K. Ch. Chatzisavvas

Information entropy, information distances, and complexity in atoms

Shannon information entropies in position and momentum spaces and their sum are calculated as functions of Z(2 < or = Z < or = 54) in atoms. Roothaan-Hartree-Fock electron wave functions are used. The universal property S = a + b ln Z is verified. In addition, we calculate the Kullback-Leibler relative entropy, the Jensen-Shannon divergence, Onicescus information energy, and a complexity measure recently proposed. Shell effects at closed-shell atoms are observed. The complexity measure shows local minima at the closed-shell atoms indicating that for the above atoms complexity decreases with respect to neighboring atoms. It is seen that complexity fluctuates around an average value, indicating that the atom cannot grow in complexity as Z increases. Onicescus information energy is correlated with the ionization potential. Kullback distance and Jensen-Shannon distance are employed to compare Roothaan-Hartree-Fock density distributions with other densities of previous works.

Net Fisher information measure versus ionization potential and dipole polarizability in atoms

Abstract The net Fisher information measure I T , defined as the product of position and momentum Fisher information measures I r and I k and derived from the non-relativistic Hartree–Fock wave functions for atoms with Z = 1 – 102 , is found to correlate well with the inverse of the experimental ionization potential. Strong direct correlations of I T are also reported for the static dipole polarizability of atoms with Z = 1 – 88 . The complexity measure, defined as the ratio of the net Onicescu information measure E T and I T , exhibits clearly marked regions corresponding to the periodicity of the atomic shell structure. The reported correlations highlight the need for using the net information measures in addition to either the position or momentum space analogues. With reference to the correlation of the experimental properties considered here, the net Fisher information measure is found to be superior than the net Shannon information entropy.

Comparison of SDL and LMC measures of complexity: Atoms as a testbed

Abstract The simple measure of complexity Γ α , β of Shiner, Davison and Landsberg (SDL) and the statistical one C , according to Lopez-Ruiz, Mancini and Calbet (LMC), are compared in atoms as functions of the atomic number Z . Shell effects i.e. local minima at the closed shells atoms are observed, as well as certain qualitative trends of Γ α , β ( Z ) and C ( Z ) . If we impose the condition that Γ and C behave similarly as functions of Z , then we can conclude that complexity increases with Z and for atoms the strength of disorder is α ≃ 0 and order is β ≃ 4 .

A simple method for the evaluation of the information content and complexity in atoms. A proposal for scalability.

Abstract We present a very simple method for the calculation of Shannon, Fisher and Onicescu entropies in atoms, as well as SDL and LMC complexity measures, as functions of the atomic number Z. Fractional occupation probabilities of electrons in atomic orbitals are employed, instead of the more complicated continuous electron probability densities in position- and momentum-spaces, used so far in the literature. Our main conclusions are compatible with the results of more sophisticated approaches and correlate fairly with experimental data. A practical way towards scalability of the quantification of complexity for systems with more components than the atom is indicated. We also discuss the issue if the complexity of the electronic structure of atoms increases with Z. A Pair ( α , β ) of Order-Disorder Indices (PODI), which can be introduced for any quantum many-body system, is evaluated in atoms ( α = 0.085 , β = 1.015 ). We conclude, by observing the trend of closed shells atoms, that “atoms are ordered systems, which grow in complexity as Z increases”.

APPLICATIONS OF DENSITY MATRICES IN A TRAPPED BOSE GAS

An overview of the Bose–Einstein condensation of correlated atoms in a trap is presented by examining the effect of interparticle correlations to one- and two-body properties of the above systems at zero temperature in the framework of the lowest order cluster expansion. Analytical expressions for the one- and two-body properties of the Bose gas are derived using Jastrow-type correlation function. In addition numerical calculations of the natural orbitals and natural occupation numbers are also carried out. Special effort is devoted for the calculation of various quantum information properties including Shannon entropy, Onicescu informational energy, Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. The above quantities are calculated for the trapped Bose gases by comparing the correlated and uncorrelated cases as a function of the strength of the short-range correlations. The Gross–Piatevskii equation is solved, giving the density distributions in position and momentum space, which are employed to calculate quantum information properties of the Bose gas.

THEORETICAL QUANTUM-INFORMATION PROPERTIES OF NUCLEI AND TRAPPED BOSE GASES

We will study fermionic systems like atomic nuclei and bosonic systems like the correlated atoms in a trap from an information-theoretical point of view. The Shannon and Onicescu information measures are calculated for the above systems by comparing the correlated and uncorrelated cases as functions of the strength of the short range correlations. One-body and two-body density and momentum distributions are employed. Thus, the effect of short-range correlations on the information content is evaluated. The magnitude of distinguishability between the correlated and uncorrelated densities is also discussed employing suitable measures for the distance of states i.e. the well known Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. We will see that the same information-theoretical properties hold for quantum many-body systems obeying Bose–Einstein and Fermi–Dirac (statistics).

Complexity and neutron star structure

We apply the statistical measure of complexity introduced by Lopez-Ruiz, Mancini and Calbet (1995) [1] to neutron star structure. We continue the recent application of Sanudo and Pacheco (2009) [2] to white dwarfs. The interplay of gravity, the short-range nuclear force and the very short-range weak interaction shows that neutron stars, under the current theoretical framework, are ordered (low complexity) systems.

Quantum-Information Content of Fractional Occupation Probabilities in Nuclei

Three measures of the information content of a probability distribution are briefly reviewed. They are applied to fractional occupation probabilities in light nuclei, taking into account short-range correlations. The effect of short-range correlations is to increase the information entropy (or disorder) of nuclei, comparing with the independent particle model. It is also indicated that the information entropy can serve as a sensitive index of order and short-range correlations in nuclei. It is concluded that increasing Z, the information entropy increases, i.e. the disorder of the nucleus increases for all measures of information considered in the present work.

Statistical measure of complexity and correlated behavior of Fermi systems.

We apply the statistical measure of complexity, introduced by López-Ruiz, Mancini, and Calbet (LMC), to uniform Fermi systems. We investigate the connection between information and complexity measures with the strongly correlated behavior of various Fermi systems as nuclear matter, electron gas, and liquid helium. We examine the possibility that LMC complexity can serve as an index quantifying correlations in the specific system and to which extent could be related with experimental quantities. Moreover, we concentrate on thermal effects on the complexity of ideal Fermi systems. We find that complexity behaves, both at low and high values of temperature, in a similar way as the specific heat.

Improving quantum gate fidelities using optimized Euler angles

An explicit algorithm for calculating the optimized Euler angles for both qubit state transfer and gate engineering given two arbitary fixed Hamiltonians is presented. It is shown how the algorithm enables us to efficiently implement single qubit gates even if the control is severely restricted and the experimentally accessible Hamiltonians are far from orthogonal. It is further shown that using the optimized Euler angles can significantly improve the fidelity of quantum operations even for systems where the experimentally accessible Hamiltonians are nearly orthogonal. Unlike schemes such as composite pulses, the proposed scheme does not significantly increase the number of local operations or gate operation times.