Three-qubit operators, the split Cayley hexagon of order two, and black holes

Lévay, Péter; Санига, Метод; Vrana, Péter

doi:10.1103/physrevd.78.124022

Cited by 68 publications

(103 citation statements)

References 44 publications

(125 reference statements)

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“…In the apparently separate world of quantum gravity, the Bekenstein-Hawking entropy [1,2] of black holes has also occupied centre stage. Despite their apparent differences, recent work [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] has demonstrated a correspondence between the two. Although we still do not know whether there are any physical reasons underlying these mathematical coincidences, there is now a growing dictionary, which translates a variety of phenomena in one language to those in the other.…”

Section: Overviewmentioning

confidence: 99%

“…In [16] the P SL(2, F 7 ) symmetry of the N = 8 black hole entropy has been related, via a special set of 63 three-qubit operators, to the generalised hexagon of order two using the dictionary constructed in section 10.2.1. They further suggested that the full G 2 (2) symmetry of the hexagon may be preserved by the black hole entropy.…”

Section: D3 Decomposition Of the 56-dimensional Representationmentioning

confidence: 99%

“…The expectation value of an observable, represented by a Hermitian matrix A, is then given by, A ψ = tr(P ψ A). To reproduce these results using the Jordan framework it is necessary to introduce a trace form [214,217], 15) where ν ∈ Z and the set {b i } forms a basis for the Jordan algebra such that any element a may be written as a = a i b i , a i ∈ R. This defines a positive definite bilinear inner product, 16)since one can always choose a basis such that b i • b j = δ ij 1 + f ijk b k . Any two idempotents 10 , P 1 and P 2 , are orthogonal, P 1 • P 2 = 0, if they are orthogonal with respect to the inner product (13.16).…”

Section: The Jordan Algebra Formulation Of Quantum Mechanicsmentioning

confidence: 99%

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Black holes, qubits and octonions

et al. 2009

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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 .

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Section: Overviewmentioning

confidence: 99%

Section: D3 Decomposition Of the 56-dimensional Representationmentioning

confidence: 99%

Section: The Jordan Algebra Formulation Of Quantum Mechanicsmentioning

confidence: 99%

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Black holes, qubits and octonions

et al. 2009

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“…This research † † The representation found for the group A 2 (2) is different from [48]. The generators x 1 and x 2 contain the Pauli spin matrices and satisfy x…”

Section: Acknowledgementsmentioning

confidence: 99%

Unitary Reflection Groups for Quantum Fault Tolerance

Planat¹,

Kibler²

2010

Jnl of Comp & Theo Nano

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Abstract.This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, J Phys A: Math Theor 41, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type B 3 and G 2 (for single qubits), D 5 and A 4 (for two qubits), E 7 and E 6 (for three qubits), the complex reflection groups G(2 l , 2, 5) and groups No 9 and 31 in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the π/4 phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 New J. of Phys. 6, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.

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“…In particular, the P SL 2 (7) subgroup of the automorphism group of the hexagon and its associated Coxeter subgeometry were found to be intricately related to the E 7 -symmetric black-hole entropy formula in string theory, given a prominent role played by Veldkamp points that answer to Klein quadrics in the ambient projective space PG(5, 2) [7]. On the other hand, the E 6 -symmetric entropy formula describing black holes and black strings in five dimensions is underlaid by the geometry of generalized quadrangle GQ (2,4) and its Veldkamp space [8]; here, the two pronounced truncations of the entropy formula with 15 and 11 charges correspond exactly to two distinct types of Veldkamp points of GQ(2, 4) [9].…”

Section: Introductionmentioning

confidence: 99%

Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

Санига

Holweck

Pracna

2017

Int. J. Geom. Methods Mod. Phys.

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Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type D n , 4 ≤ n ≤ 8, it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG(3, 2). Proper labelling of the vertices of the diagram (for 4 ≤ n ≤ 7) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG(3, 2). The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space W (3, 2) of the PG(3, 2) whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin-Peres magic square. In the case of n = 8, a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG(5, 2), the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG(3, 2) are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin-Peres square. Moreover, save for n = 4, each Veldkamp space is also endowed with some 'exceptional' point(s). Interestingly, two such points in the n = 8 case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix Y .

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Three-qubit operators, the split Cayley hexagon of order two, and black holes

Cited by 68 publications

References 44 publications

Black holes, qubits and octonions

Black holes, qubits and octonions

Unitary Reflection Groups for Quantum Fault Tolerance

Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

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