788 research outputs found
Utilitarianism with and without expected utility
We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single ‘individual preorder’. The theorems give axioms that uniquely determine a social preorder in terms of this individual preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems are also consistent with the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels. In that sense, expected utility is inessential to Harsanyi-style utilitarianism. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We then derive further results under the assumption of our basic axioms. First, the individual preorder satisfies the main expected utility axiom of strong independence if and only if the social preorder has a vector-valued expected total utility representation, covering Harsanyi’s utilitarian theorem as a special case. Second, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to strong independence. Third, if the individual preorder satisfies a ‘local expected utility’ condition popular in non-expected utility theory, then the social preorder has a ‘local expected total utility’ representation. Fourth, a wide range of non-expected utility theories nevertheless lead to social preorders of outcomes that have been seen as canonically egalitarian, such as rank-dependent social preorders. Although our aggregation theorems are stated under conditions of risk, they are valid in more general frameworks for representing uncertainty or ambiguity
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
VOA[_4]
We take a peek at a general program that associates vertex (or, chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras
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
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