The Cauchy-Schwarz Inequalities (CSI) are a basic but widespread inequality in a vector space, that bounds the (absolute value of the) inner product of two vectors by the product of their length.
It subtends many important results, such as the triangle inequality in geometry or the Heisenberg uncertainty principle in physics. The expectation value of the cross correlation $\langle I_1I_2\rangle$ between two quantities is bounded by the autocorrelations of each quantity: $$|\langle I_1I_2\rangle|\le\sqrt{\langle I_1^2\rangle\langle I_2^2\rangle}\,.$$
In statistics, it similarly relates fluctuations of cross and autocorrelations.
In quantum optics, they take the form:
$$\langle a^{\dagger}a\,b^{\dagger}b\rangle^{2} \le \langle a^{\dagger 2}a^{2}\rangle\,\langle b^{\dagger 2}b^{2}\rangle$$
This is true (or "satisfied") for classical fields. In quantum physics, we are interested in violating such inequalities, in which case the $\le$ becomes a $>$ instead.
In terms of $g^{(2)}$ (satisfied): $$\left[g^{(2)}_{ab}\right]^2 \le g^{(2)}_{aa}\, g^{(2)}_{bb}$$
In terms of populations alone, the CSI reads: $$\langle n_a n_b\rangle^2 \le \langle n_a(n_a-1)\rangle\,\langle n_b(n_b-1)\rangle .$$
Clearly, this involves only the diagonal elements $p(n_a,n_b)=\langle n_a n_b|\rho|n_a n_b\rangle$ and up to its first two factorial moments (no phase, no quantum coherence, etc.)
Note that if one mode is antibunched while $\langle n_an_b\rangle\neq0$, then the rhs is zero and CSI are automatically violated. More generally: $$g^{(2)}_{ab}>\min(g^{(2)}_{aa},g^{(2)}_{bb})$$
Importantly, CS violation also depends on the choice of basis, e.g., $|1,1\rangle$ violates CSI but the state through a BS $(|20\rangle-|02\rangle)/\sqrt2$ does not.
The violation is immune to uncorrelated linear loss ($g^{(2)}$ are loss-independent) but is destroyed by uncorrelated background counts, which inflate $\langle n_a\rangle\langle n_b\rangle$ without feeding $\langle n_an_b\rangle$.
$p(n_a,n_b)$ is a bona fide classical probability distribution, so genuine Cauchy–Schwarz does apply to it, and it can never be violated: $$\langle n_a n_b\rangle^2 \le \langle n_a^2\rangle\,\langle n_b^2\rangle$$
Since $\langle n^2\rangle = \langle n(n-1)\rangle + \langle n\rangle$, one can bound the ratio $R$ of violation by: $$R \;\le\; \left(1+\frac{1}{\langle n_a\rangle\, g^{(2)}_{aa}}\right)\left(1+\frac{1}{\langle n_b\rangle\, g^{(2)}_{bb}}\right)$$ so that as both $\langle n_a\rangle g^{(2)}_{aa}$ and $\langle n_b\rangle g^{(2)}_{bb}$ get large, the violation ratio reduces to $R\to1$ and ultimately disappears: one needs antibunching and/or low occupancy.
If there is a single mode, then from Titulaer and Glauber[1], $g^{(2)}$ is one and there is no CSI: spatial CS is a witness of the mode structure, not of the state alone.
The violation is: $$\lvert \Psi(\mathbf{r}_1,\mathbf{r}_2)\rvert^2 \;>\; \lvert \Psi(\mathbf{r}_1,\mathbf{r}_1)\rvert\,\lvert \Psi(\mathbf{r}_2,\mathbf{r}_2)\rvert$$
The resolution of the camera sets a bound to the degree of violation: the larger the pixel, the less the violation (including the whole field reduces to a single mode with no violation whatsoever).
Pauli exclusion is a maximal CS violator by construction, so fermions always violate CSI.