A quantum router for high-dimensional entanglement

In addition to being a workhorse for modern quantum technologies, entanglement plays a key role in fundamental tests of quantum mechanics. The entanglement of photons in multiple levels, or dimensions, explores the limits of how large an entangled state can be, while also greatly expanding its applications in quantum information. Here we show how a high-dimensional quantum state of two photons entangled in their orbital angular momentum can be split into two entangled states with a smaller dimensionality structure. Our work demonstrates that entanglement is a quantum property that can be subdivided into spatially separated parts. In addition, our technique has vast potential applications in quantum as well as classical communication systems.Comment: 5 pages, 5 figure

Iterative Universal Rigidity

A bar framework determined by a finite graph $G$ and configuration $\bf p$ in $d$ space is universally rigid if it is rigid in any ${\mathbb R}^D \supset {\mathbb R}^d$. We provide a characterization of universally rigidity for any graph $G$ and any configuration ${\bf p}$ in terms of a sequence of affine subsets of the space of configurations. This corresponds to a facial reduction process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure

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Department of Energy Engineering (Battery Science and Technology)The continuous throng in demand for high energy density rechargeable batteries innovatively drives technological development in cell design as well as electrochemically active materials. In that perspective metal-free batteries consisting of a flowing seawater as a cathode active material were introduced. However, the electrochemical performance of the seawater battery was restrained by NASICON (Na3Zr2Si2PO12) ceramic solid electrolyte. Here, we demonstrate a new class of fibrous nanomat hard-carbon (FNHC) anode/1D (one-dimensional) bucky paper (1DBP) cathode hybrid electrode architecture in seawater battery based on 1D building block-interweaved hetero-nanomat frameworks. Differently from conventional slurry-cast electrodes, exquisitely designed hybrid hetero-nanomat electrodes are fabricated through concurrent dual electrospraying and electrospinning for the anode, vacuum-assisted infiltration for the cathode. HC nanoparticles are closely embedded in the spatially reinforced polymeric nanofiber/CNT hetero-nanomat skeletons that play a crucial role in constructing 3D-bicontinuous ion/electron transport pathways and allow to eliminate heavy metallic aluminum foil current collectors. Eventually the FNHC/1DBP seawater full cell, driven by aforementioned physicochemical uniqueness, shows exceptional improvement in electrochemical performance (Energy density = 693 Wh kg-1), (Power density = 3341 W kg-1) removing strong stereotype of ceramic solid electrolyte, which beyond those achievable with innovative next generation battery technologies.ope

A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular $m\times n$ matrices are both equal to $mn \big(1-\frac1{m+n}\big)$, thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis (preprint 2014) as well as answering a question of Bugeaud, Cheung, and Chevallier (preprint 2016). We introduce the notion of a $template$, which generalizes the notion of a $rigid$ $system$ (Roy, 2015) to the setting of matrix approximation. Our main theorem takes the following form: for any class of templates $\mathcal F$ closed under finite perturbations, the Hausdorff and packing dimensions of the set of matrices whose successive minima functions are members of $\mathcal F$ (up to finite perturbation) can be written as the suprema over $\mathcal F$ of certain natural functions on the space of templates. Besides implying KKLM's conjecture, this theorem has many other applications including computing the Hausdorff and packing dimensions of the set of points witnessing a conjecture of Starkov (2000), and of the set of points witnessing a conjecture of Schmidt (1983).Comment: Announcemen