8 research outputs found
A SU(2) recipe for mutually unbiased bases
A simple recipe for generating a complete set of mutually unbiased bases in
dimension (2j+1)**e, with 2j + 1 prime and e positive integer, is developed
from a single matrix acting on a space of constant angular momentum j and
defined in terms of the irreducible characters of the cyclic group C(2j+1). As
two pending results, this matrix is used in the derivation of a polar
decomposition of SU(2) and of a FFZ algebra.Comment: v2: abstract enlarged, a corollary added, acknowledgments added, one
reference added, presentation improved; v3: two misprints correcte
Bases for qudits from a nonstandard approach to SU(2)
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for
quantum information and quantum computation are constructed from angular
momentum theory and su(2) Lie algebraic methods. We report on a formula for
deriving in one step the (1+p)p qupits (i.e., qudits with d = p a prime
integer) of a complete set of 1+p mutually unbiased bases in C^p. Repeated
application of the formula can be used for generating mutually unbiased bases
in C^d with d = p^e (e > or = 2) a power of a prime integer. A connection
between mutually unbiased bases and the unitary group SU(d) is briefly
discussed in the case d = p^e.Comment: From a talk presented at the 13th International Conference on
Symmetry Methods in Physics (Dubna, Russia, 6-9 July 2009) organized in
memory of Prof. Yurii Fedorovich Smirnov by the Bogoliubov Laboratory of
Theoretical Physics of the JINR and the ICAS at Yerevan State University
Clifford groups of quantum gates, BN-pairs and smooth cubic surfaces
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of
representing Clifford quantum gates in terms of unitary reflections is
revisited. In this essay, the geometry of a Clifford group G is expressed as a
BN-pair, i.e. a pair of subgroups B and N that generate G, is such that
intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group
and two extra axioms are satisfied by the double cosets acting on B. The
BN-pair used in this decomposition relies on the swap and match gates already
introduced for classically simulating quantum circuits (R Jozsa and A Miyake,
Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are
related to the configuration with 27 lines on a smooth cubic surface.Comment: 7 pages, version to appear in Journal of Physics A: Mathematical and
Theoretical (fast track communications
Variations on a theme of Heisenberg, Pauli and Weyl
The parentage between Weyl pairs, generalized Pauli group and unitary group
is investigated in detail. We start from an abstract definition of the
Heisenberg-Weyl group on the field R and then switch to the discrete
Heisenberg-Weyl group or generalized Pauli group on a finite ring Z_d. The main
characteristics of the latter group, an abstract group of order d**3 noted P_d,
are given (conjugacy classes and irreducible representation classes or
equivalently Lie algebra of dimension d**3 associated with P_d). Leaving the
abstract sector, a set of Weyl pairs in dimension d is derived from a polar
decomposition of SU(2) closely connected to angular momentum theory. Then, a
realization of the generalized Pauli group P_d and the construction of
generalized Pauli matrices in dimension d are revisited in terms of Weyl pairs.
Finally, the Lie algebra of the unitary group U(d) is obtained as a subalgebra
of the Lie algebra associated with P_d. This leads to a development of the Lie
algebra of U(d) in a basis consisting of d**2 generalized Pauli matrices. In
the case where d is a power of a prime integer, the Lie algebra of SU(d) can be
decomposed into d-1 Cartan subalgebras.Comment: Dedicated to the memory of Mosh\'e Flato on the occasion of the tenth
anniversary of his deat
A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
The basic methods of constructing the sets of mutually unbiased bases in the
Hilbert space of an arbitrary finite dimension are discussed and an emerging
link between them is outlined. It is shown that these methods employ a wide
range of important mathematical concepts like, e.g., Fourier transforms, Galois
fields and rings, finite and related projective geometries, and entanglement,
to mention a few. Some applications of the theory to quantum information tasks
are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two
more references adde
Topological Color Codes and Two-Body Quantum Lattice Hamiltonians
Topological color codes are among the stabilizer codes with remarkable
properties from quantum information perspective. In this paper we construct a
four-valent lattice, the so called ruby lattice, governed by a 2-body
Hamiltonian. In a particular regime of coupling constants, degenerate
perturbation theory implies that the low energy spectrum of the model can be
described by a many-body effective Hamiltonian, which encodes the color code as
its ground state subspace. The gauge symmetry
of color code could already be realized by
identifying three distinct plaquette operators on the lattice. Plaquettes are
extended to closed strings or string-net structures. Non-contractible closed
strings winding the space commute with Hamiltonian but not always with each
other giving rise to exact topological degeneracy of the model. Connection to
2-colexes can be established at the non-perturbative level. The particular
structure of the 2-body Hamiltonian provides a fruitful interpretation in terms
of mapping to bosons coupled to effective spins. We show that high energy
excitations of the model have fermionic statistics. They form three families of
high energy excitations each of one color. Furthermore, we show that they
belong to a particular family of topological charges. Also, we use
Jordan-Wigner transformation in order to test the integrability of the model
via introducing of Majorana fermions. The four-valent structure of the lattice
prevents to reduce the fermionized Hamiltonian into a quadratic form due to
interacting gauge fields. We also propose another construction for 2-body
Hamiltonian based on the connection between color codes and cluster states. We
discuss this latter approach along the construction based on the ruby lattice.Comment: 56 pages, 16 figures, published version