175 research outputs found
Tomographic entropic inequalities in the probability representation of quantum mechanics
A review of the tomographic-probability representation of classical and
quantum states is presented. The tomographic entropies and entropic uncertainty
relations are discussed in connection with ambiguities in the interpretation of
the state tomograms which are considered either as a set of the probability
distributions of random variables depending on extra parameters or as a single
joint probability distribution of these random variables and random parameters
with specific properties of the marginals. Examples of optical tomograms of
photon states, symplectic tomograms, and unitary spin tomograms of qudits are
given. A new universal integral inequality for generic wave function is
obtained on the base of tomographic entropic uncertainty relations.Comment: 9 pages; to be published in AIP Conference Proceedings as a
contribution to the conference "Beauty in Physics: Theory and Experiment"
(the Hacienda Cocoyoc, Morelos, Mexico, May 14--18, 2012
Entanglement Entropy In Excited States
Negli ultimi anni lâentropia di entaglement eÌ stata ampiamente studiata nel campo dellâintegrabilitaÌ. Con lâintroduzione del modello a replica eÌ stato possibile portare alla luce le proprietaÌ universali dellâ entropia di entanglement di un sistema bipartito nello stato di vuoto.
In questa tesi si eÌ investigato il problema dellâentropia di entanglement di un sistema bipartito in uno stato eccitato di singola particella. In particolare, si eÌ considerata una teoria bosonica libera in un volume finito, in modo da sfruttare al meglio le tecniche dellâintegrabilitaÌ. Nel corso di questa analisi, eÌ stato possibile rielaborare il modello a replica in un volume finito grazie ad un raddoppiamento della teoria bosonica che ha indotto una simmetria U(1) su ogni copia del modello. Tale tecnica, nota in letter- atura come doubling trick ha permesso di ricondurre il calcolo dellâentropia di Renyi a unâopportuna espansione in form factors dei campi U(1) implementanti tale simmetria e valutarne il contributo dominante nel limite in cui il volume eÌ grande.
I risultati ottenuti per la Second ReÌnyi entropy mostrano che in tale limite, lâeccesso di entanglement dovuto allo stato eccitato rispetto a quello di vuoto eÌ indipendente dallâenergia dello stato stesso e puoÌ essere interpretato come quantitaÌ che misura lâincertezza sulla localizzazione dellâeccitazione nelle due parti di cui eÌ composto il sistema
Precision-Recall Curves Using Information Divergence Frontiers
Despite the tremendous progress in the estimation of generative models, the
development of tools for diagnosing their failures and assessing their
performance has advanced at a much slower pace. Recent developments have
investigated metrics that quantify which parts of the true distribution is
modeled well, and, on the contrary, what the model fails to capture, akin to
precision and recall in information retrieval. In this paper, we present a
general evaluation framework for generative models that measures the trade-off
between precision and recall using R\'enyi divergences. Our framework provides
a novel perspective on existing techniques and extends them to more general
domains. As a key advantage, this formulation encompasses both continuous and
discrete models and allows for the design of efficient algorithms that do not
have to quantize the data. We further analyze the biases of the approximations
used in practice.Comment: Updated to the AISTATS 2020 versio
A study on black-body radiation: classical and binary photons
The present study gives a detailed analysis of the black-body radiation based
on classical random variables. It is shown that the energy of a mode of a
chaotic radiation field (Gauss variable) can be uniquely decomposed into a sum
of a discrete variable (Planck variable having the Planck-Bose distribution)
and a continuous dark variable (with a truncated exponential distribution of
finite support). The Planck variable is decomposed, on one hand, into a sum of
binary variables representing the binary photons of energies 2^s*h*nu with
s=0,1,2,etc. In this way the black-body radiation can be viewed as a mixture of
thermodinamically independent fermion gases. The Planck variable can also be
decomposed into a sum of independent Poisson components representing the
classical photo-molecules of energies m*h*nu with m=1,2,3,etc. These classical
photons have only particle-like fluctuations, on the other hand, the binary
photons have wave-particle fluctuations of fermionic character.Comment: 20 page
On Generalized Stam Inequalities and FisherâRĂ©nyi Complexity Measures
Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, âŠ) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric FisherâRĂ©nyi complexity, named ( p , ÎČ , λ ) -FisherâRĂ©nyi complexity, based on both a two-parametic extension of the Fisher information and the RĂ©nyi entropies of a probability density function Ï characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of Ï , and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the ( p , ÎČ , λ ) -Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential.The authors are very grateful to the CNRS (Steeve Zozor) and the Junta de AndalucĂa and the MINECOâFEDER under the grants FIS2014â54497 and FIS2014â59311P (JesĂșs SĂĄnchez-Dehesa) for partial financial support
Irreducible decomposition of Gaussian distributions and the spectrum of black-body radiation
It is shown that the energy of a mode of a classical chaotic field, following
the continuous exponential distribution as a classical random variable, can be
uniquely decomposed into a sum of its fractional part and of its integer part.
The integer part is a discrete random variable (we call it Planck variable)
whose distribution is just the Bose distribution yielding the Planck law of
black-body radiation. The fractional part is the dark part (we call is dark
variable) with a continuous distribution, which is, of course, not observed in
the experiments. It is proved that the Bose distribution is infinitely
divisible, and the irreducible decomposition of it is given. The Planck
variable can be decomposed into an infinite sum of independent binary random
variables representing the binary photons (more accurately photo-molecules or
photo-multiplets) of energies 2^s*h*nu with s=0,1,2... . These binary photons
follow the Fermi statistics. Consequently, the black-body radiation can be
viewed as a mixture of statistically and thermodynamically independent fermion
gases consisting of binary photons. The binary photons give a natural tool for
the dyadic expansion of arbitrary (but not coherent) ordinary photon
excitations. It is shown that the binary photons have wave-particle
fluctuations of fermions. These fluctuations combine to give the wave-particle
fluctuations of the original bosonic photons expressed by the Einstein
fluctuation formula.Comment: 29 page
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