC) invariance of the entanglement measures. Several examples are discussed, with the correspondence broadening when the supersymmetric classification of black holes is shown to match the entanglement classification of the qubit/qutrit analogues. On the microscopic front, we study the interpretation of D-brane wrapping configurations as real qubits/qutrits, including the matching of generating solutions on black hole and qubit sides. Tentative generalisations to other dimensions and qubit systems are considered. This is almost eclipsed by more recent developments linking the nilpotent U-duality orbit classi cation of black holes to the nilpotent classi cation of complex qubits. We provide preliminary results on the corresponding covariant classi cation. We explore the interesting parallel development of supersymmetric generalisations of qubits and entanglement, complete with two- and three-superqubit entanglement measures. Lastly, we briefly mention the supergravity technology of cubic Jordan algebras and Freudenthal triple systems (FTS), which are used to: 1) Relate FTS ranks to threequbit entanglement and compute SLOCC orbits. 2) Define new black hole dualities distinct from U-duality and related by a 4D/5D lift. 3) Clarify the state of knowledge of integral U-duality orbits in maximally extended supergravity in four, five, and six dimensions.
We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known as the 3-tangle, and the entropy of the 8-charge ST U black hole of N = 2 supergravity, both of which are given by the [SL(2)] 3 invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of three-qubit entanglements is related to the classification of N = 2 supersymmetric ST U black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. At the microscopic level, the black holes are described by intersecting D3-branes whose wrapping around the six compact dimensions T 6 provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, |ABC〉 (A, B, C = 0 or 1), with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge N = 8 black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan’s E7 ⊃ [SL(2)] 7 invariant. The qubits are naturally described by the seven vertices ABCDEF G of the Fano plane, which provides the multiplication table of the seven imaginary octonions,
Recent investigations have established an analogy between the entropy of four-dimensional supersymmetric black holes in string theory and entanglement in quantum information theory. Examples include: (1) N=2 STU black holes and the tripartite entanglement of three qubits (2-state systems), where the common symmetry is [SL(2)]3 and (2) N=8 black holes and the tripartite entanglement of seven qubits where the common symmetry is ${E}_{7}\ensuremath{\supset}[SL(2){]}^{7}$. Here we present another example: N=8 black holes (or black strings) in five dimensions and the bipartite entanglement of three qutrits (3-state systems), where the common symmetry is ${E}_{6}\ensuremath{\supset}[SL(3){]}^{3}$. Both the black hole (or black string) entropy and the entanglement measure are provided by the Cartan cubic E6 invariant. Similar analogies exist for magic N=2 supergravity black holes in both four and five dimensions.
Recent work has established a correspondence between the tripartite entanglement measure of three qubits and the macroscopic entropy of the four-dimensional 8-charge STU black hole of supergravity. Here we consider the configurations of intersecting D3-branes, whose wrapping around the six compact dimensions T6 provides the microscopic string-theoretic interpretation of the charges, and associate the three-qubit basis vectors |ABC>, (A, B, C=0 or 1) with the corresponding 8 wrapping cycles. In particular, we relate a well-known fact of quantum information theory, that the most general real three-qubit state can be parameterized by four real numbers and an angle, to a well-known fact of string theory, that the most general STU black hole can be described by four D3-branes intersecting at an angle.
The black-hole--qubit correspondence has been proven to be ``useful for obtaining additional insight into one of the string black hole theory and quantum information theory by exploiting approaches of the other"[Phys. Rev. D 82, 026003 (2010)]. Though different classes of stringy black holes can be related to the well-known stochastic local operations and classical communication (SLOCC) entanglement classes of pure states, the string theory requires a more detailed classification than the SLOCC classification of three qubits. In this paper, we derive the entanglement family of three qubits under local unitary operations (LU), and use the black-hole--qubit correspondence to classify \textquotedblleft large\textquotedblright\ black holes into seven inequivalent families. In particular, we show that two black holes with 4 non-vanishing charges (q0, p1, p2, and p3) are LU equivalent if their difference is only in the signs of charges. Thus, the classification of black holes is independent of the signs of charges and is only related to the ratio of the absolute values of charges. This observation simplifies the classification task, as one would only need to consider either the classification of non-BPS black holes or the classification of BPS black holes, but not both. Moreover, through the LU classification, the physical basis for this black-hole--qubit correspondence can be observed, and a relation between the black-hole entropy and the von Neumann entanglement entropy is revealed. Therefore, the LU classification offers a more straightforward physical connection than the SLOCC classification. Based on the LU classification, we further study the properties of von Neumann entanglement entropy for each of the seven families, and find the black holes with the maximal von Neumann entanglement entropy.
The black-hole--qubit correspondence has been proven to be ``useful for obtaining additional insight into one of the string black hole theory and quantum information theory by exploiting approaches of the other"[Phys. Rev. D 82, 026003 (2010)]. Though different classes of stringy black holes can be related to the well-known stochastic local operations and classical communication (SLOCC) entanglement classes of pure states, the string theory requires a more detailed classification than the SLOCC classification of three qubits. In this paper, we derive the entanglement family of three qubits under local unitary operations (LU), and use the black-hole--qubit correspondence to classify \textquotedblleft large\textquotedblright\ black holes into seven inequivalent families. In particular, we show that two black holes with 4 non-vanishing charges (q0, p1, p2, and p3) are LU equivalent if their difference is only in the signs of charges. Thus, the classification of black holes is independent of the signs of charges and is only related to the ratio of the absolute values of charges. This observation simplifies the classification task, as one would only need to consider either the classification of non-BPS black holes or the classification of BPS black holes, but not both. Moreover, through the LU classification, the physical basis for this black-hole--qubit correspondence can be observed, and a relation between the black-hole entropy and the von Neumann entanglement entropy is revealed. Therefore, the LU classification offers a more straightforward physical connection than the SLOCC classification. Based on the LU classification, we further study the properties of von Neumann entanglement entropy for each of the seven families, and find the black holes with the maximal von Neumann entanglement entropy.
As the frontiers of physics steadily progress into the 21st century we should bear in mind that the conceptual edifice of 20th-century physics has at its foundations two mutually incompatible theories; quantum mechanics and Einstein’s general theory of relativity. While general relativity refuses to succumb to quantum rule, black holes are raising quandaries that strike at the very heart of quantum theory. M-theory is a compelling candidate theory of quantum gravity. Living in eleven dimensions it encompasses and connects the five possible 10-dimensional superstring theories. However, Mtheory is fundamentally non-perturbative and consequently remains largely mysterious, offering up only disparate corners of its full structure. The physics of black holes has occupied centre stage in uncovering its non-perturbative structure. The dawn of the 21st-century has also played witness to the birth of the information age and with it the world of quantum information science. At its heart lies the phenomenon of quantum entanglement. Entanglement has applications in the emerging technologies of quantum computing and quantum cryptography, and has been used to realize quantum teleportation experimentally. The longest standing open problem in quantum information is the proper characterisation of multipartite entanglement. It is of utmost importance from both a foundational and a technological perspective. In 2006 the entropy formula for a particular 8-charge black hole appearing in M-theory was found to be given by the ’hyperdeterminant’, a quantity introduced by the mathematician Cayley in 1845. Remarkably, the hyperdeterminant also measures the degree of tripartite entanglement shared by three qubits, the basic units of quantum information. It turned out that the different possible types of three-qubit entanglement corresponded directly to the different possible subclasses of this particular black hole. This initial observation provided a link relating various black holes and quantum information systems. Since then, we have been examining this two-way dictionary between black holes and qubits and have used our knowledge of M-theory to discover new things about multipartite entanglement and quantum information theory and, vice-versa, to garner new insights into black holes and M-theory. There is now a growing dictionary, which translates a variety of phenomena in one language to those in the other. Developing these fascinating relationships, exploiting them to better understand both M-theory and quantum entanglement is the goal of this thesis. In particular, we adopt the elegant mathematics of octonions, Jordan algebras and the Freudenthal triple system as our guiding framework. In the course of this investigation we will see how these fascinating algebraic structures can be used to quantify entanglement and define new black hole dualities.