Intuitively it is clear that if I have two variables $A, B \sim \mathcal{N}(\mu, \sigma)$ and I have $\rho_{A,B}=-1$, then if some sample $a$ of $A$ is $x$, then some sample $b$ of $B$ is $\mu-x$ because $Var(A+B)=0$.
But I cannot find a general formula for predicting the value of b for different $\rho$ values. How is this "concept" called?
In the sense of "predict" that you explicated in a comment, this is not possible. Unless the correlation is $\pm1$, the value of one variable does not determine the value of the other.