We study geometrical aspects of entanglement, with the Hilbert--Schmidt norm
defining the metric on the set of density matrices. We focus first on the
simplest case of two two-level systems and show that a ``relativistic''
formulation leads to a complete analysis of the question of separability. Our
approach is based on Schmidt decomposition of density matrices for a composite
system and non-unitary transformations to a standard form. The positivity of
the density matrices is crucial for the method to work. A similar approach
works to some extent in higher dimensions, but is a less powerful tool. We
further present a numerical method for examining separability, and illustrate
the method by a numerical study of bound entanglement in a composite system of
two three-level systems.Comment: 31 pages, 6 figure
We give an explicit formula for the membrane potential of cells in terms of the intracellular and extracellular ionic concentrations, and derive equations for the ionic currents that flow through channels, exchangers and electrogenic pumps. We demonstrate that the work done by the pumps equals the change in potential energy of the cell, plus the energy lost in downhill ionic fluxes through the channels and exchangers. The theory is illustrated in a simple model of spontaneously active cells in the cardiac pacemaker. The model predicts the experimentally observed intracellular ionic concentration of potassium, calcium and sodium. Likewise, the shapes of the simulated action potential and five membrane currents are in good agreement with experiment. We do not see any drift in the values of the concentrations in a long time simulation, and we obtain the same asymptotic values when starting from the full equilibrium situation with equal intracellular and extracellular ionic concentrations.
We report here on the results of numerical searches for PPT states with specified ranks for density matrices and their partial transpose. The study includes several bipartite quantum systems of low dimensions. For a series of ranks extremal PPT states are found. The results are listed in tables and charted in diagrams. Comparison of the results for systems of different dimensions reveal several regularities. We discuss lower and upper bounds on the ranks of extremal PPT states.
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