6 research outputs found
Sharper Upper Bounds for Unbalanced Uniquely Decodable Code Pairs
Two sets form a Uniquely Decodable Code Pair
(UDCP) if every pair , yields a distinct sum , where
the addition is over . We show that every UDCP , with and , satisfies . For sufficiently small , this bound significantly
improves previous bounds by Urbanke and Li~[Information Theory Workshop '98]
and Ordentlich and Shayevitz~[2014, arXiv:1412.8415], which upper bound
by and , respectively, as approaches .Comment: 11 pages; to appear at ISIT 201
Sharper Upper Bounds for Unbalanced Uniquely Decodable Code Pairs
Two sets form a Uniquely Decodable Code Pair (UDCP) if every pair , yields a distinct sum , where the addition is over . We show that every UDCP , with and , satisfies . For sufficiently small , this bound significantly improves previous bounds by Urbanke and Li~[Information Theory Workshop '98] and Ordentlich and Shayevitz~[2014, arXiv:1412.8415], which upper bound by and , respectively, as approaches
The Heisenberg limit for laser coherence
To quantify quantum optical coherence requires both the particle- and
wave-natures of light. For an ideal laser beam [1,2,3], it can be thought of
roughly as the number of photons emitted consecutively into the beam with the
same phase. This number, , can be much larger than , the
number of photons in the laser itself. The limit on for an ideal
laser was thought to be of order [4,5]. Here, assuming nothing about
the laser operation, only that it produces a beam with certain properties close
to those of an ideal laser beam, and that it does not have external sources of
coherence, we derive an upper bound: . Moreover, using
the matrix product states (MPSs) method [6,7,8,9], we find a model that
achieves this scaling, and show that it could in principle be realised using
circuit quantum electrodynamics (QED) [10]. Thus is
only a standard quantum limit (SQL); the ultimate quantum limit, or Heisenberg
limit, is quadratically better.Comment: 6 pages, 4 figures, and 31 pages of supplemental information. v2:
This paper is now published [Nature Physics DOI:10.1038/s41567-020-01049-3
(26 October 2020)]. For copyright reasons, this arxiv paper is based on a
version of the paper prior to the accepted (21 August 2020) versio