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This expansion is derived by K. Kumar in On Expanding the Exponential, see equation (9) (with $t=1$) and section 6.

It may be helpful to consider an example that can be solved exactly; Using the Special-case closed form of the Baker-Campbell-Hausdorff formula one finds that if the commutator $[X,Y]$ evaluates to $$ …

The exponential $e^{Q}$ of any $n\times n$ upper triangular matrix $Q$ can be computed efficiently by solving a set of $n$ first-order differential equations, $u_{i}'(t)=\sum_{j}Q_{ij}u_j(t)$; these $ …

Consider a $2n\times 2n$ real matrix $M$ that satisfies the pseudo-unitarity condition $$M^{\rm T}\Lambda M=\Lambda,\;\;\Lambda=\begin{pmatrix}1_n & 0\\ 0 & -1_n\end{pmatrix}.\qquad [1]$$ (The matri …

In terms of Pauli matrices: $$U=u_1I+iu_2\sigma_3+iu_3\sigma_2+iu_4\sigma_1,\;\;u_1^2+u_2^2+u_3^2+u_4^2=1,$$ $$V=\alpha (n_1\sigma_1+n_2\sigma_2+n_3\sigma_3),\;\;n_1^2+n_2^2+n_3^2=1,$$ $$\exp(iV)=I\c …

A closed-form expression is not forthcoming, but here are the plots of $\mathbb{E}[\text{trace}\, T]$ (left plot) and $\mathbb{E}[\text{trace}\, TS]$ (right plot) as a function of $\gamma$ in the inte …

The difficulties with generalizations of the Golden-Thompson inequality to three matrices arise because the trace of a product of three positive symmetric matrices is in general not positive; unlike t …