# Explicit Lieb-Robinson Velocity Bounds

Lieb-Robinson bounds describe how effects are propagated through a system due to a local Hamiltonian. They are often described in the form $$\left|[A,B(t)]\right|\leq Ce^{vt-l},$$ where $$A$$ and $$B$$ are operators that are separated by a distance $$l$$ on a lattice where the Hamiltonian has local (e.g. nearest neighbour) interactions on that lattice, bounded by some strength $$J$$. The proofs of the Lieb Robinson bound typically show the existence of a velocity $$v$$ (that depends on $$J$$). This is often really useful for bounding properties in these systems. For example, there were some really nice results here regarding how long it takes to generate a GHZ state using a nearest-neighbour Hamiltonian.

The problem that I've had is that the proofs are sufficiently generic that it is difficult to get a tight value on what the velocity actually is for any given system.

To be specific, imagine a one dimensional chain of qubits coupled by a Hamiltonian $$H=\sum_{n=1}^N\frac{B_n}{2}Z_n+\sum_{n=1}^{N-1}\frac{J_n}{2}(X_nX_{n+1}+Y_nY_{n+1}), \tag{1}$$ where $$J_n\leq J$$ for all $$n$$. Here $$X_n$$, $$Y_n$$ and $$Z_n$$ represent a Pauli operator being applied to a given qubit $$n$$, and $$\mathbb{I}$$ everywhere else. Can you give a good (i.e. as tight as possible) upper bound for the Lieb-Robinson velocity $$v$$ for the system in Eq. (1)?

This question can be asked under two different assumptions:

• The $$J_n$$ and $$B_n$$ are all fixed in time
• The $$J_n$$ and $$B_n$$ are allowed to vary in time.

The former is a stronger assumption which may make proofs easier, while the latter is usually included in the statement of Lieb-Robinson bounds.

## Motivation

Quantum computation, and more generally quantum information, comes down to making interesting quantum states. Through works such as this, we see that information takes a certain amount of time to propagate from one place to another in a quantum system undergoing evolution due to a Hamiltonian such as in Eq. (1), and that quantum states, such as GHZ states, or states with a topological order, take a certain amount of time to produce. What the result currently shows is a scaling relation, e.g. the time required is $$\Omega(N)$$.

So, let's say I come up with a scheme that does information transfer, or produces a GHZ state etc. in a way that scales linearly in $$N$$. How good is that scheme actually? If I have an explicit velocity, I can see how closely matched the scaling coefficient is in my scheme compared to the lower bound.

If I think that one day what I want to see is a protocol implemented in the lab, then I very much care about optimising these scaling coefficients, not just the broad scaling functionality, because the faster I can implement a protocol, the less chance there is for noise to come along and mess everything up.

## Further Information


I can add a bit further to the motivation. Consider the time evolution of a single excitation starting at one end of the chain, $$\ket{1}$$, and what its amplitude is for arriving at the other end of the chain $$\ket{N}$$, a short time $$\delta t$$ later. To first order in $$\delta t$$, this is $$\bra{N}e^{-ih\delta t}\ket{1}=\frac{\delta t^{N-1}}{(N-1)!}\prod_{n=1}^{N-1}J_n+O(\delta t^{N}).$$ You can see the exponential functionality that you would expect being outside the 'light cone' defined by a Lieb-Robinson system, but more importantly, if you wanted to maximise that amplitude, you'd set all the $$J_n=J$$. So, at short times, the uniformly coupled system leads to the most rapid transfer. Trying to push this further, you can ask, as a bit of a fudge, when can $$\frac{t^{N-1}}{(N-1)!}\prod_{n=1}^{N-1}J_n\sim 1$$ Taking the large $$N$$ limit, and using Stirling's formula on the factorial leads to $$\frac{etJ}{N-1}\sim 1,$$ which suggests a maximum velocity of about $$eJ$$. Close, but hardly rigorous (as the higher order terms are non-negligible)!


• Have you computed the best LR-bound from the proofs for that model? How does it compare to the velocity you quote? – Norbert Schuch Apr 6 '18 at 13:36
• Ok, I concede it is a quantum computing question, at least the way I interpret it now: "What is the choice of $J_n$ and $B_n$ (subject to some constraints) which yields the maximum velocity for information/state/... transfer." --- Is this the right interpretation? – Norbert Schuch Apr 7 '18 at 8:53
• @NorbertSchuch Not quite. I want to be able to say "I've come up with a set of couplings that achieves a protocol with a certain scaling. That protocol is known to be constrained by Lieb-Robinson bounds. How close am I to saturating that constraint?" as a measure of how fast my protocol is. – DaftWullie Apr 9 '18 at 7:11
• @DaftWullie So - is you question: "How close am I to being optimal", or "How close am I to some kind of bound (taking the tightest possible one)"? – Norbert Schuch Apr 9 '18 at 9:00
• @user1271772 That is correct. $B(t)=e^{-iHt}B(0)e^{iHt}$ – DaftWullie May 18 '18 at 10:58