In a lecture, we were given the following example to explain the operational significance of the trace distance. Suppose that Alice prepares one of two (known) states $\rho_0$ or $\rho_1$ with equal probability, and then passes the state to Bob. Bob wishes to distinguish which state he is given, and a simple calculation shows that his maximal probability of success is given by $$p^* = \frac{1}{2} (1 + D(\rho_0,\rho_1))$$ where $D(\rho,\sigma) = \frac{1}{2}\Vert \rho-\sigma\Vert_1$ denotes the trace distance. This seems concerning since if $D(\rho_0, \rho_1) = 0$, then the success probability is $1/2$. But how can this be correct? Since the trace distance is in fact a metric, if $D(\rho_0,\rho_1) = 0$ then $\rho_0 = \rho_1$. Hence, his probability of success is 1, since he cannot guess incorrectly as there is only one possible answer.
2 Answers
Imagine they are almost the same, i.e. $\rho_0 \approx \rho_1$. It's almost impossible to distinguish them, yet they are different. So that, your success probability is $\approx 1/2$.
Also, in the case of $\rho_0 = \rho_1$ you can think that Alice prepares two identical physical systems labelled $0,1$ and chooses one to pass. Bob has to guess the label.
In this context the idea is that Alice wants to send classical information, by encoding it in quantum states. In other words, Alice encodes her choice of bit to send $i$ into the state $\rho_i$, and Bob's job is to correctly guess the $i$, by performing some measurement $\mu$, and using the measurement outcome to make a guess as to what number Alice state Alice sent. The figure you mention is the optimal average probability of Bob guessing correctly, maximising over all possible choices of measurements $\mu$ and guessing strategies.
In the case where $\rho_0=\rho_1$, there is clearly no way for Bob to infer anything at all about what Alice actually sent, because the information about Alice's choice was completely "erased" by her choice of states to send. It follows that Bob might as well just be randomly guessing Alice's state, and will only make the correct guess half of the time. In other words, the gist is that Bob doesn't want to guess the state itself, but rather the classical information that was "hidden" in the states he receives.
