Quantum Fidelity Decay of Quasi-Integrable Systems

We show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type.Comment: 11 pages, 11 figures, to be published Phys. Rev.

Spacings and pair correlations for finite Bernoulli convolutions

We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $\nu_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain algebraic $r$ the behavior is totally different.Comment: 17 pages, 6 figure

Bell's inequality with Dirac particles

We study Bell's inequality using the Bell states constructed from four component Dirac spinors. Spin operator is related to the Pauli-Lubanski pseudo vector which is relativistic invariant operator. By using Lorentz transformation, in both Bell states and spin operator, we obtain an observer independent Bell's inequality, so that it is maximally violated as long as it is violated maximally in the rest frame.Comment: 7 pages. arXiv admin note: text overlap with arXiv:quant-ph/0308156 by other author

Observing the evolution of a quantum system that does not evolve

This article deals with the problem of gathering information on the time evolution of a single metastable quantum system whose evolution is impeded by the quantum Zeno effect. It has been found it is in principle possible to obtain some information on the time evolution and, depending on the specific system, even to measure its average decay rate, even if the system does not undergo any evolution at all.Comment: Two over three PRA referees didn't like the old title... And no more quantum circuits in the new versio

No directed fractal percolation in zero area

We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1

A characterization of L<inf>2</inf> mixing and hypercontractivity via hitting times and maximal inequalities

There are several works characterizing the total-variation mixing time of a reversible Markov chain in term of natural probabilistic concepts such as stopping times and hitting times. In contrast, there is no known analog for the $L_{2}$ mixing time, $\tau_{2}$ (while there are sophisticated analytic tools to bound $\tau_2$, in general they do not determine $\tau_2$ up to a constant factor and they lack a probabilistic interpretation). In this work we show that $\tau_2$ can be characterized up to a constant factor using hitting times distributions. We also derive a new extremal characterization of the Log-Sobolev constant, $c_{\mathrm{LS}}$, as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $c_{\mathrm{LS}}$ in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $\tau_2$ is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $\tau_2$ is robust under bounded perturbations of the edge weights

The Benefits of an Alternative Approach to Analytic Number Theory

In our talk we will survey the alternative (or “pretentious”) approach to analytic number theory, which has the potential to be more flexible and broad reaching than traditional zeta-function methods, highlighting some spectacular recent developments. In this “sampler” we give some indication of the change in perspective that motivates this shift in techniqu

A variant of Peres-Mermin proof for testing noncontextual realist models

For any state in four-dimensional system, the quantum violation of an inequality based on the Peres-Mermin proof for testing noncontextual realist models has experimentally been corroborated. In the Peres-Mermin proof, an array of nine holistic observables for two two-qubit system was used. We, in this letter, present a new symmetric set of observables for the same system which also provides a contradiction of quantum mechanics with noncontextual realist models in a state-independent way. The whole argument can also be cast in the form of a new inequality that can be empirically tested.Comment: 3 pages, To be published in Euro. Phys. Let

Wigner's little group and Berry's phase for massless particles

The ``little group'' for massless particles (namely, the Lorentz transformations $\Lambda$ that leave a null vector invariant) is isomorphic to the Euclidean group E2: translations and rotations in a plane. We show how to obtain explicitly the rotation angle of E2 as a function of $\Lambda$ and we relate that angle to Berry's topological phase. Some particles admit both signs of helicity, and it is then possible to define a reduced density matrix for their polarization. However, that density matrix is physically meaningless, because it has no transformation law under the Lorentz group, even under ordinary rotations.Comment: 4 pages revte

On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_\lambda$ themselves and to natural measures $\nu_\lambda$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater than $\sqrt{1/2}$.Comment: 22 pages, 5 figure