14,426 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Capacity Theorems for Quantum Multiple Access Channels: Classical-Quantum and Quantum-Quantum Capacity Regions
We consider quantum channels with two senders and one receiver. For an
arbitrary such channel, we give multi-letter characterizations of two different
two-dimensional capacity regions. The first region is comprised of the rates at
which it is possible for one sender to send classical information, while the
other sends quantum information. The second region consists of the rates at
which each sender can send quantum information. For each region, we give an
example of a channel for which the corresponding region has a single-letter
description. One of our examples relies on a new result proved here, perhaps of
independent interest, stating that the coherent information over any degradable
channel is concave in the input density operator. We conclude with connections
to other work and a discussion on generalizations where each user
simultaneously sends classical and quantum information.Comment: 38 pages, 1 figure. Fixed typos, added new example. Submitted to IEEE
Tranactions on Information Theor
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
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