Can Lie Algebraic Decomposition Simplify Matrix Exponentials in \mathfrak{U}(N)?

[Kreizhn]

Hello all,

I have a small problem that I would appreciate some assistance with. I'm working with the Lie group [itex] \mathfrak U(N) [/itex] of unitary matrices and I have an element [itex] P \in \mathfrak U(N) [/itex]. Furthermore, I have a set
[tex] \left\{ H_1, \ldots, H_d \right\} [/tex]
that generate the Lie algebra [itex] \mathfrak u(N) [/itex] of skew-Hermitian matrices under a Lie bracket given by a commutator [itex] [A,B] = AB-BA [/itex].

My overall goal is to express P as an exponential product
[tex] P = \exp\left( \alpha_1 H_1 \right) \cdots \exp\left( \alpha_d H_d \right) [/tex]
for [itex] \alpha_i \in \mathbb R [/itex]. I know in general that this decomposition is probably not a simple one. On the other hand, I do know that I can express P as
[tex] P = \exp\left( - \mu G \right) [/tex]
for some [itex] G \in \mathfrak u(N) [/itex]. It seems to me then that maybe I can reduce this to simply finding a decomposition of G in terms of the generating set of the Lie algebra. Even if the decomposition is in terms of the Lie bracket, I should be able to exploit properties of the matrix exponential to get what I desire.

In any case, I am not certain if there is an easier way of doing this. Or if we can indeed just focus on decomposing G in terms of [itex] H_i [/itex], how would I do this?

P.S. If it helps, I can instead choose to work in the special unitary group [itex] \mathfrak{SU}(N) [/itex] and hence [itex] \mathfrak{su}(N) [/itex] the Lie algebra of traceless skew-Hermitian matrices. Also, I can write
[tex] P = \exp\left[ \alpha_1 \sum_{i \in I_1} H_i \right] \cdots \exp\left[\alpha_k \sum_{i \in I_k} H_i \right] [/tex]
For [itex] I_j \subseteq \left\{1, \ldots, d \right\} [/itex]. That is, the argument of the exponentials need not be only a single generating element, but it can also be a sum of generators.

 

[Ben Niehoff]

Unfortunately, what you are trying to do is not possible, unless your Lie group happens to be Abelian. What you need is this:

http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula

 

[Kreizhn]

Hi Ben,

Thanks for your reply. Luckily, this is for application in a numerical algorithm, and hence so long as the infinite product gradually has less and less contribution then truncation on the order of machine-epsilon will not be noticeable.

By chance, do you know how to "decompose" the element G mentioned above in terms of the set [itex] S=\left\{H_1, \ldots, H_d \right\} [/itex]. It seems like this should certainly be possible, but I'm uncertain as how to do it.

While analytically it may not be a well posed problem, it is actually possible to create numerical algorithms to find the values of [itex] \alpha_i [/itex] for fixed index sets [itex] I_j [/itex] so long as the Lie algebra generated by [itex] \left\{ H_1, \ldots, H_d \right\} [/itex] is the entire algebra [itex] \mathfrak{su}(N) [/itex]. While not presented in the exact same context, you may be interested in the result of the following paper:

C.Altafini. Controllability of quantum mechanical systems by root space decomposition of [itex] \mathfrak{su}(N) [/itex]. Journal of Mathematical Physics, 43(5): 2051-2062, 2002

Also, the Khaneja-Glaser decomposition offers an interesting way of writing special unitary matrices in terms of an exponential product using recursive Cartan decompositions of the space. Though I don't think this is really applicable to the current problem.

 

[Kreizhn]

With regard to decomposing G: The only thing I can think of is computationally generating a basis for [itex] \mathfrak{su}(N) [/itex] by computing commutators of S and then adding linearly independent elements until I have a spanning set. At that point I just vectorize all the components and solve it like any other vector space problem. This works, but is not very efficient or elegant. If anyone has a better idea, it would be much appreciated.

 

[Kreizhn]

In case anyone ever stumbles upon this problem in the future, I have solved it. One can use a similarity transformation on the original set of generators
[tex] S=\left\{ H_1, \ldots, H_d \right\} [/tex]
to create a basis for [itex] \mathfrak{su}(N) [/itex]. That is, if S does not generate the entire Lie Algebra, [itex] \exists j,k \in \left\{1, \ldots, d \right\} [/itex] such that [itex] [H_j, H_k] [/itex] is linearly independent of S. Thus [itex] H_{d+1} = e^{-iH_j \tilde t} H_k e^{iH_j \tilde t} [/itex] is linearly independent of S for arbitrarily small [itex] \tilde t[/itex].

For the sake of contradiction, assume this is not true. Then we can write
[tex] H_{d+1} = \sum_{r=1}^d a_r(\tilde t) H_r [/tex]
But differentiating with respect to [itex] \tilde t[/itex] and evaluating at zero will then give
[tex] [H_j, H_k] = \sum_{r=1}^d \frac{da_r}{dt}(0) H_r [/tex]
which is a contradiction to the assumption that S doesn't span [itex] \mathfrak{su}(N) [/itex]. We can do this until we have generated a basis for the Lie Algebra.

Now we can use successive Cartan decompositions of the [itex] \mathfrak{su}(N) [/itex] (where I forgot to mention that [itex] N=2^n [/itex] for some n ) into [itex] \mathfrak{su}(2) [/itex] as a vector space such that the map
[tex] \phi(X) = \exp(X_1) \cdots \exp(X_2) [/tex]
for [itex] X_q \in \mathfrak{su}(2), \forall q [/itex] is a diffeomorphism in a local neighbourhood V of the group identity. We can project any operator [itex] G \in \mathfrak{SU}(N) [/itex] into this neighbourhood by considering sufficiently large exponentials of a scaled principle logarithm. That is, we can write [itex] G^{1/m} = \exp(A/m) \in V[/itex]. Then the map [itex] \phi [/itex] gives us a parameterization in terms of the basis from S. Furthermore, note that elements generated by S are expressible in S since they are derived via similarity transform. That is,
[tex] s> d \quad \Rightarrow \quad \exists 0<j,k<s, H_s = e^{-iH_jt} H_k e^{iH_jt} \quad \Rightarrow \quad e^{-iH_s \tau} = e^{-iH_jt} e^{-iH_k \tau} e^{-i H_j t} [/tex]
which can be recursively done until each element is in S.

Thus every element can be written in terms of the matrix exponential of the generating set.