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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!

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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}

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  • $\begingroup$ Thanks for your answer! Unfortunately, I seem to get a $RecursionLimit error when running this code $\endgroup$ – Pentaquark Aug 6 at 13:04
  • $\begingroup$ I'm sorry if I'm being dense here, but doesn't the fact that n is set to 4 define the length of the output string? Ideally, list_List should be an arbitary number of arguments, and the function returns the amount of arguments put into that list. $\endgroup$ – Pentaquark Aug 6 at 13:15
  • $\begingroup$ The size of the output should actually only depend on ρ. So, if you write a function of ρ, you would set n = Log2[Length[ρ]] in the body of that function. $\endgroup$ – Henrik Schumacher Aug 6 at 13:22
  • $\begingroup$ Ah ok, gotcha. Thank you for your help! $\endgroup$ – Pentaquark Aug 6 at 13:25
  • $\begingroup$ You're welcome! $\endgroup$ – Henrik Schumacher Aug 6 at 15:21

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