Why are sub-normalized states studied in quantum computation?
It is often mathematically very convenient to work with sub-normalized states independently of the question whether they have a direct physical meaning.

However, you can always see them as states that are logically 'AND' with a certain event. For example, consider the c-q state

$\rho_{XB} = \sum_x |x\rangle\!\langle x|_X \otimes \rho_B^x$

where $\rho_B^x$ is sub-normalized so that the whole state is normalized. Then the trace of $\rho_B^x$ just gives the probability of finding the subsystem $B$ in the renormalized, conditional state $\rho_B^x / \text{tr}(\rho_B^x)$. 

For example, the probability of measuring a certain event (POVM element) $E_B$ AND x is simply given by $\text{tr}(\rho_B^x E_B)$ and we never needed to renormalize the state to write this down.