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Say I have a tensor product of four two dimensional spaces $\Lambda = S\otimes S\otimes S\otimes S$. The basis of $S$ is CB={{1,0},{0,1}} . I generate the representation of $\Lambda$ by

Lambda=ArrayFlatten[
 ArrayFlatten[
  ArrayFlatten[
   Table[TensorProduct[CB[[i]], CB[[j]], CB[[k]], CB[[m]]], 
   {i, 1, 2}, {j, 1, 2}, {k, 1, 2}, {m, 1, 2}]
  ]
 ]
]

The result is

$\Lambda=\left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)$

Now, I have a problem to generate the corresponding basis vectors, which I thought are given by

Flatten[ArrayFlatten[TensorProduct[CB[[i]], CB[[j]], CB[[k]], CB[[m]]]]]

for $i,j,k,m\in\{1,2\}$. But, e.g.

Lambda.Flatten[ArrayFlatten[TensorProduct[CB[[2]], CB[[2]], CB[[1]], CB[[1]]]]]

results in the null vector. How to generate the correct basis vectors using ArrayFlatten[] or Flatten[]?

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The basis vectors are

CB = {{1, 0}, {0, 1}};
Flatten@*KroneckerProduct @@@ Tuples[CB, 4]

(*    {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}}    *)

I'm not sure what your matrix $\Lambda$ is: you said it was a vector space, but matrices are rather elements of this vector space, not the space itself.

I've written a book covering this topic: see chapter 2 of Using Mathematica for Quantum Mechanics: A Student's Manual.

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  • $\begingroup$ My problem is: how do I write e.g. the vector a=TensorProduct[CB[[1]],CB[[2]],CB[[1]],C[[1]]] in the same basis as Lambda, i.e., which of the basis vectors you wrote is a? Thank you for the link, I’ll have a look! $\endgroup$ – pawel_winzig Nov 4 at 22:02
  • $\begingroup$ a = Flatten@KroneckerProduct[CB[[1]], CB[[2]], CB[[1]], CB[[1]]] gives the basis vector {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}. $\endgroup$ – Roman Nov 5 at 21:29
  • $\begingroup$ Shoudn't $\Lambda$ be in this case not a diagonal matrix? $\endgroup$ – pawel_winzig Nov 6 at 7:45
  • $\begingroup$ Again, Λ is a vector space, not a matrix. $\endgroup$ – Roman Nov 6 at 13:55
  • $\begingroup$ It seems my description of the question is misleading: See $\Lambda$ as an operator acting on four spins. The matrix is the representation of this operator. $\endgroup$ – pawel_winzig Nov 6 at 16:24

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