We need to find the probabilities of the sum and the difference of two quantum dice. What is the probability of their sum to be 2? it can be accomplished only when both dice are 1. the probability of their sum to be 3? It can be accomplished with two ways: one of them 1 and the other 2 and the other way round. We need to express those probabilities. But, we are talking about quantum dice and not clasic dice. Does anyone has a clue?
Thank you very much!
I'll assume that by quantum dice is meant they are not distinguishable, which is maybe vague. But it could be made precise by saying that, for a roll of two "quantum" dice, the outcomes are multisets of size $2$, each equally likely. Then there are the six "doubles" $\{k,k\}$ with $1 \le k \le 6$, and the $15$ usual sets of size 2 from $\{1,2,3,4,5,6\}$ such as $\{1,2\},/ \{1,3\}$ and so on. This gives the sample space size of $6+15=21$ rather than the usual $36$ for "nonquantum" dice. Now there is only one multiset of size $2$ with sum $3$, namely $\{1,2\}.$ Note how using sets instead of ordered pairs for the outcomes takes care of the dice being indistinguishable. So this way the probability of sum $3$ comes out $1/21$ [number of ways over size of sample space].