= Solution
The <Flavor eigenstates>[flavour states] $K^0=d\bar s$ and $\bar K^0=s\bar d$ have definite <Strangeness> and are distinguished by flavour-sensitive production or decay. The <mass eigenstates>[propagation states] diagonalize the <Effective Hamiltonian for neutral-meson mixing>[effective Hamiltonian] $H_{\mathrm{eff}}=M-i\Gamma/2$; their complex <eigenvalues> give a pole <mass> and a <decay width>. The common <Flavor eigenstate>[flavour] combinations specified in the source assume the usual <CPT symmetry>[CPT-symmetric] two-state description and are normalized with $|p|^2+|q|^2=1$.
Choose the <phase> convention $CP|K^0\rangle=|\bar K^0\rangle$. If mixing conserves <CP>, $p=q$ can be chosen, and the symmetric and antisymmetric combinations are <CP> even and odd. More generally the equality must be understood with the corresponding flavour-state <phase> convention; $|p|=|q|$ alone is not a statement fixing an arbitrary written <phase>.
The almost <CP eigenstate>[CP-even] state decays rapidly to <Pion>[two pions], whereas the almost <CP eigenstate>[CP-odd] state is largely restricted to slower channels such as <Pion>[three pions]. Their very different <mean lifetimes>[lifetimes] are easier to distinguish than their tiny <mass> splitting, motivating the names $K_S^0$ and $K_L^0$. In the source's first notation the subscript $L$ means \i[light], not \i[long-lived]: the light state is identified with the short-lived <kaon>, and the heavy state with the long-lived <kaon>. Below, $\ell,h$ label light/heavy to avoid that collision.
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