What does this kind of notation mean? Trying to understand quantum information. Need some help :( 
What does this notation $$ \langle\alpha|\hat{n}|\alpha\rangle $$ mean? Here $$|\alpha\rangle$$ is a coherent state and $$\hat{n}$$ is photon-number operator. I know that $$ \langle\alpha|\hat{n}|\alpha\rangle = {|\alpha|}^2 ,$$ but how do I calculate that?
 A: The previous answer gives the meaning, here is the calculation.
The coherent state $\mid \alpha\rangle$ is an eigenvectior of the annihilation operator 
$\mathsf{a}$, with 
$$\mathsf{a}\mid \alpha\rangle= \alpha \mid \alpha\rangle .$$
The Hermitian conjugate of this equation is:
$$\big(\mathsf{a}\mid \alpha\rangle\big)^+
= \langle\alpha\mid\mathsf{a}^+ = \alpha^* \langle\alpha\mid,$$
where $^*$ denote complex conjugate and $^+$ the adjoint.
Hence
$$\langle\alpha\mid \mathsf{n}\mid \alpha\rangle
= (\langle\alpha\mid \mathsf{a}^+)(\mathsf{a}\mid \alpha\rangle)
= \alpha^*  \alpha \langle\alpha\mid \alpha\rangle
= | \alpha|^2$$
If you would to follow the path explained in the previous answer, you'd could use the expansion:
$$
\mid \alpha\rangle=e^{-|\alpha|^2/2}
\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} \mid n \rangle
$$
And the equalities
$$
\mathsf{n}\mid n \rangle = n \mid n \rangle , \quad
\langle n'\mid\mathsf{n}\mid n \rangle = \delta_{n,n'} n
$$
 of the number states (or Fock states).
One has:
\begin{align*}
\langle\alpha\!\mid\!\mathsf{n}\!\mid\! \alpha\rangle 
&=e^{-|\alpha|^2}
\sum_{n=0}^\infty \!
\sum_{n'=0}^\infty\frac{\alpha^{*n'}\alpha^n}{\sqrt{n'!n!}} 
\langle n'\!\mid\!\mathsf{n}\!\mid\! n \rangle \\
&=     e^{-|\alpha|^2}
\sum_{n=0}^\infty n \frac{|\alpha|^{2n}}{n!} = |\alpha|^2
\end{align*}
where the final equality is the standard result on the average of the Poissonian distribution of parameter $|\alpha|^2$. 
(Or in elementary terms,  taking advantage of the vanishing of the term for $n=0$, one can factorize 
$|\alpha|^2$, and simplify $n/n!=1/(n-1)!$. Then the sum is  $e^{|\alpha|^2}$ which exactly cancels the prefactor.)
A: If you had many systems in the state $|\alpha\rangle$ and you were to make measurements of the photon number for each of those systems, then $\langle\alpha|\hat n|\alpha\rangle$ is the expectation value of those measurements.
This can be calculated using inner products if you can directly calculate $\hat n|\alpha\rangle$ and $\langle\alpha|\left(\hat n|\alpha\rangle\right)$. Or you can work in some other complete basis $|m\rangle$ so that
$$\langle\alpha|\hat n|\alpha\rangle=\sum_m\sum_{m'}\langle\alpha|m'\rangle\langle m'|\hat n|m\rangle\langle m|\alpha\rangle$$
assuming you know components $\langle m|\alpha\rangle$ and matrix elements $\langle m'|\hat n|m\rangle$
