7,131 research outputs found
Notes on the firewall paradox, complexity, and quantum theory
We investigate what it means to apply the solution, proposed to the firewall
paradox by Harlow and Hayden, to the famous quantum paradoxes of Sch\"odinger's
Cat and Wigner's Friend if ones views these as posing a thermodynamic decoding
problem (as does Hawking radiation in the firewall paradox). The implications
might point to a relevance of the firewall paradox for the axiomatic and set
theoretic foundations underlying mathematics. We reconsider in this context the
results of Benioff on the foundational challenges posed by the randomness
postulate of quantum theory. A central point in our discussion is that one can
mathematically not naturally distinguish between computational complexity (as
central to the approach of Harlow and Hayden and further developed by Susskind)
and proof theoretic complexity (since they represent the same concept on a
Turing machine), with the latter being related to a finite bound on Kolmogorov
entropy (due to Chaitin incompleteness).Comment: 31 pages; further references adde
Geometric Analogue of Holographic Reduced Representation
Holographic reduced representations (HRR) are based on superpositions of
convolution-bound -tuples, but the -tuples cannot be regarded as vectors
since the formalism is basis dependent. This is why HRR cannot be associated
with geometric structures. Replacing convolutions by geometric products one
arrives at reduced representations analogous to HRR but interpretable in terms
of geometry. Variable bindings occurring in both HRR and its geometric analogue
mathematically correspond to two different representations of
(the additive group of binary -tuples with addition
modulo 2). As opposed to standard HRR, variable binding performed by means of
geometric product allows for computing exact inverses of all nonzero vectors, a
procedure even simpler than approximate inverses employed in HRR. The formal
structure of the new reduced representation is analogous to cartoon
computation, a geometric analogue of quantum computation.Comment: typos in eqs. (57-58) are correcte
A pedagogical explanation for the non-renormalizability of gravity
We present a short and intuitive argument explaining why gravity is
non-renormalizable. The argument is based on black-hole domination of the high
energy spectrum of gravity and not on the standard perturbative irrelevance of
the gravitational coupling. This is a pedagogical note, containing textbook
material that is widely appreciated by experts and is by no means original.Comment: 10 pages, 4 figures, Latex. V2: typos corrected, some emphasis and
clarifications adde
Quantum Walks on Necklaces and Mixing
We analyze continuous-time quantum walks on necklace graphs - cyclical graphs
consisting of many copies of a smaller graph (pearl). Using a Bloch-type ansatz
for the eigenfunctions, we block-diagonalize the Hamiltonian, reducing the
effective size of the problem to the size of a single pearl. We then present a
general approach for showing that the mixing time scales (with growing size of
the necklace) similarly to that of a simple walk on a cycle. Finally, we
present results for mixing on several necklace graphs.Comment: 11 pages, 5 figures, typos corrected, acknowledgements update
Entanglement beyond subsystems
We present a notion of generalized entanglement which goes beyond the
conventional definition based on quantum subsystems. This is accomplished by
directly defining entanglement as a property of quantum states relative to a
distinguished set of observables singled out by Physics. While recovering
standard entanglement as a special case, our notion allows for substantially
broader generality and flexibility, being applicable, in particular, to
situations where existing tools are not directly useful.Comment: 14 pages, no figures. Invited contribution to the Proceedings of the
Coding Theory and Quantum Computing Workshop, AMS series in Contemporary
Mathematics (2004
Disjunctive Quantum Logic in Dynamic Perspective
In arXiv: math.LO/0011208 we proposed the {\sl intuitionistic or disjunctive
representation of quantum logic}, i.e., a representation of the property
lattice of physical systems as a complete Heyting algebra of logical
propositions on these properties, where this complete Heyting algebra goes
equipped with an additional operation, the {\sl operational resolution}, which
identifies the properties within the logic of propositions. This representation
has an important application ``towards dynamic quantum logic'', namely in
describing the temporal indeterministic propagation of actual properties of
physical systems. This paper can as such by conceived as an addendum to
``Quantum Logic in Intuitionistic Perspective'' that discusses spin-off and
thus provides an additional motivation. We derive a quantaloidal semantics for
dynamic disjunctive quantum logic and illustrate it for the particular case of
a perfect (quantum) measurement.Comment: 13 Pages; camera ready version; some corrections are made;
indications and references on current progress on the matter have been
include
Quantum Reasoning using Lie Algebra for Everyday Life (and AI perhaps...)
We investigate the applicability of the formalism of quantum mechanics to
everyday life. It seems to be directly relevant for situations in which the
very act of coming to a conclusion or decision on one issue affects one's
confidence about conclusions or decisions on another issue. Lie algebra theory
is argued to be a very useful tool in guiding the construction of quantum
descriptions of such situations. Tests, extensions and speculative applications
and implications, including for the encoding of thoughts in neural networks,
are discussed. It is suggested that the recognition and incorporation of such
mathematical structure into machine learning and artificial intelligence might
lead to significant efficiency and generality gains in addition to ensuring
probabilistic reasoning at a fundamental level.Comment: 17 page
A consistent flow of entropy
A common approach to evaluate entropy in quantum systems is to solve a
master-Bloch equation to determine density matrix and substitute it in entropy
definition. However, this method has been recently understood to lack many
energy correlators. The new correlators make entropy evaluation to be different
from the substitution method described above. The reason for such complexity
lies in the nonlinearity of entropy. In this paper we present a pedagogical
approach to evaluate the new correlators and explain their contribution in the
analysis. We show that the inherent nonlinearity in entropy makes the second
law of thermodynamics to carry new terms associated to the new correlators. Our
results show important new remarks on quantum black holes. Our formalism
reveals that the notion of degeneracy of states at the event horizon makes an
indispensable deviation from black hole entropy in the leading order.Comment: Contribution to Special issue in Fortschritte der Physik for the
Frontiers of Quantum and Mesoscopic Thermodynamics Conference , 7 pages, 2
figure
Distributed Computation as Hierarchy
This paper presents a new distributed computational model of distributed
systems called the phase web that extends V. Pratt's orthocurrence relation
from 1986. The model uses mutual-exclusion to express sequence, and a new kind
of hierarchy to replace event sequences, posets, and pomsets. The model
explicitly connects computation to a discrete Clifford algebra that is in turn
extended into homology and co-homology, wherein the recursive nature of objects
and boundaries becomes apparent and itself subject to hierarchical recursion.
Topsy, a programming environment embodying the phase web, is available from
www.cs.auc.dk/topsy.Comment: 16 pages, 3 figure
Entanglement in Weakly Coupled Lattice Gauge Theories
We present a direct lattice gauge theory computation that, without using
dualities, demonstrates that the entanglement entropy of Yang-Mills theories
with arbitrary gauge group contains a generic logarithmic term at
sufficiently weak coupling . In two spatial dimensions, for a region of
linear size , this term equals
and it dominates the universal part of the entanglement entropy. Such
logarithmic terms arise from the entanglement of the softest mode in the
entangling region with the environment. For Maxwell theory in two spatial
dimensions, our results agree with those obtained by dualizing to a compact
scalar with spontaneous symmetry breaking.Comment: 35 pages, one figure; v2 with more detailed explanations, published
in JHE
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