By regarding gravity as the convolution of left and right Yang-Mills theories together with a spectator scalar field in the bi-adjoint representation, we derive in linearised approximation the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincaré. As a concrete example we focus on the new-minimal (12+12) off-shell version of simple four-dimensional supergravity obtained by tensoring the off-shell Yang-Mills multiplets (4 + 4, NL = 1) and (3 + 0, NR = 0).PACS numbers: 11.25.-w Keywords: super-Yang-Mills, supergravity, local symmetries In Einstein's general theory of relativity the requirement that the laws of physics be the same to all observers is embodied in the principle of general covariance: the equations must be invariant under arbitrary coordinate transformations. Fermions require in addition local Lorentz invariance and many models of interest also incorporate local supersymmetry and local p-form gauge invariance. The purpose of this paper is to derive, in linearised approximation, all these gravitational symmetries starting from those of flat-space Yang-Mills, namely local gauge invariance and global (super)-Poincaré.Early attempts to derive gravity from Yang-Mills were based on gauging spacetime symmetries such as Lorentz, Poincaré or de Sitter [1][2][3][4][5]. More recently, the AdS/CFT correspondence has provided a different link between gravity and gauge theories [6][7][8]. However, our approach will differ from both in important respects. We appeal to the idea of "Gravity as the square of Yang-Mills" by tensoring left and right multiplets with arbitrary nonAbelian gauge groups G L and G R . Squaring Yang-Mills is a recurring theme in attempts to understand the quantum theory of gravity and appears in several different forms: Closed states from products of open states and KLT relations in string theory [9][10][11] 3,4,5,6,8,9,10,12,16. This 4×4 square in D = 3 is the base of a "magic pyramid" with a 3 × 3 square in D = 4, a 2 × 2 square in D = 6 and Type II supergravity at the apex in D = 10 [28]. In this paper we focus instead on the Yang-Mills origin of the local gravitational symmetries of general covariance, local Lorentz invariance, local supersymmetry and p-form gauge invariance acting on the classical fields.Although much of the squaring literature invokes taking a product of left and right Yang-MiIls fields

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 E 7 ⊃ [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, reflecting the fact that E 7 has a natural structure of an O-graded algebra. This in turn provides a novel imaginary octonionic interpretation of the 56 = 7 × 8 charges of N = 8: the 24 = 3 × 8 NS-NS charges correspond to the three imaginary quaternions and the 32 = 4 × 8 R-R to the four complementary imaginary octonions. We contrast this approach with that based on Jordan algebras and the Freudenthal triple system. N = 8 black holes (or black strings) in five dimensions are also related to the bipartite entanglement of three qutrits (3-state systems), where the analogous measure is Cartan's E 6 ⊃ [SL(3)] 3 invariant. Similar analogies exist for magic N = 2 supergravity black holes in both four and five dimensions. Despite the ubiquity of octonions, our analogy between black holes and quantum information theory is based on conventional quantum mechanics but for completeness we also provide a more exotic one based on octonionic quantum mechanics. Finally, we note some intriguing, but still mysterious, assignments of entanglements to cosets, such as the 4-way entanglement of eight qubits to E 8 /[SL (2)] 8 .

The quantised charges x of four dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U-duality and whose U-invariant quartic norm ∆(x) determines the lowest order entropy. Here we introduce a Freudenthal duality x →x, for whichx = −x. Although distinct from U-duality it nevertheless leaves ∆(x) invariant. However, the requirement thatx be integer restricts us to the subset of black holes for which ∆(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantised charges A of five dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N (A) determines the lowest order entropy. We introduce an analogous Jordan dual A ⋆ , with N (A) necessarily a perfect cube, for which A ⋆⋆ = A and which leaves N (A) invariant. The two dualities are related by a 4D/5D lift.

The Becchi-Rouet-Stora-Tyutin (BRST) transformations and equations of motion of a gravitytwo-form-dilaton system are derived from the product of two Yang-Mills theories in a BRST covariant form, to linear approximation. The inclusion of ghost fields facilitates the separation of the graviton and dilaton. The gravitational gauge fixing term is uniquely determined by those of the Yang-Mills factors which can be freely chosen. Moreover, the resulting gravity-two-form-dilaton Lagrangian is anti-BRST invariant and the BRST and anti-BRST charges anti commute as a direct consequence of the formalism.

We invoke the black hole/qubit correspondence to derive the classification of four-qubit entanglement. The U-duality orbits resulting from timelike reduction of string theory from D = 4 to D = 3 yield 31 entanglement families, which reduce to nine up to permutation of the four qubits.PACS numbers: 11.25.Mj, 03.65.Ud, 04.70.Dy Keywords: black hole, U-duality, qubit, entanglement Recent work has established some intriguing correspondences between two very different areas of theoretical physics: the entanglement of qubits in quantum information theory (QIT) and black holes in string theory. See [1] for a review. In particular, there is a one-to-one correspondence between the classification of three qubit entanglement [2] and the classification of extremal black holes in the ST U supergravity theory [3,4] that appears in the compactification of string theory from D = 10 to D = 4 dimensions. Moreover, the Bekenstein-Hawking black hole entropy is provided by the three-way entanglement measure.The purpose of this paper is to use this black hole/qubit correspondence to address the much more difficult problem of classifying four-qubit entanglement, currently an active area of research in QIT as experimentalists now control entanglement with four qubits [5]. Although two and three qubit entanglement is wellunderstood, the literature on four qubits can be confusing and seemingly contradictory, as illustrated in Table I. This is due in part to genuine calculational disagreements, but in part to the use of distinct (but in principle consistent and complementary) perspectives on the criteria for classification. On the one hand there is the "covariant" approach which distinguishes the orbits of the equivalence group of Stochastic Local Operations and Classical Communication (SLOCC) by the vanishing or not of covariants/invariants. This philosophy is adopted for the three-qubit case in [2,13], for example, where it was shown that three qubits can be tripartite entangled in two inequivalent ways, denoted W and GHZ (Greenberger-Horne-Zeilinger). The analogous four-qubit case was treated, with partial results, in [14]. On the other hand, there is the "normal form" approach which considers "families" of orbits. Any given state may be transformed into a unique normal form. If the normal form depends on some of the algebraically independent SLOCC invariants it constitutes a family of orbits parametrized by these invariants. On the other hand a parameter-independent family contains a single orbit. This philosophy is adopted for the four-qubit case |Ψ = a ABCD |ABCD A, B, C, D = 0, 1 in [11,12]. Up to permutation of the four qubits, these authors found 6 parameter-dependent families calledFor example, a family of orbits parametrized by all four of the algebraically independent SLOCC invariants is given by the normal form G abcd :(1)To illustrate the difference between these two approaches, consider the separable EPR-EPR state (|00 + |11 ) ⊗ (|00 + |11 ). Since this is obtained by setting b = c = d = 0 in (1) it belongs to the G abcd fa...