27 research outputs found
MUBs: From finite projective geometry to quantum phase enciphering
This short note highlights the most prominent mathematical problems and
physical questions associated with the existence of the maximum sets of
mutually unbiased bases (MUBs) in the Hilbert space of a given dimensionComment: 5 pages, accepted for AIP Conf Book, QCMC 2004, Strathclyde, Glasgow,
minor correction
Distributed Storage Systems based on Equidistant Subspace Codes
Distributed storage systems based on equidistant constant dimension codes are
presented. These equidistant codes are based on the Pl\"{u}cker embedding,
which is essential in the repair and the reconstruction algorithms. These
systems posses several useful properties such as high failure resilience,
minimum bandwidth, low storage, simple algebraic repair and reconstruction
algorithms, good locality, and compatibility with small fields
Abstract algebra, projective geometry and time encoding of quantum information
Algebraic geometrical concepts are playing an increasing role in quantum
applications such as coding, cryptography, tomography and computing. We point
out here the prominent role played by Galois fields viewed as cyclotomic
extensions of the integers modulo a prime characteristic . They can be used
to generate efficient cyclic encoding, for transmitting secrete quantum keys,
for quantum state recovery and for error correction in quantum computing.
Finite projective planes and their generalization are the geometric counterpart
to cyclotomic concepts, their coordinatization involves Galois fields, and they
have been used repetitively for enciphering and coding. Finally the characters
over Galois fields are fundamental for generating complete sets of mutually
unbiased bases, a generic concept of quantum information processing and quantum
entanglement. Gauss sums over Galois fields ensure minimum uncertainty under
such protocols. Some Galois rings which are cyclotomic extensions of the
integers modulo 4 are also becoming fashionable for their role in time encoding
and mutual unbiasedness.Comment: To appear in R. Buccheri, A.C. Elitzur and M. Saniga (eds.),
"Endophysics, Time, Quantum and the Subjective," World Scientific, Singapore.
16 page
Equidistant Codes in the Grassmannian
Equidistant codes over vector spaces are considered. For -dimensional
subspaces over a large vector space the largest code is always a sunflower. We
present several simple constructions for such codes which might produce the
largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker
embedding, for 1-intersecting codes of -dimensional subspaces over \F_q^n,
, where the code size is is
presented. Finally, we present a related construction which generates
equidistant constant rank codes with matrices of size
over \F_q, rank , and rank distance .Comment: 16 page
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
A Geometric View of the Service Rates of Codes Problem and its Application to the Service Rate of the First Order Reed-Muller Codes
Service rate is an important, recently introduced, performance metric
associated with distributed coded storage systems. Among other interpretations,
it measures the number of users that can be simultaneously served by the
storage system. We introduce a geometric approach to address this problem. One
of the most significant advantages of this approach over the existing
approaches is that it allows one to derive bounds on the service rate of a code
without explicitly knowing the list of all possible recovery sets. To
illustrate the power of our geometric approach, we derive upper bounds on the
service rates of the first order Reed-Muller codes and simplex codes. Then, we
show how these upper bounds can be achieved. Furthermore, utilizing the
proposed geometric technique, we show that given the service rate region of a
code, a lower bound on the minimum distance of the code can be obtained