I wouldn’t say the tensor product is how “and” works. “and” usually refers to the Boolean and operator. The tensor product is how combining two systems works. In classical computing, if you have:

int a; int b;

The set of possible states is the Cartesian product of the states of a with the states of b. In quantum computing, it’s the tensor product instead. It’s not terribly complicated to work with the matrices — they’re just huge. Fortunately, it’s not usually necessary to write out the matrices at all.

I read the string diagram paper. It’s supposed to be comprehensible to kids, but I couldn’t make heads or tails of it.

(I did my Ph.D. on quantum computing.)