Infinitesimal local operations and differential conditions for entanglement monotones

Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the {\it entanglement monotone}: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by {\it infinitesimal local operations}, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of {\it convexity}. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.Comment: 21 pages, RevTeX format, no figures, three minor corrections, including a factor of two in the differential conditions, the tracelessness of the matrix in the convexity condition, and the proof that the local purity is a monotone under local measurements. The conclusions of the paper are unaffecte

Measuring non-linear functionals of quantum harmonic oscillator states

Using only linear interactions and a local parity measurement we show how entanglement can be detected between two harmonic oscillators. The scheme generalizes to measure both linear and non-linear functionals of an arbitrary oscillator state. This leads to many applications including purity tests, eigenvalue estimation, entropy and distance measures - all without the need for non-linear interactions or complete state reconstruction. Remarkably, experimental realization of the proposed scheme is already within the reach of current technology with linear optics.Comment: 5 pages, 2 figures. Minor corrections and some new references adde

Quantum state diffusion with a moving basis: computing quantum-optical spectra

Quantum state diffusion (QSD) as a tool to solve quantum-optical master equations by stochastic simulation can be made several orders of magnitude more efficient if states in Hilbert space are represented in a moving basis of excited coherent states. The large savings in computer memory and time are due to the localization property of the QSD equation. We show how the method can be used to compute spectra and give an application to second harmonic generation.Comment: 8 pages in RevTeX, 1 uuencoded postscript figure, submitted to Phys. Rev.

Quantum state diffusion, localization and computation

Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wave packets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle, we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.Comment: 16 pages in LaTeX, 2 uuencoded postscript figures, submitted to J. Phys.

Dirac model of electronic transport in graphene antidot barriers

In order to use graphene for semiconductor applications, such as transistors with high on/off ratios, a band gap must be introduced into this otherwise semimetallic material. A promising method of achieving a band gap is by introducing nanoscale perforations (antidots) in a periodic pattern, known as a graphene antidot lattice (GAL). A graphene antidot barrier (GAB) can be made by introducing a 1D GAL strip in an otherwise pristine sheet of graphene. In this paper, we will use the Dirac equation (DE) with a spatially varying mass term to calculate the electronic transport through such structures. Our approach is much more general than previous attempts to use the Dirac equation to calculate scattering of Dirac electrons on antidots. The advantage of using the DE is that the computational time is scale invariant and our method may therefore be used to calculate properties of arbitrarily large structures. We show that the results of our Dirac model are in quantitative agreement with tight-binding for hexagonal antidots with armchair edges. Furthermore, for a wide range of structures, we verify that a relatively narrow GAB, with only a few antidots in the unit cell, is sufficient to give rise to a transport gap

An observable entanglement measure for unknown mixed quantum states

We show how an unknown mixed quantum state's entanglement can be quantified by a suitable, local parity measurement on its two-fold copy.Comment: in press in PR

Electronic and optical properties of graphene antidot lattices: Comparison of Dirac and tight-binding models

The electronic properties of graphene may be changed from semimetallic to semiconducting by introducing perforations (antidots) in a periodic pattern. The properties of such graphene antidot lattices (GALs) have previously been studied using atomistic models, which are very time consuming for large structures. We present a continuum model that uses the Dirac equation (DE) to describe the electronic and optical properties of GALs. The advantages of the Dirac model are that the calculation time does not depend on the size of the structures and that the results are scalable. In addition, an approximation of the band gap using the DE is presented. The Dirac model is compared with nearest-neighbour tight-binding (TB) in order to assess its accuracy. Extended zigzag regions give rise to localized edge states, whereas armchair edges do not. We find that the Dirac model is in quantitative agreement with TB for GALs without edge states, but deviates for antidots with large zigzag regions.Comment: 15 pages, 7 figures. Accepted by Journal of Physics: Condensed matte

Breaking anchored droplets in a microfluidic Hele-Shaw cell

We study microfluidic self digitization in Hele-Shaw cells using pancake droplets anchored to surface tension traps. We show that above a critical flow rate, large anchored droplets break up to form two daughter droplets, one of which remains in the anchor. Below the critical flow velocity for breakup the shape of the anchored drop is given by an elastica equation that depends on the capillary number of the outer fluid. As the velocity crosses the critical value, the equation stops admitting a solution that satisfies the boundary conditions; the drop breaks up in spite of the neck still having finite width. A similar breaking event also takes place between the holes of an array of anchors, which we use to produce a 2D array of stationary drops in situ.Comment: 5 pages, 4 figures, to appear in Phys. Rev. Applie

Rayleigh-Taylor instability under an inclined plane

We revisit the canonical Rayleigh-Taylor instability and investigate the case of a thin film of fluid upon the underside of an inclined plane. The presence of a natural flow along the plane competes with the conventional droplet forming instability. In particular, experiments reveal that no drops form for inclinations greater than a critical value. These features are rationalized in the context of the absolute/convective analysis conducted in this article

Quantum Walks driven by many coins

Quantum random walks have been much studied recently, largely due to their highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum random walk on the line: the use of multiple quantum ``coins'' in order to diminish the effects of interference between paths. We find solutions to this system in terms of the single coin random walk, and compare the asymptotic limit of these solutions to numerical simulations. We find exact analytical expressions for the time-dependence of the first two moments, and show that in the long time limit the ``quantum mechanical'' behavior of the one-coin walk persists. We further show that this is generic for a very broad class of possible walks, and that this behavior disappears only in the limit of a new coin for every step of the walk.Comment: 36 pages RevTeX 4.0 + 5 figures (encapsulated Postscript). Submitted to Physical Review