4,986 research outputs found
The SIC Question: History and State of Play
Recent years have seen significant advances in the study of symmetric
informationally complete (SIC) quantum measurements, also known as maximal sets
of complex equiangular lines. Previously, the published record contained
solutions up to dimension 67, and was with high confidence complete up through
dimension 50. Computer calculations have now furnished solutions in all
dimensions up to 151, and in several cases beyond that, as large as dimension
844. These new solutions exhibit an additional type of symmetry beyond the
basic definition of a SIC, and so verify a conjecture of Zauner in many new
cases. The solutions in dimensions 68 through 121 were obtained by Andrew
Scott, and his catalogue of distinct solutions is, with high confidence,
complete up to dimension 90. Additional results in dimensions 122 through 151
were calculated by the authors using Scott's code. We recap the history of the
problem, outline how the numerical searches were done, and pose some
conjectures on how the search technique could be improved. In order to
facilitate communication across disciplinary boundaries, we also present a
comprehensive bibliography of SIC research.Comment: 16 pages, 1 figure, many references; v3: updating bibliography,
dimension eight hundred forty fou
Reliable Physical Layer Network Coding
When two or more users in a wireless network transmit simultaneously, their
electromagnetic signals are linearly superimposed on the channel. As a result,
a receiver that is interested in one of these signals sees the others as
unwanted interference. This property of the wireless medium is typically viewed
as a hindrance to reliable communication over a network. However, using a
recently developed coding strategy, interference can in fact be harnessed for
network coding. In a wired network, (linear) network coding refers to each
intermediate node taking its received packets, computing a linear combination
over a finite field, and forwarding the outcome towards the destinations. Then,
given an appropriate set of linear combinations, a destination can solve for
its desired packets. For certain topologies, this strategy can attain
significantly higher throughputs over routing-based strategies. Reliable
physical layer network coding takes this idea one step further: using
judiciously chosen linear error-correcting codes, intermediate nodes in a
wireless network can directly recover linear combinations of the packets from
the observed noisy superpositions of transmitted signals. Starting with some
simple examples, this survey explores the core ideas behind this new technique
and the possibilities it offers for communication over interference-limited
wireless networks.Comment: 19 pages, 14 figures, survey paper to appear in Proceedings of the
IEE
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