Tensor product distributive property?

Question

Is it true that for vectors $a$, $b$, $c$, $d$ we have
$$|a\rangle \otimes |b\rangle \langle c| \otimes \langle d|= |a \rangle \langle c| \otimes |b \rangle \langle d|?$$ So does this kind of distributive law hold for tensor products?

Answer 1

There are situations where a rule like this applies: tensor product of two functions applied to the tensor product of two vectors; this yields the tensor product of the first function to the first vector with the result of the second function applied to the second vector.

Here it appears that you have the tensor product of two kets, $|a>|b>=|ab>$, multiplied by the tensor product of the two bras, $<c|<d|=<cd|$. Their product as an operator, the outer product |a$b><cd|$.

So is $|ab><cd| =|a><c|⊗|b><d|$? We can test this by applying it to the product ket $|xy>=|x>|y>=|x>⊗|y>$. When you do this you will find that they give the same result: $<c|x><d|y>|ab>$.

So the answer is yes, they are equivalent.

But no, it is not a distributive law; instead it is simply the tensor space at work: it keeps the functions & vectors nicely separated while simultaneously keeping them in the correct order. This is why, if you preserve the order, as I have done in the alternative notations, you can drop the tensor product operator symbol, $⊗$, without fear of making any mistakes ... just as we drop the ordinary multiplication symbol in algebra. It's all a matter of getting used to the concepts along with the notation.