Lieb-Robinson bounds

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 ${\displaystyle \nu }$-dimensional lattice (${\displaystyle \nu =1,2}$ or ${\displaystyle 3}$) ${\displaystyle \Gamma }$, such as the square lattice ${\displaystyle \Gamma =\mathbb {Z} ^{\nu }}$.

A Hilbert space of states ${\displaystyle {\mathcal {H}}_{x}}$ is associated with each point ${\displaystyle x\in \Gamma }$. The dimension of this space is finite, but this was generalized in 2008 to include infinite dimensions (see below). This is called quantum spin system.

For every finite subset of the lattice, ${\displaystyle X\subset \Gamma }$, the associated Hilbert space is given by the tensor product

${\displaystyle {\mathcal {H}}_{X}=\bigotimes _{x\in X}{\mathcal {H}}_{x}}$.

An observable ${\displaystyle A}$ supported on (i.e., depends only on) a finite set ${\displaystyle X\subset \Gamma }$ is a linear operator on the Hilbert space ${\displaystyle {\mathcal {H}}_{X}}$.

When ${\displaystyle {\mathcal {H}}_{x}}$ is finite dimensional, choose a finite basis of operators that span the set of linear operators on ${\displaystyle {\mathcal {H}}_{x}}$. Then any observable on ${\displaystyle {\mathcal {H}}_{x}}$ can be written as a sum of basis operators on ${\displaystyle {\mathcal {H}}_{x}}$.

The Hamiltonian of the system is described by an interaction ${\displaystyle \Phi (\cdot )}$. The interaction is a function from the finite sets ${\displaystyle X\subset \Gamma }$ to self-adjoint observables ${\displaystyle \Phi (X)}$ supported in ${\displaystyle X}$. The interaction is assumed to be finite range (meaning that ${\displaystyle \Phi (X)=0}$ if the size of ${\displaystyle X}$ exceeds a certain prescribed size) and translation invariant. These requirements were lifted later.[2][9]

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 ${\displaystyle R}$ such that for any set ${\displaystyle X}$ with diameter greater than ${\displaystyle R}$ the interaction is zero, i.e., ${\displaystyle \Phi (X)=0}$. Again, this requirement was lifted later.[2][9]

The Hamiltonian of the system with interaction ${\displaystyle \Phi }$ is defined formally by:

${\displaystyle H_{\Phi }=\sum _{X\subset \Gamma }\Phi (X)}$.

The laws of quantum mechanics say that corresponding to every physically observable quantity there is a self-adjoint operator ${\displaystyle A}$. For every observable ${\displaystyle A}$ with a finite support Hamiltonian defines a continuous one-parameter group ${\displaystyle \tau _{t}}$ of transformations of the observables ${\displaystyle \tau _{t}}$ given by

${\displaystyle A(t)=e^{itH_{\Phi }}Ae^{-itH_{\Phi }}.}$

Here, ${\displaystyle t}$ has a physical meaning of time. (Technically speaking, this time evolution is defined by a power-series expansion that is known to be a norm-convergent series ${\displaystyle A(t)=A+it[H,A]+{\frac {(it)^{2}}{2!}}[H,[H,A]]+\cdots }$, see,[10] Theorem 7.6.2, which is an adaptation from.[11] More rigorous details can be found in.[1])

The bound in question was proved in[1] and is the following: For any observables ${\displaystyle A}$ and ${\displaystyle B}$ with finite supports ${\displaystyle X\subset \Gamma }$ and ${\displaystyle Y\subset \Gamma }$, respectively, and for any time ${\displaystyle t\in \mathbb {R} }$ the following holds for some positive constants ${\displaystyle a,c}$ and ${\displaystyle v}$:

${\displaystyle \|\,[A(t),B]\,\|\leq c\,\exp\{-a(d(X,Y)-v|t|)\},}$

(1)

where ${\displaystyle d(X,Y)}$ denotes the distance between the sets ${\displaystyle X}$ and ${\displaystyle Y}$. The operator ${\displaystyle [A,B]=AB-BA}$ is called the commutator of the operators ${\displaystyle A}$ and ${\displaystyle B}$, while the symbol ${\displaystyle \|O\|}$ denotes the norm, or size, of an operator ${\displaystyle O}$. It is very important to note that the bound has nothing to do with the state of the quantum system, but depends only on the Hamiltoninan governing the dynamics. Once this operator bound is established it necessarily carries over to any state of the system.

A positive constant ${\displaystyle c}$ depends on the norms of the observables ${\displaystyle A}$ and ${\displaystyle B}$, the sizes of the supports ${\displaystyle X}$ and ${\displaystyle Y}$, the interaction, the lattice structure and the dimension of the Hilbert space ${\displaystyle {\mathcal {H}}_{x}}$. A positive constant ${\displaystyle v}$ depends on the interaction and the lattice structure only. The number ${\displaystyle a>0}$ can be chosen at will provided ${\displaystyle d(X,Y)/v|t|}$ is chosen sufficiently large. In other words, the further out one goes on the light cone, ${\displaystyle d(X,Y)-v|t|}$, the sharper the exponential decay rate is. (In later works authors tended to regard ${\displaystyle a}$ as a fixed constant.) The constant ${\displaystyle v}$ is called the group velocity or Lieb-Robinson velocity.

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 ${\displaystyle v_{LR}}$.[1] This more explicit form (1) can be seen from the proof of the bound[1]

Lieb-Robinson bound shows that for times ${\displaystyle |t|<d(X,Y)/v}$ the norm on the right-hand side is exponentially small. This is the exponentially small error mentioned above.

The reason for considering the commutator on the left-hand side of the Lieb–Robinson bounds is the following:

The commutator between observables ${\displaystyle A}$ and ${\displaystyle B}$ is zero if their supports are disjoint.

The converse is also true: if observable ${\displaystyle A}$ is such that its commutator with any observable ${\displaystyle B}$ supported outside some set ${\displaystyle X}$ is zero, then ${\displaystyle A}$ has a support inside set ${\displaystyle X}$.

This statement is also approximately true in the following sense:[12] suppose that there exists some ${\displaystyle \epsilon >0}$ such that ${\displaystyle \|[A,B]\|\leq \epsilon \|B\|}$ for some observable ${\displaystyle A}$ and any observable ${\displaystyle B}$ that is supported outside the set ${\displaystyle X}$. Then there exists an observable ${\displaystyle A(\epsilon )}$ with support inside set ${\displaystyle X}$ that approximates an observable ${\displaystyle A}$, i.e. ${\displaystyle \|A-A(\epsilon )\|\leq \epsilon }$.

Thus, Lieb-Robinson bounds say that the time evolution of an observable ${\displaystyle A}$ with support in a set ${\displaystyle X}$ is supported (up to exponentially small errors) in a ${\displaystyle \delta }$-neighborhood of set ${\displaystyle X}$, where ${\displaystyle \delta <v|t|}$ with ${\displaystyle v}$ being the Lieb-Robinson velocity. Outside this set there is no influence of ${\displaystyle A}$. In other words, this bounds assert that the speed of propagation of perturbations in quantum spin systems is bounded.

In[13] 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,[14] Chapter 6. No great interest was shown in the Lieb-Robinson bounds until 2004 when Hastings[15] applied them to the Lieb–Schultz–Mattis theorem. Subsequently, Nachtergaele and Sims[16] extended the results of[13] 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[2][9][17] 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 ${\displaystyle c}$ in (1) were made.[18] In 2008 the Lieb-Robinson bound was extended to the case in which each ${\displaystyle H_{x}}$ is infinite dimensional.[19] In[19] it was shown that on-site unbounded perturbations do not change the Lieb-Robinson bound. That is, Hamiltonians of the following form can be considered on a finite subset ${\displaystyle \Lambda \subset \Gamma }$:

${\displaystyle H_{\Lambda }=\sum _{x\in \Lambda }H_{x}+\sum _{X\subset \Lambda }\Phi (X),}$

where ${\displaystyle H_{x}}$ is a self-adjoint operator over ${\displaystyle {\mathcal {H}}_{x}}$, which needs not to be bounded.

Lieb–Robinson bounds are used in many areas of mathematical physics. Among the main applications of the bound there is the error bounds on quantum simulation algorithms, the existence of the thermodynamic limit, the exponential decay of correlations and the Lieb–Schultz–Mattis theorem.

The first experimental observation of the Lieb–Robinson velocity was done by Cheneau et al.[35]