I know that, given an arbitrary unitary matrix A, it can be written as the product of several "local" unitary matrices Ai -- local in the sense that they only act on a small constant number of vector components, in fact it is sufficient to take 2 as that constant, which is best possible. For example,(adsbygoogle = window.adsbygoogle || []).push({});

[tex] A=\left(\begin{array}{cccc} a & 0 & b & 0 \\0 & 1 & 0 & 0 \\

c & 0 & d & 0\\ 0 & 0 & 0 & 1\end{array}\right)[/tex]

is such a local matrix, because it only acts on the basis element e1 and e3. Another way to think of it is that it's only nontrivial over a subspace of dimension 2.

So the picture is that there are these gigantic linear transformations on C^n(n is very large), being built out of smaller transforms which act on small constant-dimension subspaces. So as n grows, the number of local Ai's needed will grow too. I have an upper bound on the number of Ai needed in the decomposition, in terms of n. But that's not what I'm interested in. What I would like to know, is what is the least number of Ai's needed to decompose a specific matrix, or a class of matrices. I know of no machinery that makes this easy, and looking around online for a bit didn't help.

So the question is, if I have a specific unitary matrix, how do I go about calculating the minimum number of Ais needed in the decomposition, and if I have a class of matrices, how do I figure it out. Is there some reading on this subject I can find online?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Local decomposition of unitary matrices

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**