LiebRobinsontype

Lieb-Robinson type refers to a class of mathematical bounds that describe the effective speed at which information and correlations can propagate in quantum many-body systems, particularly on lattices or graphs. Named after Elliott Lieb and Derek Robinson, these bounds establish a quasi-locality property for the time evolution of local observables in non-relativistic quantum spin systems.

Typically, the bounds take the form of an exponential decay bound on the norm of the commutator of time-evolved and spatially separated local operators. A common statement is that there exist constants c, μ, and a finite velocity v (the Lieb-Robinson velocity) such that for local operators A_x and B_y supported near sites x and y, one has ||[A_x(t), B_y]|| ≤ c exp[-μ(d(x,y) − v|t|)], where d(x,y) is a distance measure on the underlying lattice. This implies a light-cone-like region outside which correlations and perturbations are strongly suppressed.

The concept has broad extensions beyond strict nearest-neighbor interactions. It applies to systems with decaying interactions that are sufficiently short-range and to various geometries, including continuous spaces and graphs. Researchers have developed generalized Lieb-Robinson type bounds for power-law decays, higher dimensions, and disordered or time-dependent Hamiltonians, often with modified velocity or decay terms.

Significance and applications include establishing quasi-locality of dynamics, proving area laws for entanglement, bounding propagation of information in quantum circuits, and underpinning rigorous results in quantum many-body theory and quantum information science.

See also: Lieb-Robinson bounds, locality in quantum systems, quantum spin models, entanglement propagation.