282 research outputs found
Unitary designs and codes
A unitary design is a collection of unitary matrices that approximate the
entire unitary group, much like a spherical design approximates the entire unit
sphere. In this paper, we use irreducible representations of the unitary group
to find a general lower bound on the size of a unitary t-design in U(d), for
any d and t. We also introduce the notion of a unitary code - a subset of U(d)
in which the trace inner product of any pair of matrices is restricted to only
a small number of distinct values - and give an upper bound for the size of a
code of degree s in U(d) for any d and s. These bounds can be strengthened when
the particular inner product values that occur in the code or design are known.
Finally, we describe some constructions of designs: we give an upper bound on
the size of the smallest weighted unitary t-design in U(d), and we catalogue
some t-designs that arise from finite groups.Comment: 25 pages, no figure
Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier
In quantum mechanics some properties are maximally incompatible, such as the
position and momentum of a particle or the vertical and horizontal projections
of a 2-level spin. Given any definite state of one property the other property
is completely random, or unbiased. For N-level systems, the 6-level ones are
the smallest for which a tomographically efficient set of N+1 mutually unbiased
bases (MUBs) has not been found. To facilitate the search, we numerically
extend the classification of unbiased bases, or Hadamards, by incrementally
adjusting relative phases in a standard basis. We consider the non-unitarity
caused by small adjustments with a second order Taylor expansion, and choose
incremental steps within the 4-dimensional nullspace of the curvature. In this
way we prescribe a numerical integration of a 4-parameter set of Hadamards of
order 6.Comment: 5 pages, 2 figure
On the high density behavior of Hamming codes with fixed minimum distance
We discuss the high density behavior of a system of hard spheres of diameter
d on the hypercubic lattice of dimension n, in the limit n -> oo, d -> oo,
d/n=delta. The problem is relevant for coding theory. We find a solution to the
equations describing the liquid up to very large values of the density, but we
show that this solution gives a negative entropy for the liquid phase when the
density is large enough. We then conjecture that a phase transition towards a
different phase might take place, and we discuss possible scenarios for this
transition. Finally we discuss the relation between our results and known
rigorous bounds on the maximal density of the system.Comment: 15 pages, 6 figure
A simple construction of complex equiangular lines
A set of vectors of equal norm in represents equiangular lines
if the magnitudes of the inner product of every pair of distinct vectors in the
set are equal. The maximum size of such a set is , and it is conjectured
that sets of this maximum size exist in for every . We
describe a new construction for maximum-sized sets of equiangular lines,
exposing a previously unrecognized connection with Hadamard matrices. The
construction produces a maximum-sized set of equiangular lines in dimensions 2,
3 and 8.Comment: 11 pages; minor revisions and comments added in section 1 describing
a link to previously known results; correction to Theorem 1 and updates to
reference
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Transmutations and spectral parameter power series in eigenvalue problems
We give an overview of recent developments in Sturm-Liouville theory
concerning operators of transmutation (transformation) and spectral parameter
power series (SPPS). The possibility to write down the dispersion
(characteristic) equations corresponding to a variety of spectral problems
related to Sturm-Liouville equations in an analytic form is an attractive
feature of the SPPS method. It is based on a computation of certain systems of
recursive integrals. Considered as families of functions these systems are
complete in the -space and result to be the images of the nonnegative
integer powers of the independent variable under the action of a corresponding
transmutation operator. This recently revealed property of the Delsarte
transmutations opens the way to apply the transmutation operator even when its
integral kernel is unknown and gives the possibility to obtain further
interesting properties concerning the Darboux transformed Schr\"{o}dinger
operators.
We introduce the systems of recursive integrals and the SPPS approach,
explain some of its applications to spectral problems with numerical
illustrations, give the definition and basic properties of transmutation
operators, introduce a parametrized family of transmutation operators, study
their mapping properties and construct the transmutation operators for Darboux
transformed Schr\"{o}dinger operators.Comment: 30 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1111.444
MUBs inequivalence and affine planes
There are fairly large families of unitarily inequivalent complete sets of
N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The
number of such sets is not bounded above by any polynomial as a function of N.
While it is standard that there is a superficial similarity between complete
sets of MUBs and finite affine planes, there is an intimate relationship
between these large families and affine planes. This note briefly summarizes
"old" results that do not appear to be well-known concerning known families of
complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical
Physics 53, 032204 (2012) except for format changes due to the journal's
style policie
Symmetric Informationally Complete Quantum Measurements
We consider the existence in arbitrary finite dimensions d of a POVM
comprised of d^2 rank-one operators all of whose operator inner products are
equal. Such a set is called a ``symmetric, informationally complete'' POVM
(SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d.
SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and
foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions
two, three, and four. We further conjecture that a particular kind of
group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical
results up to dimension 45 to bolster this claim.Comment: 8 page
Discrete phase space based on finite fields
The original Wigner function provides a way of representing in phase space
the quantum states of systems with continuous degrees of freedom. Wigner
functions have also been developed for discrete quantum systems, one popular
version being defined on a 2N x 2N discrete phase space for a system with N
orthogonal states. Here we investigate an alternative class of discrete Wigner
functions, in which the field of real numbers that labels the axes of
continuous phase space is replaced by a finite field having N elements. There
exists such a field if and only if N is a power of a prime; so our formulation
can be applied directly only to systems for which the state-space dimension
takes such a value. Though this condition may seem limiting, we note that any
quantum computer based on qubits meets the condition and can thus be
accommodated within our scheme. The geometry of our N x N phase space also
leads naturally to a method of constructing a complete set of N+1 mutually
unbiased bases for the state space.Comment: 60 pages; minor corrections and additional references in v2 and v3;
improved historical introduction in v4; references to quantum error
correction in v5; v6 corrects the value quoted for the number of similarity
classes for N=
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