Asymptotic Entropy Research Articles

Binegativity and geometry of entangled states in two qubits

We prove that the binegativity is always positive for any two-qubit state. As a result, and as suggested by previous work, the asymptotic relative entropy of entanglement in two qubits does not exceed the Rains bound, and the positive partial transposed-entanglement cost for any two-qubit state is determined to be the logarithmic negativity of the state. Further, the proof reveals some geometrical characteristics of the entangled states, and shows that partial transposition can give another separable approximation of the entangled state in two qubits.

Entropy production in the cyclic lattice Lotka-Volterra model

The cyclic Lotka-Volterra model in a $D$-dimensional regular lattice is considered. Its ``nucleus growth'' mode is analyzed under the scope of Tsallis' entropies $S_q=(1-\sum_i p_i^q)/(q-1)$, $q\in \mathbb{R}$. It is shown both numerically and by means of analytical considerations that a linear increase of entropy with time, meaning finite asymptotic entropy rate, is achieved for the entropic index $q_c=1-1/D$. Although the lattice exhibits fractal patterns along its evolution, the characteristic value of $q$ can be interpreted in terms of very simple features of the dynamics.

Magnetic Protoneutron Star Winds and [CLC][ITAL]r[/ITAL][/CLC]-Process Nucleosynthesis

Because of their neutron richness and association with supernovae, postexplosion protoneutron star winds are thought to be a likely astrophysical site for rapid neutron-capture nucleosynthesis (the r-process). However, the most recent models of spherical neutrino-driven protoneutron star winds do not produce robust r-process nucleosynthesis for canonical neutron stars with a gravitational mass of 1.4 M☉ and coordinate radius of 10 km. These models fail variously; the flow entropy is too low, the electron fraction is too high, or the dynamical expansion timescale is too long. To date, no models have included the effects of an ordered dipole magnetic field. We show that a strong magnetic field can trap the outflow in the neutrino heating region, thus leading to much higher matter entropy. We estimate both the trapping timescale and the resulting entropy amplification. For sufficiently large energy deposition rates, the trapped matter emerges dynamically from the region of closed magnetic field lines and escapes to infinity. We find that ordered dipoles with surface fields of 6 × 1014 G increase the asymptotic entropy sufficiently for robust r-process nucleosynthesis.

Numerical simulations of surface convection in a late M-dwarf

Based on detailed 2D and 3D numerical radiation-hydrodynamics (RHD) simulations of time-dependent compressible convection, we have studied the dynamics and thermal structure of the convective surface layers of a prototypical late-type M-dwarf (Teff~2800K log(g)=5.0, solar chemical composition). The RHD models predict stellar granulation qualitatively similar to the familiar solar pattern. Quantitatively, the granular cells show a convective turn-over time scale of ~100s, and a horizontal scale of 80km; the relative intensity contrast of the granular pattern amounts to 1.1%, and root-mean-square vertical velocities reach 240m/s at maximum. Deviations from radiative equilibrium in the higher, formally convectively stable atmospheric layers are found to be insignificant allowing a reliable modeling of the atmosphere with 1D standard model atmospheres. A mixing-length parameter of alpha=2.1 provides the best representation of the average thermal structure of the RHD model atmosphere while alternative values are found when fitting the asymptotic entropy encountered in deeper layers of the stellar envelope alpha=1.5, or when matching the vertical velocity field alpha=3.5. The close correspondence between RHD and standard model atmospheres implies that presently existing discrepancies between observed and predicted stellar colors in the M-dwarf regime cannot be traced back to an inadequate treatment of convection in the 1D standard models. The RHD models predict a modest extension of the convectively mixed region beyond the formal Schwarzschild stability boundary which provides hints for the distribution of dust grains in cooler (brown dwarf) atmospheres.

Asymptotic relative entropy of entanglement for orthogonally invariant states

For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement $E_R^\infty$ with respect to states having a positive partial transpose (PPT). This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $O\otimes O$, where $O$ is any orthogonal matrix. We show that in this case $E_R^\infty$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of new results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to $E_R^\infty$; (iii) for states for which the relative entropy of entanglement $E_R$ is additive, the Rains bound is equal to $E_R$.

Asymptotic relative entropy of entanglement.

We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. This is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated.

Asymptotic symmetries and statistical entropy of 2-dimensional gravity

The asymptotic symmetries of two-dimensional Anti-de Sitter space are generated by a central extension of a Virasoro algebra. Using a canonical realization of this symmetry and Cardy's formula we calculate the statistical entropy of 2D black holes, and show that it agrees, up to a numerical factor, with the thermodynamical result.

Hausdorff dimension of the harmonic measure on trees

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

Asymptotic entropy, periodic orbits, and pseudo-Anosov maps

In this paper we derive some properties of a variety of entropy that measures rotational complexity of annulus homeomorphisms, called asymptotic or rotational entropy. In previous work [KS] the authors showed that positive asymptotic entropy implies the existence of infinitely many periodic orbits corresponding to an interval of rotation numbers. In our main result, we show that a Hölder $C^1$ diffeomorphism with nonvanishing asymptotic entropy is isotopic rel a finite set to a pseudo-Anosov map. We also prove that the closure of the set of recurrent points supports positive asymptotic entropy for a ($C^0$) homeomorphism with nonzero asymptotic entropy.

Entropic uncertainty relations for a quantum oscillator

We calculated the Shannon entropy of position and momentum for the stationary quantum states of the harmonic oscillator as a function of its energy and determined the corresponding entropic uncertainty relations for them. We found an approximate phenomenological function for the dependence of position and momentum entropies on the large quantum numbers and the corresponding asymptotic entropy - energy relation for the stationary harmonic oscillator. We also studied the time evolution of the position and momentum entropies of the non-stationary harmonic oscillator for the coherent states, squeezed vacuum and Schrodinger cat states.

Entropy, character theory and centrality of finite quasigroups

AbstractThe paper introduces concepts of entropy and asymptotic entropy for finite quasigroups. A quasigroup is abelian if and only if its entropy is maximal. It is a З-quasigroup if and only if its asymptotic entropy is maximal.

Asymptotic freedom and entropy in a perpetually oscillating universe

A model of a perpetually oscillating universe is considered. In this model the universe, when contracting to Planck dimensions, transforms from the Friedmann to the De Sitter phase with one and the same Λ ∼ 1 l 2 P1 , irrespective of the value of the total bare mass of the Friedmann universe.