Jan Myrheim

A theory for the membrane potential of living cells.

Abstract 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.

Geometrical aspects of entanglement

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 nonunitary 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.

Thermodynamics for fractional exclusion statistics

Abstract We discuss the thermodynamics of a gas of free particles obeying Haldanes exclusion statistics, deriving low-temperature and low-density expansions. For gases with a constant density of states, we derive an exact equation of state and find that temperature-dependent quantities are independent of the statistics parameter.

Dimensional reduction in anyon systems

Fractional statistics in one space dimension can be defined in two inequivalent ways: (i) By restricting the wave function for the relative two-body problem to the halfline x ⩾ 0, and imposing the boundary condition ψx = ηψ at x = 0. (ii) By quantizing the sp(1, R) algebra of observables x2 ± p2 and xp + px, and noticing that the irreducible hermitian representations are labelled by a real parameter μ. We show that both these cases can be obtained by a dimensional reduction of a system of anyons in two dimensions. Case one corresponds to restricting the motion of the anyons to a line by a confining potential, and we give η as a function of the statistics parameter θ for two different potentials. The second case corresponds to anyons in a magnetic field restricted to the first Landau level, and we find a linear relationship between μ and θ. We also construct coherent states corresponding to anyons in the lowest Landau level, and calculate the corresponding Berry connection. The statistics phase θ is shown to equal the Berry phase corresponding to an interchange of two anyons, thus generalizing previous results for bosons and fermions.

Universality for period n-tuplings in complex mappings

Abstract The theory of period doublings for real iterative mappings is generalized to period n -tupling for complex iterative mappings. We find an infinity of universal functions associated with different sequences of period n -tuplings.

Extreme points of the set of density matrices with positive partial transpose

We present a necessary and sufficient condition for a finite-dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.

NUMERICAL STUDY OF CHARGE AND STATISTICS OF LAUGHLIN QUASIPARTICLES

We present numerical calculations of the charge and statistics, as extracted from Berry phases, of the Laughlin quasiparticles, near filling fraction 1/3, and for system sizes of up to 200 electrons. For the quasiholes our results confirm that the charge and statistics parameter are e/3 and 1/3, respectively. For the quasielectron charge we find a slow convergence towards the expected value of -e/3, with a finite size correction for N electrons of approximately -0.13e/N. The statistics parameter for the quasielectrons has no well defined value even for 200 electrons, but might possibly converge to 1/3. The anyon model works well for the quasiholes, but requires singular two-anyon wave functions for modelling two Laughlin quasielectrons.

The Third virial coefficient of free anyons

Abstract We use a path integral representation for the partition function of non-interacting anyons confined in a harmonic oscillator potential in order to prove that the third virial coefficient of free anyons is finite, and to calculate it numerically. Our results together with previously known results are consistent with a rapidly converging Fourier series in the statistics angle.

Unextendible product bases and extremal density matrices with positive partial transpose

In bipartite quantum systems of dimension 3x3 entangled states that are positive under partial transposition (PPT) can be constructed with the use of unextendible product bases (UPB). As discussed in a previous publication all the lowest rank entangled PPT states of this system seem to be equivalent, under special linear product transformations, to states that are constructed in this way. Here we consider a possible generalization of the UPB constuction to low-rank entangled PPT states in higher dimensions. The idea is to give up the condition of orthogonality of the product vectors, while keeping the relation between the density matrix and the projection on the subspace defined by the UPB. We examine first this generalization for the 3x3 system where numerical studies indicate that one-parameter families of such generalized states can be found. Similar numerical searches in higher dimensional systems show the presence of extremal PPT states of similar form. Based on these results we suggest that the UPB construction of the lowest rank entangled states in the 3x3 system can be generalized to higher dimensions, with the use of non-orthogonal UPBs.

Low rank positive partial transpose states and their relati on to product vectors

It is known that entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. In a previous paper we presented a classification of rank four entangled PPT states which we believe to be complete. In the present paper we continue our investigations of the low rank entangled PPT states. We use perturbation theory in order to construct rank five entangled PPT states close to the known rank four states, and in order to compute dimensions and study the geometry of surfaces of low rank PPT states. We exploit the close connection between low rank PPT states and product vectors. In particular, we show how to reconstruct a PPT state from a sufficient number of product vectors in its kerne l. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.