100 research outputs found
Born Rule and Noncontextual Probability
The probabilistic rule that links the formalism of Quantum Mechanics (QM) to
the real world was stated by Born in 1926. Since then, there were many attempts
to derive the Born postulate as a theorem, Gleason's being the most prominent.
The Gleason derivation, however, is generally considered rather intricate and
its physical meaning, in particular in relation with the noncontextuality of
probability (NP), is not quite evident. More recently, we are witnessing a
revival of interest in possible demonstrations of the Born rule, like Zurek's
and Deutsch's based on the decoherence and on the theory of decisions,
respectively. Despite an ongoing debate about the presence of hidden
assumptions and circular reasonings, these have the merit of prompting more
physically oriented approaches to the problem. Here we suggest a new proof of
the Born rule based on the noncontextuality of probability. Within the theorem
we also demonstrate the continuity of probability with respect to the
amplitudes, which has been suggested to be a gap in Zurek's and Deutsch's
approaches, and we show that NP is implicitly postulated also in their
demonstrations. Finally, physical motivations of NP are given based on an
invariance principle with respect to a resolution change of measurements and
with respect to the principle of no-faster-than-light signalling.Comment: 10 page
Nonlinear Tight-Binding Approximation for Bose-Einstein Condensates in a Lattice
The dynamics of Bose-Einstein condensates trapped in a deep optical lattice
is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity
and the intersite hopping rates are retrieved from a nonlinear tight-binding
approximation taking into account the effective dimensionality of each
condensate. We derive analytically the Bloch and the Bogoliubov excitation
spectra, and the velocity of sound waves emitted by a traveling condensate.
Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of
localized wavepackets. We calculate the ground-state atomic distribution in the
presence of an harmonic confining potential, and the frequencies of small
amplitude dipole and quadrupole oscillations. We finally quantize the DNL,
recovering an extended Bose-Hubbard model
Entanglement and squeezing in continuous-variable systems
We introduce a multi-mode squeezing coefficient to characterize entanglement
in -partite continuous-variable systems. The coefficient relates to the
squeezing of collective observables in the -dimensional phase space and can
be readily extracted from the covariance matrix. Simple extensions further
permit to reveal entanglement within specific partitions of a multipartite
system. Applications with nonlinear observables allow for the detection of
non-Gaussian entanglement.Comment: 11 pages, 2 figure
On the dispute between Boltzmann and Gibbs entropy
Very recently, the validity of the concept of negative temperature has been
challenged by several authors since they consider Boltzmann's entropy (that
allows negative temperatures) inconsistent from a mathematical and statistical
point of view, whereas they consider Gibbs' entropy (that does not admit
negative temperatures) the correct definition for microcanonical entropy.
In the present paper we prove that for systems with equivalence of the
statistical ensembles Boltzmann entropy is the correct microcanonical entropy.
Analytical results on two systems supporting negative temperatures, confirm the
scenario we propose. In addition, we corroborate our proof by numeric
simulations on an explicit lattice system showing that negative temperature
equilibrium states are accessible and obey standard statistical mechanics
prevision.Comment: To appear in Annals of Physic
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