Lieb-Robinson bound

A Lieb-Robinson bound limits the commutator of initially separated local observables by $\Vert [A(t),B]\Vert \le C\Vert A\Vert \Vert B\Vert e^{-\mu (d-v|t|)}$. It gives local lattice dynamics an effective causal cone with Lieb-Robinson velocity $v$.

A depth-$D$ circuit of gates with range at most $r$ expands the support of a local observable by at most $rD$. States connected by a depth bounded independently of system size are regarded as belonging to the same short-range-entangled phase when no required symmetry is violated.

The backward light cone of an output observable is the set of input degrees of freedom that can influence it. In a depth-$D$ circuit with gate range at most $r$, it lies within distance $rD$ of the observable's output support.

The GHZ state has nonzero connected correlations between arbitrarily distant sites. A local circuit starting from a product state can create such a correlation only when the two backward light cones overlap, giving depth at least proportional to the system's linear size.