Spatial positions of cells in tissues strongly influence function, yet a high-throughput, genome-wide readout of gene expression with cellular resolution is lacking. We developed Slide-seq, a method for transferring RNA from tissue sections onto a surface covered in DNA-barcoded beads with known positions, allowing the locations of the RNA to be inferred by sequencing. Using Slide-seq, we localized cell types identified by single-cell RNA sequencing datasets within the cerebellum and hippocampus, characterized spatial gene expression patterns in the Purkinje layer of mouse cerebellum, and defined the temporal evolution of cell type–specific responses in a mouse model of traumatic brain injury. These studies highlight how Slide-seq provides a scalable method for obtaining spatially resolved gene expression data at resolutions comparable to the sizes of individual cells.

The spatial organization of cells in tissue has a profound influence on their function, yet a highthroughput, genome-wide readout of gene expression with cellular resolution is lacking. Here, we introduce Slide-seq, a highly scalable method that enables facile generation of large volumes of unbiased spatial transcriptomes with 10 µm spatial resolution, comparable to the size of individual cells. In Slide-seq, RNA is transferred from freshly frozen tissue sections onto a surface covered in DNA-barcoded beads with known positions, allowing the spatial locations of the RNA to be inferred by sequencing. To demonstrate Slide-seq's utility, we localized cell types identified by large-scale scRNA-seq datasets within the cerebellum and hippocampus. We next systematically characterized spatial gene expression patterns in the Purkinje layer of mouse cerebellum, identifying new axes of variation across Purkinje cell compartments. Finally, we used Slide-seq to define the temporal evolution of cell-type-specific responses in a mouse model

Lithographic nanofabrication is often limited to successive fabrication of two-dimensional layers. We present a strategy for the direct assembly of three-dimensional nanomaterials consisting of metals, semiconductors, and biomolecules arranged in virtually any three-dimensional geometry. We use hydrogels as scaffolds for volumetric deposition of materials at defined points in space. We then optically pattern these scaffolds in three dimensions, attach one or more functional materials, and then shrink and dehydrate them in a controlled way to achieve nanoscale feature sizes in a solid substrate. We demonstrate this process, Implosion Fabrication (ImpFab), by directly writing highly conductive, 3D silver nanostructures within an acrylic scaffold using a volumetric silver deposition process, achieving resolutions in the tens of nanometers and complex, non-self-supporting 3D geometries of interest for optical metamaterials.

We generalize the notion of the best separable approximation (BSA) and best W-class approximation (BWA) to arbitrary pure state entanglement measures, defining the best zero-E approximation (BEA). We show that for any polynomial entanglement measure E, any mixed state ρ admits at least one "S-decomposition," i.e., a decomposition in terms of a mixed state on which E is equal to zero, and a single additional pure state with (possibly) non-zero E. We show that the BEA is not in general the optimal S-decomposition from the point of view of bounding the entanglement of ρ, and describe an algorithm to construct the entanglement-minimizing S-decomposition for ρ and place an upper bound on E(ρ). When applied to the three-tangle, the cost of the algorithm is linear in the rank d of the density matrix and has accuracy comparable to a steepest descent algorithm whose cost scales as d 8 log d. We compare the upper bound to a lower bound algorithm given by Eltschka and Siewert for the three-tangle, and find that on random rank-two three-qubit density matrices, the difference between the upper and lower bounds is 0.14 on average. We also find that the three-tangle of random full-rank three qubit density matrices is less than 0.023 on average.Non-classical correlations in quantum states such as entanglement distinguish quantum from classical information theory. The ability to calculate entanglement of mixed quantum states is relevant for the analysis of tomography data for systems of multiple qubits in several implementations [1][2][3]. Multipartite systems can contain multiple inequivalent types of entanglement that cannot be converted into one another by local operations and classical communication [4].One approach to characterizing pure-state entanglement in a system of qubits associates a polynomial function that is invariant under determinant 1 local operations with each type of entanglement [5][6][7]. Examples of such polynomial invariants include the concurrence for two qubits [8] and the three-tangle, which quantifies the amount of entanglement in a three-qubit system that cannot be accounted for by entanglement between pairs of the qubits [9].A polynomial invariant E is extended to mixed states by way of the convex roof, given for a rank-d density matrix ρ by:where E = {p i , |ψ i } is a pure-state ensemble for ρ and Υ ρ is the set of all such ensembles. Caratheodory's theorem allows us to restrict the optimization to ensembles containing no more than d 2 elements [10]. An ensemble that minimizes eq. (1) is said to be minimal. We consider the rank d of the density matrix d, rather than the dimension of the Hilbert space on which it acts, because d is the parameter that determines the computational difficulty of the convex roof minimization. A number of special cases of computation of the convex roof have been solved for cases of restricted rank [11][12][13].Minimal ensembles have been found analytically for the concurrence of arbitrary two-qubit mixed states [8], and for the three-tangle of rank-two mixtures of gener...