Jon Magne Leinaas

ELECTRONS AS ACCELERATED THERMOMETERS

The possibility of using accelerated electrons to exhibit the quantum field theoretic relation between acceleration and temperature is considered. In principle, the depolarization of electrons in a magnetic field could be used to give temperature reading. The effect is examined for linearly accelerated electrons, but the result is that the relevant orders of magnitude are too small for real experiments in linear accelerators. For electrons in storage rings sufficiently large accelerations can be obtained, and the residual depolarization which has been found theoretically and experimentally is shown to be an effect closely related to the thermal effect of linearly accelerated electrons.

The Unruh effect and quantum fluctuations of electrons in storage rings

The quantum fluctuation of electron orbits in ideal storage rings is a sort of Fulling-Unruh effect (heating by acceleration in vacuum). To spell this out, the effect is analyzed in an appropriate comoving, and so accelerating and rotating, co-ordinate system. The depolarization of the electrons is a related effect, but is greatly complicated by spin-orbit coupling. This analysis confirms the standard result for the polarization, except in the neighbourhood of a narrow resonance.

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.

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.

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.

Ghosts and geometry

Abstract We comment on recent attempts to give a geometrical interpretation of the Faddeev-Popov ghost fields.

Incompressible Quantum Liquids and New Conservation Laws

In this Letter, we investigate a class of Hamiltonians which, in addition to the usual center-of-mass momentum conservation, also have center-of-mass position conservation. We find that, regardless of the particle statistics, the energy spectrum is at least q-fold degenerate when the filling factor is p/q, where and are coprime integers. Interestingly, the simplest Hamiltonian respecting this type of symmetry encapsulates two prominent examples of novel states of matter, namely, the fractional quantum Hall liquid and the quantum dimer liquid. We discuss the relevance of this class of Hamiltonian to the search for featureless Mott insulators.

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.

Numerical studies of entangled PPT states in composite quantum systems

AbstractWe report here on the results of numerical searches for PPT states with specified ranks for densitymatrices and their partial transpose. The study includes several bipartite quantum systems of low dimen-sions. For a series of ranks extremal PPT states are found. The results are listed in tables and charted indiagrams. Comparison of the results for systems of different dimensions reveal several regularities. Wediscuss lower and upper bounds on the ranks of extremal PPT states. 1 Introduction In recent years the study of entanglement in composite quantum systems has taken several different direc-tions. One direction is the study of entanglement from a geometrical point of view [1, 2, 3, 4, 5]. This hasled to questions concerning the relations between different convex sets of Hermitian matrices, where thefull set of density matrices is one of them. The main motivation for the interest in these convex sets is theinformation they give about the general question of how to identify entanglement in a composite system.Unless the system is in a pure quantum state, the knowledge of the corresponding density matrix does notreadily disclose the state as being entangled or non-entangled. And the complexity of the correspondingproblem, known as the separability problem, increases rapidly with the dimensionality of the quantumsystem [6].The density operators are the positive, normalized, semidefinite Hermitian operators that act on theHilbert space of the quantum system, and in the following we shall use the notation Dfor this set. Anotherconvex set is the set of non-entangled states, usually referred to as separable states, and we use Sas notationfor this subset of D. Since the set of entangled states is the complement to the set of separable states, withinthe full set of density matrices, the question of identifying the entangled states can be reformulated as thequestion of finding the boundaries of the convex set of separable states S.For a bipartite system there is furthermore a convex subset of the density matrices, here referred toas P, which is closely related to the set of separable matrices. This is the subset of density matrices thatremain positive semidefinite under the operation of partial transposition of the matrix with respect to oneof the two subsystems of the composite system. For short these states are called PPT states. A necessarycondition for separability of a density matrix is that it remains positive under partial transposition, andthus the set of separable states is included in the set of PPT states, SˆP[7]. For bipartite systems ofdimensions 2x2 and 2x3 the two sets are in fact identical [8], but in higher dimensions the separable statesform a proper subset of the set of PPT states. However, numerical studies have shown that for systems oflow dimensions, like the 3x3 system, the set Pis only slightly larger than S[4, 9].The necessary condition that the separable density matrices remain positive under partial transpositionis important, since this condition is easy to check. It effectively reduces the separability problem to aquestion of identifying the PPT states that are entangled, i.e., that do not belong to S. These states are also1It is known how to construct, in a bipartite quantum system, a unique low rank entangled mixed state with positive partial transpose (a PPT state) fr om an unextendible product basis (a UPB), defined as an unextendible set of orthogonal product ve ctors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by non-singular product transformations, unitary or non-unitary. The characteristic prop

Charge and statistics of quantum Hall quasi-particles — a numerical study of mean values and fluctuations

Abstract We present Monte Carlo studies of charge expectation values and charge fluctuations for quasi-particles in the quantum Hall system. We have studied the Laughlin wave functions for quasi-hole and quasi-electron, and also Jains definition of the quasi-electron wave function. The considered systems consist of from 50 to 200 electrons, and the filling fraction is 1 3 . For all quasi-particles our calculations reproduce well the expected values of charge; −1 3 times the electron charge for the quasi-hole, and 1 3 for the quasi-electron. Regarding fluctuations in the charge, our results for the quasi-hole and Jain quasi-electron are consistent with the expected value zero in the bulk of the system, but for the Laughlin quasi-electron we find small, but significant, deviations from zero throughout the whole electron droplet. We also present Berry phase calculations of charge and statistics parameter for the Jain quasi-electron, calculations which supplement earlier studies for the Laughlin quasi-particles. We find that the statistics parameter, calculated as a function of distance, is more well behaved for the Jain quasi-electron than it is for the Laughlin quasi-electron. However, the sign of the parameter is opposite of what is expected from qualitative arguments.