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The speed of light $c$ sets a strict upper bound on the speed of informationtransfer in both classical and quantum systems. In nonrelativistic quantumsystems, the Lieb-Robinson Theorem imposes an emergent speed limit $v\hspace{-0.2mm} \ll \hspace{-0.2mm} c$, establishing locality under unitaryevolution and constraining the time needed to perform useful quantum tasks. Weextend the Lieb-Robinson Theorem to quantum dynamics with measurements. Incontrast to the expectation that measurements can arbitrarily violate spatiallocality, we find at most an $(M \hspace{-0.5mm} +\hspace{-0.5mm} 1)$-foldenhancement to the speed $v$ of quantum information, provided the outcomes ofmeasurements in $M$ local regions are known. This holds even when classicalcommunication is instantaneous, and extends beyond projective measurements toweak measurements and other nonunitary channels. Our bound is asymptoticallyoptimal, and saturated by existing measurement-based protocols. We tightlyconstrain the resource requirements for quantum computation, error correction,teleportation, and generating entangled resource states (Bell, GHZ,quantum-critical, Dicke, W, and spin-squeezed states) fromshort-range-entangled initial states. Our results impose limits on the use ofmeasurements and active feedback to speed up quantum information processing,resolve fundamental questions about the nature of measurements in quantumdynamics, and constrain the scalability of a wide range of proposed quantumtechnologies.

Locality and error correction in quantum dynamics with measurement