Baker-Campbell-Hausdorff series

While each term in the resulting series has the form required by equation 12,29, the caveat is that as each commutator connects sites x and x that are a distance d apart, the terms of order n in Ai and Aj yield contributions to H x) that obey the third condition in equation 12.29 only for d = nd. In other words, we also need to require that the Baker-Campbell-Hausdorff series has a sufficient degree of convergence, t... 

This is the start of the celebrated Baker-Campbell-Hausdorff series [21] (often referred to as the BCH Lemma for short) which gives an explicit formula for the product of exponentials of non-commuting operators. 

Now we introduce curvy steps. An intuitive interpretation of what is done here is to expand the Taylor series of the exponential transformation to higher orders, such that the step directions are no longer straight lines, but instead they are curved. Invoking the Baker-(Campbell-)Hausdorff lemma (see, e.g.. Ref. 123), the unitary transformation of the density matrix can be written as... 

---