Computing Higher Order Tensor of Variable Rank

Question

The following code takes a vector x of variable length, computes the outer product of the vector with itself to form the matrix $\rho$ of dimension $2^n \times 2^n$. The function T[i_, list_List] then computes elements of a tensor $\mathcal{T}$ of rank $n$ according to

$$T_{\mu_1,...,\mu_n}=\text{Tr}(\rho \;\; \sigma_{\mu_1}\otimes...\otimes\sigma_{\mu_n})$$

with $\mu_1,...,\mu_n=1,2,3$ and $\sigma_i$ being the three Pauli Matrices.

OuterVectorProduct[x_] := KroneckerProduct[x, x] 

T[i_, list_List] := 
 FullSimplify[Tr[i.KroneckerProduct @@ PauliMatrix[list]]] 

That is: T[rho,{1,1}]outputs the $T_{11}$ element of the Tensor $\mathcal T$ with respect to some matrix $\rho$.

I would now like to write a function that outputs the entire tensor. To do so, I need to extract the number of arguments within the list in the function T[i_,list_List].

That is, in our example T[rho,{1,1}], I need to extract the number of arguments in the curly braces.

How does one do that?

Thanks!

Answer 1

Probably not the most efficient way to do so, but for descently sized n, this should work well:

T[i_, list_List] := Tr[i.KroneckerProduct @@ PauliMatrix[list]]
n = 4;
ρ = RandomReal[{-1, 1}, 2^n];
A = ArrayReshape[
   T[ρ, #] & /@ Tuples[Range[3], n], 
   ConstantArray[3, n]
   ];
A // Dimensions

{3, 3, 3, 3}