Boundary behavior of the Kobayashi distance in pseudoconvex Reinhardt domains

We prove that the Kobayashi distance near boundary of a pseudoconvex Reinhardt domain $D$ increases asymptotically at most like $-\log d_D+C$. Moreover, for boundary points from $\text{int}\bar{D}$ the growth does not exceed $1/2\log(-\log d_D)+C$. The lower estimate by $-1/2\log d_D+C$ is obtained under additional assumptions of $\mathcal C^1$-smoothness of a domain and a non-tangential convergence.Comment: 16 pages. To appear in Mich. Math.

(Weak) mm-extremals and mm-geodesics

We present a collection of results on (weak) $m$-extremals and $m$-geodesics, concerning general properties, the planar case, quasi-balanced pseudoconvex domains, complex ellipsoids, the Euclidean ball and boundary properties. We prove $3$-geodesity of $3$-extremals in the Euclidean ball. Equivalence of weak $m$-extremality and $m$-extremality in some class of convex complex ellipsoids, containing symmetric ones and $\mathcal C^2$-smooth ones is showed. Moreover, first examples of $3$-extremals being not $3$-geodesics in convex domains are given.Comment: 25 pages. In this version equivalence of weak m-extremality and m-extremality is proved for a bigger family of convex complex ellipsoid

Open quantum systems are harder to track than open classical systems

For a Markovian open quantum system it is possible, by continuously monitoring the environment, to know the stochastically evolving pure state of the system without altering the master equation. In general, even for a system with a finite Hilbert space dimension $D$, the pure state trajectory will explore an infinite number of points in Hilbert space, meaning that the dimension $K$ of the classical memory required for the tracking is infinite. However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed that tracking of a qubit ($D=2$) is always possible with a bit ($K=2$), and gave a heuristic argument implying that a finite $K$ should be sufficient for any $D$, although beyond $D=2$ it would be necessary to have $K>D$. Our paper is concerned with rigorously investigating the relationship between $D$ and $K_{\rm min}$, the smallest feasible $K$. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with $D>2$, $K_{\rm min}>D$, by a computational proof (via Hilbert Nullstellensatz certificates of infeasibility). That is, beyond $D=2$, $D$-dimensional open quantum systems are provably harder to track than $D$-dimensional open classical systems. Moreover, we develop, and better justify, a new heuristic to guide our expectation of $K_{\rm min}$ as a function of $D$, taking into account the number $L$ of Lindblad operators as well as symmetries in the problem. The use of invariant subspace and Wigner symmetries makes it tractable to conduct a numerical search, using the method of polynomial homotopy continuation, to find finite physically realizable ensembles (as they are known) in $D=3$. The results of this search support our heuristic. We thus have confidence in the most interesting feature of our heuristic: in the absence of symmetries, $K_{\rm min} \sim D^2$, implying a quadratic gap between the classical and quantum tracking problems.Comment: 35 pages, 3 figures, Accepted in Quantum Journal, minor change

Symmetries and physically realizable ensembles for open quantum systems

A $D$-dimensional Markovian open quantum system will undergo stochastic evolution which preserves pure states, if one monitors without loss of information the bath to which it is coupled. If a finite ensemble of pure states satisfies a particular set of constraint equations then it is possible to perform the monitoring in such a way that the (discontinuous) trajectory of the conditioned system state is, at all long times, restricted to those pure states. Finding these physically realizable ensembles (PREs) is typically very difficult, even numerically, when the system dimension is larger than 2. In this paper, we develop symmetry-based techniques that potentially greatly reduce the difficulty of finding a subset of all possible PREs. The two dynamical symmetries considered are an invariant subspace and a Wigner symmetry. An analysis of previously known PREs using the developed techniques provides us with new insights and lays the foundation for future studies of higher dimensional systems.Comment: 30 pages, 4 figures, comments welcome. Published versio

Unpinning triggers for superfluid vortex avalanches

The pinning and collective unpinning of superfluid vortices in a decelerating container is a key element of the canonical model of neutron star glitches and laboratory spin-down experiments with helium II. Here the dynamics of vortex (un)pinning is explored using numerical Gross-Pitaevskii calculations, with a view to understanding the triggers for catastrophic unpinning events (vortex avalanches) that lead to rotational glitches. We explicitly identify three triggers: rotational shear between the bulk condensate and the pinned vortices, a vortex proximity effect driven by the repulsive vortex-vortex interaction, and sound waves emitted by moving and repinning vortices. So long as dissipation is low, sound waves emitted by a repinning vortex are found to be sufficiently strong to unpin a nearby vortex. For both ballistic and forced vortex motion, the maximum inter-vortex separation required to unpin scales inversely with pinning strength.Comment: 16 pages, 18 figure

Geometric properties of semitube domains

In the paper we study the geometry of semitube domains in $\mathbb C^2$. In particular, we extend the result of Burgu\'es and Dwilewicz for semitube domains dropping out the smoothness assumption. We also prove various properties of non-smooth pseudoconvex semitube domains obtaining among others a relation between pseudoconvexity of a semitube domain and the number of connected components of its vertical slices. Finally, we present an example showing that there is a non-convex domain in $\mathbb C^n$ such that its image under arbitrary isometry is pseudoconvex.Comment: 6 page

Dynamical parameter estimation using realistic photodetection

We investigate the effect of imperfections in realistic detectors upon the problem of quantum state and parameter estimation by continuous monitoring of an open quantum system. Specifically, we have reexamined the system of a two-level atom with an unknown Rabi frequency introduced by Gambetta and Wiseman [Phys. Rev. A 64, 042105 (2001)]. We consider only direct photodetection and use the realistic quantum trajectory theory reported by Warszawski, Wiseman, and Mabuchi [Phys. Rev. A 65, 023802 (2002)]. The most significant effect comes from a finite bandwidth, corresponding to an uncertainty in the response time of the photodiode. Unless the bandwidth is significantly greater than the Rabi frequency, the observer's ability to obtain information about the unknown Rabi frequency, and about the state of the atom, is severely compromised. This result has implications for quantum control in the presence of unknown parameters for realistic detectors, and even for ideal detectors, as it implies that most of the information in the measurement record is contained in the precise timing of the detections.Comment: 8 pages, 6 figure