Information geometry of density matrices and state estimation

Given a pure state vector |x> and a density matrix rho, the function p(x|rho)= defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher-Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.Comment: published versio

Selective decay by Casimir dissipation in fluids

The problem of parameterizing the interactions of larger scales and smaller scales in fluid flows is addressed by considering a property of two-dimensional incompressible turbulence. The property we consider is selective decay, in which a Casimir of the ideal formulation (enstrophy in 2D flows, helicity in 3D flows) decays in time, while the energy stays essentially constant. This paper introduces a mechanism that produces selective decay by enforcing Casimir dissipation in fluid dynamics. This mechanism turns out to be related in certain cases to the numerical method of anticipated vorticity discussed in \cite{SaBa1981,SaBa1985}. Several examples are given and a general theory of selective decay is developed that uses the Lie-Poisson structure of the ideal theory. A scale-selection operator allows the resulting modifications of the fluid motion equations to be interpreted in several examples as parameterizing the nonlinear, dynamical interactions between disparate scales. The type of modified fluid equation systems derived here may be useful in modelling turbulent geophysical flows where it is computationally prohibitive to rely on the slower, indirect effects of a realistic viscosity, such as in large-scale, coherent, oceanic flows interacting with much smaller eddies

Martingale Models for Quantum State Reduction

Stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory. Here we examine the correspondence of the two theories in detail, making a systematic use of the methods of martingale theory. An analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred. We consider the general projection postulate of L\"uders applicable in the case of a possibly degenerate eigenvalue spectrum, and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state. We also analyse the associated Lindblad equation for the evolution of the density matrix, and obtain an exact time-dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the L\"uders density matrix. Finally, we apply Girsanov's theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric Brownian motions, thereby constructing an explicit unravelling of the Lindblad equation.Comment: 30 pages LaTeX. Submitted to Journal of Physics

Generalized seniority from random Hamiltonians

We investigate the generic pairing properties of shell-model many-body Hamiltonians drawn from ensembles of random two-body matrix elements. Many features of pairing that are commonly attributed to the interaction are in fact seen in a large part of the ensemble space. Not only do the spectra show evidence of pairing with favored J=0 ground states and an energy gap, but the relationship between ground state wave functions of neighboring nuclei show signatures of pairing as well. Matrix elements of pair creation/annihilation operators between ground states tend to be strongly enhanced. Furthermore, the same or similar pair operators connect several ground states along an isotopic chain. This algebraic structure is reminiscent of the generalized seniority model. Thus pairing may be encoded to a certain extent in the Fock space connectivity of the interacting shell model even without specific features of the interaction required.Comment: 10 pages, 7 figure

Quantum noise and stochastic reduction

In standard nonrelativistic quantum mechanics the expectation of the energy is a conserved quantity. It is possible to extend the dynamical law associated with the evolution of a quantum state consistently to include a nonlinear stochastic component, while respecting the conservation law. According to the dynamics thus obtained, referred to as the energy-based stochastic Schrodinger equation, an arbitrary initial state collapses spontaneously to one of the energy eigenstates, thus describing the phenomenon of quantum state reduction. In this article, two such models are investigated: one that achieves state reduction in infinite time, and the other in finite time. The properties of the associated energy expectation process and the energy variance process are worked out in detail. By use of a novel application of a nonlinear filtering method, closed-form solutions--algebraic in character and involving no integration--are obtained for both these models. In each case, the solution is expressed in terms of a random variable representing the terminal energy of the system, and an independent noise process. With these solutions at hand it is possible to simulate explicitly the dynamics of the quantum states of complicated physical systems.Comment: 50 page

Translationally-invariant coupled-cluster method for finite systems

The translational invariant formulation of the coupled-cluster method is presented here at the complete SUB(2) level for a system of nucleons treated as bosons. The correlation amplitudes are solution of a non-linear coupled system of equations. These equations have been solved for light and medium systems, considering the central but still semi-realistic nucleon-nucleon S3 interaction.Comment: 16 pages, 2 Postscript figures, to be published in Nucl. Phys.

Revealing and Resolving the Restrained Enzymatic Cleavage of DNA Self-Assembled Monolayers on Gold: Electrochemical Quantitation and ESI-MS Confirmation

Herein we report a combined electrochemical and ESI-MS study of the enzymatic hydrolysis efficiency of DNA self-assembled monolayers (SAMs) on gold, platform systems for understanding nucleic acid surface chemistry and for constructing DNA-based biosensors. Our electrochemical approach is based on the comparison of the amounts of surface-tethered DNA nucleotides before and after Exonuclease I (Exo I) incubation using electrostatically bound [Ru(NH3)6]3+ as redox indicators. It is surprising to reveal that the hydrolysis efficiency of ssDNA SAMs does not depend on the packing density and base sequence, and that the cleavage ends with surface-bound shorter strands (9-13 mers). The ex-situ ESI-MS observations confirmed that the hydrolysis products for ssDNA SAMs (from 24 to 56 mers) are dominated with 10-15 mer fragments, in contrast to the complete digestion in solution. Such surface-restrained hydrolysis behavior is due to the steric hindrance of the underneath electrode to the Exo I/DNA binding, which is essential for the occurrence of Exo I-catalyzed processive cleavage. More importantly, we have shown that the hydrolysis efficiency of ssDNA SAMs can be remarkably improved by adopting long alkyl linkers (locating DNA strands further away from the substrates)

REACTIVITY OF CHLOROPHYLL a/b-PROTEINS AND MICELLAR TRITON X-100 COMPLEXES OF CHLOROPHYLLS a OR b WITH BOROHYDRIDE

The reaction of several plant chlorophyll-protein complexes with NaBH4 has been studied by absorption spectroscopy. In all the complexes studied, chlorophyll b is more reactive than Chi a, due to preferential reaction of its formyl substituent at C-7. The complexes also show large variations in reactivity towards NaBH4 and the order of reactivity is: LHCI > PSII complex > LHCII > PSI > P700 (investigated as a component of PSI). Differential pools of the same type of chlorophyll have been observed in several complexes. Parallel work was undertaken on the reactivity of micellar complexes of chlorophyll a and of chlorophyll b with NaBH4 to study the effect of aggregation state on this reactivity. In these complexes, both chlorophyll a and b show large variations in reactivity in the order monomer > oligomer > polymer with chlorophyll b generally being more reactive than chlorophyll a. It is concluded that aggregation decreases the reactivity of chlorophylls towards NaBH4 in vitro, and may similarly decrease reactivity in naturally-occurring chlorophyll-protein complexes

Partial Dynamical Symmetry and Mixed Dynamics

Partial dynamical symmetry describes a situation in which some eigenstates have a symmetry which the quantum Hamiltonian does not share. This property is shown to have a classical analogue in which some tori in phase space are associated with a symmetry which the classical Hamiltonian does not share. A local analysis in the vicinity of these special tori reveals a neighbourhood of phase space foliated by tori. This clarifies the suppression of classical chaos associated with partial dynamical symmetry. The results are used to divide the states of a mixed system into ``chaotic'' and ``regular'' classes.Comment: 10 pages, Revtex, 3 figures, Phys. Rev. Lett. in pres

Continuous Equilibrium in Affine and Information-Based Capital Asset Pricing Models

We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents' optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents' risk aversion on the implied volatility of simultaneously-traded European-style options.Comment: 24 pages, 4 figure