173 research outputs found
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
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
Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
We study the commutation relations within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. Illustrative low dimensional examples are the quartit ()
and two-qubit () systems, the octit (), qubit/quartit () and three-qubit () systems, and so on. In the single qudit case,
e.g. , one defines a bijection between the maximal
commuting sets [with the sum of divisors of ] of Pauli
observables and the maximal submodules of the modular ring ,
that arrange into the projective line and a independent set
of size [with the Dedekind psi function]. In the
multiple qudit case, e.g. , the Pauli graphs rely on
symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if
) and GQ(3,3) (if ). More precisely, in dimension ( a
prime) of the Hilbert space, the observables of the Pauli group (modulo the
center) are seen as the elements of the -dimensional vector space over the
field . In this space, one makes use of the commutator to define
a symplectic polar space of cardinality , that
encodes the maximal commuting sets of the Pauli group by its totally isotropic
subspaces. Building blocks of are punctured polar spaces (i.e. a
observable and all maximum cliques passing to it are removed) of size given by
the Dedekind psi function . For multiple qudit mixtures (e.g.
qubit/quartit, qubit/octit and so on), one finds multiple copies of polar
spaces, ponctured polar spaces, hypercube geometries and other intricate
structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo
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
Qudits of composite dimension, mutually unbiased bases and projective ring geometry
The Pauli operators attached to a composite qudit in dimension may
be mapped to the vectors of the symplectic module
( the modular ring). As a result, perpendicular vectors
correspond to commuting operators, a free cyclic submodule to a maximal
commuting set, and disjoint such sets to mutually unbiased bases. For
dimensions , and 18, the fine structure and the incidence
between maximal commuting sets is found to reproduce the projective line over
the rings , , ,
and ,
respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A
Mathematical and Theoretical (2008) accepte
Pauli graphs, Riemann hypothesis, Goldbach pairs
Let consider the Pauli group with unitary quantum
generators (shift) and (clock) acting on the vectors of the
-dimensional Hilbert space via and , with
. It has been found that the number of maximal mutually
commuting sets within is controlled by the Dedekind psi
function (with a prime)
\cite{Planat2011} and that there exists a specific inequality , involving the Euler constant , that is only satisfied at specific low dimensions . The set is closely related to
the set of integers that are totally Goldbach, i.e.
that consist of all primes ) is equivalent to Riemann hypothesis.
Introducing the Hardy-Littlewood function (with the twin prime constant),
that is used for estimating the number of
Goldbach pairs, one shows that the new inequality is also equivalent to Riemann hypothesis. In this paper,
these number theoretical properties are discusssed in the context of the qudit
commutation structure.Comment: 11 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
Projective Ring Line of a Specific Qudit
A very particular connection between the commutation relations of the
elements of the generalized Pauli group of a -dimensional qudit, being a
product of distinct primes, and the structure of the projective line over the
(modular) ring \bZ_{d} is established, where the integer exponents of the
generating shift () and clock () operators are associated with submodules
of \bZ^{2}_{d}. Under this correspondence, the set of operators commuting
with a given one -- a perp-set -- represents a \bZ_{d}-submodule of
\bZ^{2}_{d}. A crucial novel feature here is that the operators are also
represented by {\it non}-admissible pairs of \bZ^{2}_{d}. This additional
degree of freedom makes it possible to view any perp-set as a {\it
set-theoretic} union of the corresponding points of the associated projective
line
On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
The Bell-Kochen-Specker theorem (BKS) theorem rules out realistic {\it
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 real rays and
-dimensional bases of -qubits (). Specifically, we look at the
parity proof 18-9 with two qubits (A. Cabello, 1996), the parity proof 36-11
with three qubits (M. Kernaghan & A. Peres, 1995 \cite{Kernaghan1965}) and a
newly discovered non-parity proof 80-21 with four qubits (that improves work of
P. K Aravind's group in 2008). The rays in question arise as real eigenstates
shared by some maximal commuting sets (bases) of operators in the -qubit
Pauli group. One finds characteristic signatures of the distances between the
bases, which carry various symmetries in their graphs.Comment: version to appear in European Physical Journal Plu
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