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How can we simplify tensor expressions in Mathematica 9 using the mixed-product identity $(A\otimes B)(C \otimes D) \equiv AC \otimes BD$ ?

Is it possible to implement this kind of evaluations using the new Mathematica 9 tensor capabilities?

The following expression is false:

TensorProduct[a, b].TensorProduct[c, d] === TensorProduct[ a.c, b.d]

I don't really need to prove this identity, but rather to use it for simplifying some expressions. In my case, $a$ and $c$ are some (unknown) symbolic matrices, while $b$ and $d$ are explicit integer $2\times 2$ matrices. I'd like Mathematica to evaluate the matrix product between $b$ and $d$ explicitly.

For instance, when

b=PauliMatrix[1]; d=PauliMatrix[3];

as the product of two Pauli matrices gives another Pauli matrix, I'd like to obtain the simplified result

TensorProduct[a.c, -I PauliMatrix[2]]
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2 Answers

up vote 4 down vote accepted

You could also do

$Assumptions = (a | b | c | d) \[Element] Matrices[{k, k}]

KroneckerProduct[a, b] . KroneckerProduct[c, d] // TensorExpand

which returns

KroneckerProduct[a.c, b.d]
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That is exactly what I was looking for. Thanks –  benkj Feb 3 at 16:13
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  1. The mixed-product identity

    $$ (A\otimes B)(C \otimes D) \equiv AC \otimes BD$$

    is written in terms of the Kronecker product (see here) not TensorProduct. They have different dimensions

    A = RandomReal[1.0, {10, 10}];
    KroneckerProduct[A, A] // Dimensions
    TensorProduct[A, A] // Dimensions
    
    {100, 100}
    {10, 10, 10, 10}
    
  2. lhs === rhs gives exact comparison of lhs and rhs. You need to use ==.

  3. Unfortunately tensors operations with KroneckerProduct and Dot is not fully implemented in Mathematica 9. However, for example, Dot[a,b] is equivalent to TensorContract[a\[TensorProduct]b, {{2, 3}}].

Finally, you can rewrite your identity as

$Assumptions = (a | b | c | d) \[Element] Matrices[{k, k}];

TensorTranspose[
   TensorContract[a\[TensorProduct]b\[TensorProduct]c\[TensorProduct]d, {{2, 5}, {4, 7}}], 
 {1, 3, 2, 4}] == 
 TensorContract[a\[TensorProduct]c, {{2, 3}}]\[TensorProduct]TensorContract[
   b\[TensorProduct]d, {{2, 3}}] // TensorReduce

True

Unfortunately, I can't explain derivation of this formula because I don't understand it completely.

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Thanks, your answer makes me understand a little bit better the new Tensor capabilities of Mathematica 9. However, I'm not able to use your commands for simplifying expression. This was indeed, my main problem. I edited the question to better clarify this point. –  benkj Oct 16 '13 at 7:44
    
@benkj I think it is much simpler to implement it with pattern replacements or do it manually. –  ybeltukov Oct 16 '13 at 9:55
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