# Why are sub-normalized states studied in quantum computation?

By basic postulates of QM, any state of a system is described by a normalised density operator. Now i fail to see why people study sub-normalized states ( e.g.: In generalised fidelity etc). I'd be glad to know any physical interpretations of the same.

• I am sorry! It seems i don't have permission to add this topic to quantum computing. I request those who have the accesses please add it. – Vamsi Krishna Oct 26 '14 at 16:09
• Subnormalized density operators are not only very useful in studying quantum computation, but in studying quantum physics as well. – Peter Shor Oct 26 '14 at 16:33

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