394 research outputs found

### Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces

We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz,
are Leibniz quantum compact metric spaces and that they form a continuous
family over the group of invertible matrices with entries in the commutant of
the quantum tori in the regular representation, when this group is endowed with
a natural length function.Comment: 16 Pages, v3: accepted in Journal of Math. Physic

### Reply to Norsen's paper "Are there really two different Bell's theorems?"

Yes. That is my polemical reply to the titular question in Travis Norsen's
self-styled "polemical response to Howard Wiseman's recent paper." Less
polemically, I am pleased to see that on two of my positions --- that Bell's
1964 theorem is different from Bell's 1976 theorem, and that the former does
not include Bell's one-paragraph heuristic presentation of the EPR argument ---
Norsen has made significant concessions. In his response, Norsen admits that
"Bell's recapitulation of the EPR argument in [the relevant] paragraph leaves
something to be desired," that it "disappoints" and is "problematic". Moreover,
Norsen makes other statements that imply, on the face of it, that he should
have no objections to the title of my recent paper ("The Two Bell's Theorems of
John Bell"). My principle aim in writing that paper was to try to bridge the
gap between two interpretational camps, whom I call 'operationalists' and
'realists', by pointing out that they use the phrase "Bell's theorem" to mean
different things: his 1964 theorem (assuming locality and determinism) and his
1976 theorem (assuming local causality), respectively. Thus, it is heartening
that at least one person from one side has taken one step on my bridge. That
said, there are several issues of contention with Norsen, which we (the two
authors) address after discussing the extent of our agreement with Norsen. The
most significant issues are: the indefiniteness of the word 'locality' prior to
1964; and the assumptions Einstein made in the paper quoted by Bell in 1964 and
their relation to Bell's theorem.Comment: 13 pages (arXiv version) in http://www.ijqf.org/archives/209

### Connes distance by examples: Homothetic spectral metric spaces

We study metric properties stemming from the Connes spectral distance on
three types of non compact noncommutative spaces which have received attention
recently from various viewpoints in the physics literature. These are the
noncommutative Moyal plane, a family of harmonic Moyal spectral triples for
which the Dirac operator squares to the harmonic oscillator Hamiltonian and a
family of spectral triples with Dirac operator related to the Landau operator.
We show that these triples are homothetic spectral metric spaces, having an
infinite number of distinct pathwise connected components. The homothetic
factors linking the distances are related to determinants of effective Clifford
metrics. We obtain as a by product new examples of explicit spectral distance
formulas. The results are discussed.Comment: 23 pages. Misprints corrected, references updated, one remark added
at the end of the section 3. To appear in Review in Mathematical Physic

### The beat of a fuzzy drum: fuzzy Bessel functions for the disc

The fuzzy disc is a matrix approximation of the functions on a disc which
preserves rotational symmetry. In this paper we introduce a basis for the
algebra of functions on the fuzzy disc in terms of the eigenfunctions of a
properly defined fuzzy Laplacian. In the commutative limit they tend to the
eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of
the first kind, thus deserving the name of fuzzy Bessel functions.Comment: 30 pages, 8 figure

### An Obstruction to Quantization of the Sphere

In the standard example of strict deformation quantization of the symplectic
sphere $S^2$, the set of allowed values of the quantization parameter $\hbar$
is not connected; indeed, it is almost discrete. Li recently constructed a
class of examples (including $S^2$) in which $\hbar$ can take any value in an
interval, but these examples are badly behaved. Here, I identify a natural
additional axiom for strict deformation quantization and prove that it implies
that the parameter set for quantizing $S^2$ is never connected.Comment: 23 page. v2: changed sign conventio

### Strict Deformation Quantization for a Particle in a Magnetic Field

Recently, we introduced a mathematical framework for the quantization of a
particle in a variable magnetic field. It consists in a modified form of the
Weyl pseudodifferential calculus and a C*-algebraic setting, these two points
of view being isomorphic in a suitable sense. In the present paper we leave
Planck's constant vary, showing that one gets a strict deformation quantization
in the sense of Rieffel. In the limit h --> 0 one recovers a Poisson algebra
induced by a symplectic form defined in terms of the magnetic field.Comment: 23 page

### Dynamical Logical Qubits in the Bacon-Shor Code

The Bacon-Shor code is a quantum error correcting subsystem code composed of
weight 2 check operators that admits a single logical qubit, and has distance
$d$ on a $d \times d$ square lattice. We show that when viewed as a Floquet
code, by choosing an appropriate measurement schedule of the check operators,
it can additionally host several dynamical logical qubits. Specifically, we
identify a period 4 measurement schedule of the check operators that preserves
logical information between the instantaneous stabilizer groups. Such a
schedule measures not only the usual stabilizers of the Bacon-Shor code, but
also additional stabilizers that protect the dynamical logical qubits against
errors. We show that the code distance of these Floquet-Bacon-Shor codes scales
as $\Theta(d/\sqrt{k})$ on a $d \times d$ lattice with $k$ dynamical logical
qubits, along with the logical qubit of the parent subsystem code. Moreover,
several errors are shown to be self-corrected purely by the measurement
schedule itself.Comment: 10 pages + 3 page appendix, 3 figures, 1 tabl

### On the role of twisted statistics in the noncommutative degenerate electron gas

We consider the problem of a degenerate electron gas in the background of a
uniformly distributed positive charge, ensuring overall neutrality of the
system, in the presence of non-commutativity. In contrast to previous
calculations that did not include twisted statistics, we find corrections to
the ground state energy already at first order in perturbation theory when the
twisted statistics is taken into account. These corrections arise since the
interaction energy is sensitive to two particle correlations, which are
modified for twisted anti-commutation relations

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