Quantum information (sum is 4 or 6)

[schulzy]

Homework Statement

We have two people, Alice and Bob. They get numbers random, but we have two cases.
Firs case: $$\sum6$$
If:
Bob get 6 then Alice get 0
Bob get 5 then Alice get 1
Bob get 4 then Alice get 2
Bob get 3 then Alice get 3
Bob get 2 then Alice get 4

Second case $$\sum4$$
If
Bob get 4 then Alice get 0
Bob get 3 then Alice get 1
Bob get 2 then Alice get 2
Bob get 1 then Alice get 3
Bob get 0 then Alice get 4

They don't know, what number do have by other and they can communicate nobody. They can use whatever gate (e.g. Hadamard, CNOT, fCNOT). Alice and Bob can just 1-1 bit sending out. A judge get these bits and he can decide, how it is a first case $$\left(\sum6\right)$$ or a second case$$\left(\sum4\right)$$ . Also we need just know, the sum was 4 or 6.
Both can use any Bell state.
I append a drawing, how it is looking out.

Homework Equations

Bell states
$$\left|\beta_{00}\right\rangle= \frac{1}{\sqrt{2}}\left(\left|0\right\rangle\ \left|0\right\rangle\ +\left|1\right\rangle\ \left|1\right\rangle\ \right)$$
$$\left|\beta_{01}\right\rangle= \frac{1}{\sqrt{2}}\left(\left|0\right\rangle\ \left|1\right\rangle\ +\left|1\right\rangle\ \left|0\right\rangle\ \right)$$
$$\left|\beta_{10}\right\rangle= \frac{1}{\sqrt{2}}\left(\left|0\right\rangle\ \left|0\right\rangle\ -\left|1\right\rangle\ \left|1\right\rangle\ \right)$$
$$\left|\beta_{11}\right\rangle= \frac{1}{\sqrt{2}}\left(\left|0\right\rangle\ \left|1\right\rangle\ -\left|1\right\rangle\ \left|0\right\rangle\ \right)$$

The Attempt at a Solution

I would like to know, how can I make this problem or what name can I find this solution.

 

[BigJDubs]

I don't need a solution to this problem. I would like to know, what is the name of this type of problems or what do I search on google.