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.
Contents
- Set up
- Improvements of the Lieb Robinson bounds
- Harmonic and Anharmonic Hamiltonians
- Irreversible dynamics
- Some applications
- Thermodynamic limit of the dynamics
- Exponential decay of correlations
- Lieb Schultz Mattis theorem
- Discretisation of the Continuum via Gauss Quadrature Rules
- Experiments
- References
In the study of quantum systems such as quantum optics, quantum information theory, atomic physics, and condensed matter physics, it is important to know that there is a finite speed with which information can propagate. The theory of relativity shows that no information, or anything else for that matter, can travel faster than the speed of light. When non-relativistic mechanics is considered, however, (Newton's equations of motion or Schrödinger's equation of quantum mechanics) it had been thought that there is then no limitation to the speed of propagation of information. This is not so for certain kinds of quantum systems of atoms arranged in a lattice, often called quantum spin systems. This is important conceptually and practically, because it means that, for short periods of time, distant parts of a system act independently.
The surprising existence of such a finite limit to the speed of propagation, up to exponentially small error terms, was discovered mathematically in 1972. It turns the locality properties of physical systems into the existence of an upper bound for this speed. This bound is known as the Lieb-Robinson bound and the speed is known as the Lieb-Robinson velocity. The velocity is not universal, because it depends on the details of the system under consideration, but, for each system, there is a finite velocity.
One of the practical applications of Lieb-Robinson bounds is quantum computing. 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.
Review articles can be found in the following references, for example,
Set up
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.
Lieb-Robinson bounds are considered on a
A Hilbert space of states
For every finite subset of the lattice,
An observable
When
The Hamiltonian of the system is described by an interaction
Although translation invariance is usually assumed, it is not necessary to do so. It is enough to assume that the interaction is bounded above and below on its domain. Thus, the bound is quite robust in the sense that it is tolerant of changes of the Hamiltonian. A finite range is essential, however. An interaction is said to be of finite range if there is a finite number
The Hamiltonian of the system with interaction
The laws of quantum mechanics say that corresponding to every physically observable quantity there is a self-adjoint operator
Here,
The bound in question was proved in and is the following: For any observables
where
A positive constant
The bound (1) is presented slightly differently from the equation in the original paper. This more explicit form (1) can be seen from the proof of the bound
Lieb-Robinson bound shows that for times
The reason for considering the commutator on the left-hand side of the Lieb–Robinson bounds is the following:
The commutator between observables
The converse is also true: if observable
This statement is also approximately true in the following sense: suppose that there exists some
Thus, Lieb-Robinson bounds say that the time evolution of an observable
Improvements of the Lieb-Robinson bounds
In Robinson generalized the bound (1) by considering exponentially decaying interactions (that need not be translation invariant), i.e., for which the strength of the interaction decays exponentially with the diameter of the set. This result is discussed in detail in, Chapter 6. No great interest was shown in the Lieb-Robinson bounds until 2004 when Hastings applied them to the Lieb–Schultz–Mattis theorem. Subsequently Nachtergaele and Sims extended the results of to include models on vertices with a metric and to derive exponential decay of correlations. From 2005–2006 interest in Lieb–Robinson bounds strengthened with additional applications to exponential decay of correlations (see 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.
Several further improvements of the constant
where
Harmonic and Anharmonic Hamiltonians
The Lieb-Robinson bounds were extended to certain continuous quantum systems, that is to a general harmonic Hamiltonian, which, in a finite volume
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, Further generalizations of the harmonic lattice were discussed,
Irreversible dynamics
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. The dissipative part is described by terms of Lindblad form, so that the dynamics
Lieb-Robinson bounds for the irreversible dynamics were considered by in the classical context and by 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, where they also proved the existence of the infinite dynamics as a strongly continuous cocycle of unit preserving completely positive maps.
Some applications
Lieb–Robinson bounds are used in many areas of mathematical physics. Among the main applications of the bound there is the existence of the thermodynamic limit, the exponential decay of correlations and the Lieb–Schultz–Mattis theorem.
Thermodynamic limit of the dynamics
One of the important properties of any model meant to describe properties of bulk matter is the existence of the thermodynamic limit. This says that intrinsic properties of the system should be essentially independent of the size of the system which, in any experimental setup, is finite.
The static thermodynamic limit from the equilibrium point of view was settled much before the Lieb–Robinson bound was proved, see for example. In certain cases one can use a Lieb–Robinson bound to establish the existence of a thermodynamic limit of the dynamics,
Robinson was the first to show the existence of the thermodynamic limit for exponentially decaying interactions. Later, Nachtergaele et al. showed the existence of the infinite volume dynamics for almost every type of interaction described in the section "Improvements of Lieb–Robinson bounds" above.
Exponential decay of correlations
Let
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
holds for observables
Alternatively the state
Such a decay was long known for relativistic dynamics, but only guessed for Newtonian dynamics. The Lieb–Robinson bounds succeed in replacing the relativistic symmetry by local estimates on the Hamiltonian.
Lieb-Schultz-Mattis theorem
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.
For one-dimensional and quasi-one-dimensional systems of even length and with half-integral spin Affleck and Lieb, generalizing the original result by Lieb, Schultz, and Mattis, proved that the gap
where
The Lieb–Robinson bound was utilized by Hastings and by Nachtergaele-Sims in a proof of the Lieb–Schultz–Mattis Theorem for higher-dimensional cases. The following bound on the gap was obtained:
Discretisation of the Continuum via Gauss-Quadrature Rules
In 2015, it was shown that the Lieb-Robinson bound can also have applications outside of the context of local Hamiltonians as we now explain. The Spin-Boson model describes the dynamics of a spin coupled to a continuum of oscillators. It has been studied in great detail and explains quantum dissipative effects in a wide range of quantum systems. Let
where
Experiments
The first experimental observation of the Lieb–Robinson velocity was done by Cheneau et al.