34,782 research outputs found
On reverse hypercontractivity
We study the notion of reverse hypercontractivity. We show that reverse
hypercontractive inequalities are implied by standard hypercontractive
inequalities as well as by the modified log-Sobolev inequality. Our proof is
based on a new comparison lemma for Dirichlet forms and an extension of the
Strook-Varapolos inequality.
A consequence of our analysis is that {\em all} simple operators L=Id-\E as
well as their tensors satisfy uniform reverse hypercontractive inequalities.
That is, for all and every positive valued function for we have . This should
be contrasted with the case of hypercontractive inequalities for simple
operators where is known to depend not only on and but also on the
underlying space.
The new reverse hypercontractive inequalities established here imply new
mixing and isoperimetric results for short random walks in product spaces, for
certain card-shufflings, for Glauber dynamics in high-temperatures spin systems
as well as for queueing processes. The inequalities further imply a
quantitative Arrow impossibility theorem for general product distributions and
inverse polynomial bounds in the number of players for the non-interactive
correlation distillation problem with -sided dice.Comment: Final revision. Incorporates referee's comments. The proof of
appendix B has been corrected. A shorter version of this article will appear
in GAF
The simplest causal inequalities and their violation
In a scenario where two parties share, act on and exchange some physical
resource, the assumption that the parties' actions are ordered according to a
definite causal structure yields constraints on the possible correlations that
can be established. We show that the set of correlations that are compatible
with a definite causal order forms a polytope, whose facets define causal
inequalities. We fully characterize this causal polytope in the simplest case
of bipartite correlations with binary inputs and outputs. We find two families
of nonequivalent causal inequalities; both can be violated in the recently
introduced framework of process matrices, which extends the standard quantum
formalism by relaxing the implicit assumption of a fixed causal structure. Our
work paves the way to a more systematic investigation of causal inequalities in
a theory-independent way, and of their violation within the framework of
process matrices.Comment: 7 + 4 pages, 2 figure
Shifting the Quantum-Classical Boundary: Theory and Experiment for Statistically Classical Optical Fields
The growing recognition that entanglement is not exclusively a quantum
property, and does not even originate with Schr\"odinger's famous remark about
it [Proc. Camb. Phil. Soc. 31, 555 (1935)], prompts examination of its role in
marking the quantum-classical boundary. We have done this by subjecting
correlations of classical optical fields to new Bell-analysis experiments, and
report here values of the Bell parameter greater than . This
is many standard deviations outside the limit established by the
Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [Phys. Rev. Lett. 23, 880
(1969)], in agreement with our theoretical classical prediction, and not far
from the Tsirelson limit . These results cast a new light
on the standard quantum-classical boundary description, and suggest a
reinterpretation of it.Comment: Comments and Remarks are warmly welcome! arXiv admin note: text
overlap with arXiv:1406.333
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