22,471 research outputs found
About the Dedekind psi function in Pauli graphs
We study the commutation structure within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. The simplest illustrative examples are the quartit ()
and two-qubit () systems. It is shown how the sum of divisor function
and the Dedekind psi function enter
into the theory for counting the number of maximal commuting sets of the qudit
system. In the case of a multiple qudit system (with and a prime),
the arithmetical functions and count the
cardinality of the symplectic polar space that endows the
commutation structure and its punctured counterpart, respectively. Symmetry
properties of the Pauli graphs attached to these structures are investigated in
detail and several illustrative examples are provided.Comment: Proceedings of Quantum Optics V, Cozumel to appear in Revista
Mexicana de Fisic
Quantum States Arising from the Pauli Groups, Symmetries and Paradoxes
We investigate multiple qubit Pauli groups and the quantum states/rays
arising from their maximal bases. Remarkably, the real rays are carried by a
Barnes-Wall lattice (). We focus on the smallest subsets of rays
allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem
rules out realistic non-contextual theories by resorting to impossible
assignments of rays among a selected set of maximal orthogonal bases. We
investigate the geometrical structure of small BKS-proofs involving
rays and -dimensional bases of -qubits. Specifically, we look at the
classes of parity proofs 18-9 with two qubits (A. Cabello, 1996), 36-11 with
three qubits (M. Kernaghan & A. Peres, 1995) and related classes. One finds
characteristic signatures of the distances among the bases, that carry various
symmetries in their graphs.Comment: The XXIXth International Colloquium on Group-Theoretical Methods in
Physics, China (2012
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
Quantum Entanglement and Projective Ring Geometry
The paper explores the basic geometrical properties of the observables
characterizing two-qubit systems by employing a novel projective ring geometric
approach. After introducing the basic facts about quantum complementarity and
maximal quantum entanglement in such systems, we demonstrate that the
1515 multiplication table of the associated four-dimensional matrices
exhibits a so-far-unnoticed geometrical structure that can be regarded as three
pencils of lines in the projective plane of order two. In one of the pencils,
which we call the kernel, the observables on two lines share a base of Bell
states. In the complement of the kernel, the eight vertices/observables are
joined by twelve lines which form the edges of a cube. A substantial part of
the paper is devoted to showing that the nature of this geometry has much to do
with the structure of the projective lines defined over the rings that are the
direct product of copies of the Galois field GF(2), with = 2, 3 and 4.Comment: 13 pages, 6 figures Fig. 3 improved, typos corrected; Version 4:
Final Version Published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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