Finding the Analytic Expression for Exponential of a Tensor Product Matrix

[NaturePaper]

Will anyone help me to find out the analytic expression
of the following $$2^N\times2^N$$ exponential?

$$exp[t(X\otimes X\otimes I\ldots\otimes I+I\otimes X\otimes X\otimes I\ldots\otimes I+\ldots+I\otimes I\otimes\ldots I \otimes X \otimes X+X\otimes I\ldots I\otimes X)]$$,

where

$$I= \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]$$

and
$$X=\left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right]$$.

[Note that the parenthesis in the `exponential' contains sum of N+1 terms each of which is a tensor product of 2 Xs and (N-2) of Is in some order.]

I've evaluated (via Mathematica) for N=3,4,5,6. But I need an analytic expression for it.

Thanks and Regards.

 

[fresh_42]

I'm not quite sure how you arrange the tensor product to get a matrix form again. You can work with a standard basis $E_{ij} = \begin{cases} 1&\text{ at position }(i,j) \\0&\text{ elsewhere }\end{cases}$, and write the combined matrix accordingly. But this looks as if you want to calculate the derivative at $t=0$ for some purpose, in which case it would probably make more sense to work with $I,X$ where I would write $X=P$ as the permutation it is to make it more obvious.

Investigation of small $n$ and look for an induction should help, too.