The Lieb–Robinson bound is a theoretical upper limit on the speed at which information can propagate in non-relativistic quantum systems.
It demonstrates that information cannot travel instantaneously in quantum theory, even when the relativity limits of the speed of light are ignored.
The existence of such a finite speed was discovered mathematically by Elliott H. Lieb and Derek W. Robinson in 1972.
[1] It turns the locality properties of physical systems into the existence of, and upper bound for this speed.
For finite-range, e.g. nearest-neighbor, interactions, this velocity is a constant independent of the distance travelled.
In long-range interacting systems, this velocity remains finite, but it can increase with the distance travelled.
The theory of relativity shows that no information, or anything else for that matter, can travel faster than the speed of light.
This is important conceptually and practically, because it means that, for short periods of time, distant parts of a system act independently.
Current proposals to construct quantum computers built out of atomic-like units mostly rely on the existence of this finite speed of propagation to protect against too rapid dispersal of information.
[4][3] To define the bound, it is necessary to first describe basic facts about quantum mechanical systems composed of several units, each with a finite dimensional Hilbert space.
is defined formally by: The laws of quantum mechanics say that corresponding to every physically observable quantity there is a self-adjoint operator
(Technically speaking, this time evolution is defined by a power-series expansion that is known to be a norm-convergent series
[citation needed] Once this operator bound is established it necessarily carries over to any state of the system.
The bound (1) is presented slightly differently from the equation in the original paper which derived velocity-dependent decay rates along spacetime rays with velocity greater than
Subsequently, Nachtergaele and Sims[12] extended the results of[9] to include models on vertices with a metric and to derive exponential decay of correlations.
From 2005 to 2006 interest in Lieb–Robinson bounds strengthened with additional applications to exponential decay of correlations (see[2][5][13] and the sections below).
New proofs of the bounds were developed and, in particular, the constant in (1) was improved making it independent of the dimension of the Hilbert space.
The Lieb–Robinson bounds were extended to certain continuous quantum systems, that is to a general harmonic Hamiltonian,[15] which, in a finite volume
are positive integers, takes the form: where the periodic boundary conditions are imposed and
Anharmonic Hamiltonians with on-site and multiple-site perturbations were considered and the Lieb–Robinson bounds were derived for them,[15][16] Further generalizations of the harmonic lattice were discussed,[17][18] Another generalization of the Lieb–Robinson bounds was made to the irreversible dynamics, in which case the dynamics has a Hamiltonian part and also a dissipative part.
Lieb–Robinson bounds for the irreversible dynamics were considered by[13] in the classical context and by[19] for a class of quantum lattice systems with finite-range interactions.
Lieb–Robinson bounds for lattice models with a dynamics generated by both Hamiltonian and dissipative interactions with suitably fast decay in space, and that may depend on time, were proved by,[20] where they also proved the existence of the infinite dynamics as a strongly continuous cocycle of unit preserving completely positive maps.
[23] Thus, the Lieb–Robinson bounds for power-law interactions typically yield a sub-linear light cone that is asymptotically linear in the limit
Later, the algorithm is generalized to power-law interactions and subsequently used to derive a stronger Lieb–Robinson bound.
The static thermodynamic limit from the equilibrium point of view was settled much before the Lieb–Robinson bound was proved, see[6] for example.
[9] Later, Nachtergaele et al.[5][16][20] showed the existence of the infinite volume dynamics for almost every type of interaction described in the section "Improvements of Lieb–Robinson bounds" above.
Lieb–Robinson bounds are used to show that the correlations decay exponentially in distance for a system with an energy gap above a non-degenerate ground state
Lieb–Schultz–Mattis theorem implies that the ground state of the Heisenberg antiferromagnet on a bipartite lattice with isomorphic sublattices, is non-degenerate, i.e., unique, but the gap can be very small.
, The Lieb–Robinson bound was utilized by Hastings[11] and by Nachtergaele-Sims[28] in a proof of the Lieb–Schultz–Mattis Theorem for higher-dimensional cases.
One can also bound the error induced by local Fock space truncation of the harmonic oscillators[30] The first experimental observation of the Lieb–Robinson velocity was done by Cheneau et al.[31]