5 research outputs found
The role of supersymmetry in the black hole/qubit correspondence
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
Black Holes, Qubits and Octonions
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, known as the 3-tangle, and the entropy
of the 8-charge STU 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. There are further relationships between the attractor mechanism and
local distillation protocols. 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 supset [SL(2)]^7 invariant. The
qubits are naturally described by the seven vertices ABCDEFG 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 x
8 charges of N=8: the 24=3 x 8 NS-NS charges correspond to the three imaginary
quaternions and the 32=4 x 8 R-R to the four complementary imaginary octonions.
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 supset [SL(3)]^3 invariant.Comment: Version to appear in Physics Reports, including previously omitted
new results on small STU black hole charge orbits and expanded bibliography.
145 pages, 15 figures, 41 table
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size