Lieb-Robinson bound

A Lieb-Robinson bound limits how quickly a local observable can influence a distant region under quantum dynamics. A standard setting is a lattice of finite-dimensional quantum systems with a Hamiltonian made of uniformly bounded, finite-range interactions. For observables $A_X,B_Y$ with disjoint finite operator supports $X,Y$, a typical bound iswhere $\Vert \cdot \Vert$ is the operator norm, $\mu >0$, $v$ is a finite velocity depending on interaction strengths and range, and the support-dependent prefactor is independent of total system size. On a regular lattice, suitably controlled exponentially decaying interactions also admit this form. ^[ref]

The estimate gives an effective causal cone: the commutator is exponentially small when the separation exceeds $v|t|$. It can have small nonzero tails, so the estimate does not impose a sharp relativistic light cone. An operational consequence follows by perturbing a quantum state with a distant local unitary operator $U_Y$:Thus the same Lieb-Robinson bound controls the change in a distant expectation value. This permits quantitative local approximations to many-body unitary time evolution; long-range interactions require estimates matched to their actual decay.

A Duhamel comparison can bound truncation error using commutators evolved under the full Hamiltonian. This avoids assuming that a black-box Lieb-Robinson bound for $H$ also holds for every truncation. Polynomial shell weights and exponential distance suppression make the sum over omitted shells convergent. Comparing two truncations through the full evolution bounds their difference.

Iterating the integral commutator inequality gives ordered interaction chains with factors $(2|t|{)}^n/n!$. A per-site interaction weight at most $s{e}^{-\mu }$ bounds each successive chain extension by $ks{e}^{-\mu }$. No chain shorter than the interaction distance connects separated supports. Reversing chains supplies the smaller support prefactor, producing the Lieb-Robinson bound. Overlapping supports require retaining the equal-time commutator term.

The interaction distance between disjoint supports can count the minimum number of interacting hyperedges in a chain connecting them. Overlapping supports have distance zero in the usual support convention. Distances between interaction terms can instead assign shell zero only to the central term and positive shells to distinct terms. The convention must be specified when using a truncated neighborhood Hamiltonian: ordinary zero support distance includes distinct overlapping terms.

Apply Haar twirling conditional expectation to the complement of a neighbourhood around the initial operator support. A Lieb-Robinson bound on commutators with arbitrary complement-supported unitaries then bounds the approximation error without summing over sites. For a bound proportional to $e^{-\mu d}(e^{2kst}-1)$, choosing $v=4ks/\mu$ gives an error at most $\mu vt|X|\Vert A_X\Vert e^{-\mu l/2}$ outside radius $vt+l$.

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.