On the Curvature of Monotone Metrics and a Conjecture Concerning the Kubo-Mori Metric

It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its expected quantum statistical meaning. The scalar curvature is explained in more detail for three examples, the Bures metric, the largest monotone metric and the Kubo-Mori metric. In particular, we show an important conjecture of Petz concerning the Kubo-Mori metric up to a formal proof of the concavity of a certain function on R_+^3. This concavity seems to be numerically evident. The conjecture of Petz asserts that the scalar curvature of the Kubo-Mori metric increases if one goes to more mixed states.Comment: 20 pages, 4 figure

Connections and Metrics Respecting Standard Purification

Standard purification interlaces Hermitian and Riemannian metrics on the space of density operators with metrics and connections on the purifying Hilbert-Schmidt space. We discuss connections and metrics which are well adopted to purification, and present a selected set of relations between them. A connection, as well as a metric on state space, can be obtained from a metric on the purification space. We include a condition, with which this correspondence becomes one-to-one. Our methods are borrowed from elementary *-representation and fibre space theory. We lift, as an example, solutions of a von Neumann equation, write down holonomy invariants for cyclic ones, and ``add noise'' to a curve of pure states.Comment: Latex, 27 page

The Scalar Curvature of the Bures Metric on the Space of Density Matrices

The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we determine its scalar curvature and Ricci tensor and prove a lower bound for the curvature on the submanifold of trace one matrices. This bound is achieved for the maximally mixed state, a further hint for the quantum statistical meaning of the scalar curvature.Comment: Latex, 9 page

Geodesic distances on density matrices

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.Comment: 10 page

On the Apparent Orbital Inclination Change of the Extrasolar Transiting Planet TrES-2b

On June 15, 2009 UT the transit of TrES-2b was detected using the University of Arizona's 1.55 meter Kuiper Telescope with 2.0-2.5 millimag RMS accuracy in the I-band. We find a central transit time of $T_c = 2454997.76286 \pm0.00035$ HJD, an orbital period of $P = 2.4706127 \pm 0.0000009$ days, and an inclination angle of $i = 83^{\circ}.92 \pm 0.05$, which is consistent with our re-fit of the original I-band light curve of O'Donovan et al. (2006) where we find $i = 83^{\circ}.84 \pm0.05$. We calculate an insignificant inclination change of $\Delta i = -0^{\circ}.08 \pm 0.07$ over the last 3 years, and as such, our observations rule out, at the $\sim 11 \sigma$ level, the apparent change of orbital inclination to $i_{predicted} = 83^{\circ}.35 \pm0.1$ as predicted by Mislis and Schmitt (2009) and Mislis et al. (2010) for our epoch. Moreover, our analysis of a recently published Kepler Space Telescope light curve (Gilliland et al. 2010) for TrES-2b finds an inclination of $i = 83^{\circ}.91 \pm0.03$ for a similar epoch. These Kepler results definitively rule out change in $i$ as a function of time. Indeed, we detect no significant changes in any of the orbital parameters of TrES-2b.Comment: 19 pages, 1 table, 7 figures. Re-submitted to ApJ, January 14, 201

Multi-field modelling and simulation of large deformation ductile fracture

In the present contribution we focus on a phase-field approach to ductile fracture applied to large deformation contact problems. Phase-field approaches to fracture allow for an efficient numerical investigation of complex three-dimensional fracture problems, as they arise in contact and impact situations. To account for large deformations the underlying formulation is based on a multiplicative decomposition of the deformation gradient into an elastic and plastic part. Moreover, we make use of a fourth-order crack regularization combined with gradient plasticity. Eventually, a demonstrative example shows the capability of the proposed framework