Distinguishing two quantum states practically Suppose we have two states
$$|x\rangle =   1 |0\rangle +   0 |1\rangle$$
and
$$|y\rangle =   \sqrt{1-\epsilon^2} |0> +   \epsilon |1>$$
where 
say $\epsilon = 10^{-20}$
Can we distinguish between $|x\rangle$ and $|y\rangle$ practically? 
How many times will we have to repeat the measurement experiment in order to be sure that $|x\rangle \neq |y\rangle$? 
Because $|x\rangle = |0\rangle$ and $|y\rangle$ is very very very close to $|0\rangle$.
 A: For the state $|x\rangle$, if you repeat the same experiment with this initial state $N$ times, you never get $|1\rangle$ as the outcome. However, if you repeat the same experiment with $|y\rangle$ as the initial state, the probability to get $|1\rangle$ in each copy of the experiment is $\epsilon^2$. The results $|1\rangle$ will appear infrequently, distributed as in a Poisson process.
To prove that the coefficient in front of $|1\rangle$ is nonzero, you need to get the outcome $|1\rangle$ at least once. It will approximately occur after $1/\epsilon^2$ repetitions of the experiment, in your case $10^{40}$ repetitions (clearly, your numbers mean that $|x\rangle$ and $|y\rangle$ are indistinguishable in practice because $10^{40}$ is very large). The probability that you will never get $|1\rangle$ after many more experiments than $10^{40}$ is dropping to zero exponentially.
If you want to measure $\epsilon$ with a relative error $\delta$, you need roughly $1/\delta^2\epsilon^2$ copies of the experiment because the relative errors in a measured quantity goes down like $1/\sqrt{N}$.
One should use all these situations for emphasizing the main difference between classical and quantum physics: in classical physics, small changes of the state imply small changes in the things we can measure. In quantum physics, small changes of the state vector may bring large changes in the outcome of a single experiment, but the probability of such a large change is very small.
