  Speed of state transitions in macroscopic systems is a crucial concept for
foundations of nonequilibrium statistical mechanics as well as various
applications in quantum technology represented by optimal quantum control.
While extensive studies have made efforts to obtain rigorous constraints on
dynamical processes since Mandelstam and Tamm, speed limits that provide tight
bounds for macroscopic transitions have remained elusive. Here, by employing
the local conservation law of probability, the fundamental principle in
physics, we develop a general framework for deriving qualitatively tighter
speed limits for macroscopic systems than many conventional ones. We show for
the first time that the speed of the expectation value of an observable defined
on an arbitrary graph, which can describe general many-body systems, is bounded
by the "gradient" of the observable, in contrast with conventional speed limits
depending on the entire range of the observable. This framework enables us to
derive novel quantum speed limits for macroscopic unitary dynamics. Unlike
previous bounds, the speed limit decreases when the expectation value of the
transition Hamiltonian increases; this intuitively describes a new tradeoff
relation between time and quantum phase difference. Our bound is dependent on
instantaneous quantum states and thus can achieve the equality condition, which
is conceptually distinct from the Lieb-Robinson bound. Our work elucidates
novel speed limits on the basis of local conservation law, providing
fundamental limits to various types of nonequilibrium quantum macroscopic
phenomena.
