27 research outputs found
Quantum Goethals-Preparata Codes
We present a family of non-additive quantum codes based on Goethals and
Preparata codes with parameters ((2^m,2^{2^m-5m+1},8)). The dimension of these
codes is eight times higher than the dimension of the best known additive
quantum codes of equal length and minimum distance.Comment: Submitted to the 2008 IEEE International Symposium on Information
Theor
Generalized Concatenation for Quantum Codes
We show how good quantum error-correcting codes can be constructed using
generalized concatenation. The inner codes are quantum codes, the outer codes
can be linear or nonlinear classical codes. Many new good codes are found,
including both stabilizer codes as well as so-called nonadditive codes.Comment: 5 pages, to be presented at ISIT 200
Quantum phase uncertainty in mutually unbiased measurements and Gauss sums
Mutually unbiased bases (MUBs), which are such that the inner product between
two vectors in different orthogonal bases is constant equal to the inverse
, with the dimension of the finite Hilbert space, are becoming
more and more studied for applications such as quantum tomography and
cryptography, and in relation to entangled states and to the Heisenberg-Weil
group of quantum optics. Complete sets of MUBs of cardinality have been
derived for prime power dimensions using the tools of abstract algebra
(Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions
the cardinality is much less. The bases can be reinterpreted as quantum phase
states, i.e. as eigenvectors of Hermitean phase operators generalizing those
introduced by Pegg & Barnett in 1989. The MUB states are related to additive
characters of Galois fields (in odd characteristic p) and of Galois rings (in
characteristic 2). Quantum Fourier transforms of the components in vectors of
the bases define a more general class of MUBs with multiplicative characters
and additive ones altogether. We investigate the complementary properties of
the above phase operator with respect to the number operator. We also study the
phase probability distribution and variance for physical states and find them
related to the Gauss sums, which are sums over all elements of the field (or of
the ring) of the product of multiplicative and additive characters. Finally we
relate the concepts of mutual unbiasedness and maximal entanglement. This
allows to use well studied algebraic concepts as efficient tools in our quest
of minimal uncertainty in quantum information primitives.Comment: 11 page
Equiangular lines, mutually unbiased bases, and spin models
We use difference sets to construct interesting sets of lines in complex
space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in
C^k when k-1 is a prime power. Using semiregular relative difference sets with
parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k.
We show how to construct these difference sets from commutative semifields and
that several known maximal sets of mutually unbiased bases can be obtained in
this way, resolving a conjecture about the monomiality of maximal sets. We also
relate mutually unbiased bases to spin models.Comment: 23 pages; no figures. Minor correction as pointed out in
arxiv.org:1104.337