$C$, $P$ and $CP$ in particle physics We know that the charge conjugation $C$ and parity $P$, both are violated maximally in particle physics. Then what is the need for looking for $CP$ symmetry in order to distinguish matter from antimatter? Why is $CP$ symmetry important?
 A: Imagine a process where particle $A$ decays to particle $B$. This decay is described by its decay amplitude $M_{A\to B}$ (basically some coefficient in a Lagrangian). For the antiparticles, $\overline A$ and $\overline B$, this decay amplitude is $M_{\overline A\to \overline B}$. 
If CP is not violated, these two decay amplitudes are the same complex number: $$M_{A\to B} = M_{\overline A\to \overline B} =: |M|\text{e}^{\text i\varphi} $$
Now we introduce the CKM-matrix and its phase factor $\text e^{\text i \Omega}$. The two amplitudes pick up this factor and its conjugate:
$$ M_{A\to B} \to M_{A\to B}\,\text e^{\text i \Omega}\quad \text{and}\quad  M_{\overline A\to \overline B}\to M_{\overline A\to \overline B}\, \text e^{-\text i \Omega}$$
Any physical observable like a cross section is proportional to $|M|^2$, so until now, these two would still behave the same. But what if there's an intermediate state, $X$: $A\to X\to B$ and $\overline A\to \overline X\to \overline B$. The decay amplitudes are now given by
\begin{align*}
M_{A\to B} &= M_{A\to X} + M_{X\to B} = |M_1|\text{e}^{\text i\varphi_1}\text e^{\text i \Omega_1} + |M_2|\text{e}^{\text i\varphi_2}\text e^{\text i \Omega_2},\\
M_{\overline A\to \overline B} &= M_{\overline A\to \overline X} + M_{\overline X\to \overline B} = |M_1|\text{e}^{\text i\varphi_1}\text e^{-\text i \Omega_1} + |M_2|\text{e}^{\text i\varphi_2}\text e^{-\text i \Omega_2}.
\end{align*}
When we calculate the difference of the absolute squares, we get:
$$
|M_{A\to B}|^2 - |M_{\overline A\to \overline B}|^2 = -4|M_1||M_2|\sin(\varphi_1-\varphi_2)\sin(\Omega_1-\Omega_2).
$$
Now there's a difference in the decay of particles and antiparticles. If we eliminate the CKM-phase factor ($\Omega_{1,2}\to 0$), the above difference would vanish and particles would again behave the same as antiparticles. 
