In quantum many-body systems with local interactions, quantum information and entanglement cannot spread outside of a linear light cone, which expands at an emergent velocity analogous to the speed of light. Local operations at sufficiently separated spacetime points approximately commute-given a many-body state jψi, O x ðtÞO y jψi ≈ O y O x ðtÞjψi with arbitrarily small errors-so long as jx − yj ≳ vt, where v is finite. Yet, most nonrelativistic physical systems realized in nature have long-range interactions: Two degrees of freedom separated by a distance r interact with potential energy VðrÞ ∝ 1=r α. In systems with long-range interactions, we rigorously establish a hierarchy of linear light cones: At the same α, some quantum information processing tasks are constrained by a linear light cone, while others are not. In one spatial dimension, this linear light cone exists for every many-body state jψi when α > 3 (Lieb-Robinson light cone); for a typical state jψi chosen uniformly at random from the Hilbert space when α > 5 2 (Frobenius light cone); and for every state of a noninteracting system when α > 2 (free light cone). These bounds apply to time-dependent systems and are optimal up to subalgebraic improvements. Our theorems regarding the Lieb-Robinson and free light cones-and their tightness-also generalize to arbitrary dimensions. We discuss the implications of our bounds on the growth of connected correlators and of topological order, the clustering of correlations in gapped systems, and the digital simulation of systems with long-range interactions. In addition, we show that universal quantum state transfer, as well as manybody quantum chaos, is bounded by the Frobenius light cone and, therefore, is poorly constrained by all Lieb-Robinson bounds.

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Glutathione and multidrug resistance protein (MRP) play an important role on the metabolism of a variety of drugs. Bismuth drugs have been used to treat gastrointestinal disorder and Helicobacter pylori infection for decades without exerting acute toxicity. They were found to interact with a wide variety of biomolecules, but the major metabolic pathway remains unknown. For the first time (to our knowledge), we systematically and quantitatively studied the metabolism of bismuth in human cells. Our data demonstrated that over 90% of bismuth was passively absorbed, conjugated to glutathione, and transported into vesicles by MRP transporter. Mathematical modeling of the system reveals an interesting phenomenon. Passively absorbed bismuth consumes intracellular glutathione, which therefore activates de novo biosynthesis of glutathione. Reciprocally, sequestration by glutathione facilitates the passive uptake of bismuth and thus completes a self-sustaining positive feedback circle. This mechanism robustly removes bismuth from both intra-and extracellular space, protecting critical systems of human body from acute toxicity. It elucidates the selectivity of bismuth drugs between human and pathogens that lack of glutathione, such as Helicobacter pylori, opening new horizons for further drug development.bismuth | drug selectivity | glutathione | MRP | positive feedback