# An upper bound on sample complexity for state identification given ensemble distinction problem

I am trying to derive Fact 5. in paper 1:

Let $$\mathscr{E}=\{\sigma_1,.., \sigma_m\}$$ be an ensemble of quantum states in $$\mathbb{C}^n$$. If there is a POVM $$\mathscr{M}$$ for the state distinction problem with distinguishing power $$\delta$$* for $$\mathscr{E}$$, then there is a single register state identification procedure $$\mathscr{A}$$ for $$\mathscr{E}$$ that needs $$t=\mathscr{O}\big(\frac{\log m}{\delta^2}\big)$$ copies of the unknown $$\sigma\in\mathscr{E}$$.

*A POVM $$\mathscr{M}$$ has distinguishing power $$\delta$$ for the ensemble $$\mathscr{E}$$ if $$||\mathscr{M}(\sigma_i)-\mathscr{M}(\sigma_j)||_1 \leq \delta$$ for all $$1 \leq i < j \leq m$$.

There is a proof sketch in the paper but it seems fairly dense to me...

Basically, first we apply the measurement $$\mathscr{M}$$ on each of the $$t$$ copies of the unknown state $$\sigma$$ which could be any of the states in the given ensemble $$\mathscr{E}$$ to obtain the measurement statistics/data. Then, assuming that the unknown state is either of a pair of states $$\sigma_i,~ \sigma_j$$, where $$1 \leq i < j \leq m$$, we perform a maximum likelihood estimation procedure on the measurement data. And repeat this classical post-processing (on the data obtained by measuring the unknown state) $$m-1$$ times.

At each iterative step of maximum likelihood estimation over a pair of states, the probability of correctly identifying the state is said to be at least $$1-\frac{1}{4m}$$. The author says that one can obtain this probability via a standard Chernoff bound but I am unable to see how except that since $$t=\mathscr{O}\big(\frac{\log m}{\delta^2}\big)$$, it implies $$|t|\leq M\frac{\log m}{\delta^2}$$, for $$M>0$$. This would give

$$-t \geq -M\frac{\log m}{\delta^2} \implies 1-\frac{t\delta^2}{4} \geq 1-M \frac{\log m}{4}$$.

As $$\log m > \frac{1}{m}$$ for all $$m\geq 2$$ and $$M>0$$, we have

$$1-\frac{t\delta^2}{4} \geq 1-\frac{1}{4m}$$.

The left hand side above looks like an approximation for $$e^{-\frac{t\delta^2}{4}}$$ which might be seen as some kind of Chernoff bound...

The question is what is the event space when talking of a Chernoff bound.