  Let $\varepsilon>0$. An $n$-tuple $(p_i)_{i=1}^n$ of probability vectors is
called $\varepsilon$-differentially private ($\varepsilon$-DP) if
$e^\varepsilon p_j-p_i$ has no negative entries for all $i,j=1,\ldots,n$. An
$n$-tuple $(\rho_i)_{i=1}^n$ of density matrices is called classical-quantum
$\varepsilon$-differentially private (CQ $\varepsilon$-DP) if
$e^\varepsilon\rho_j-\rho_i$ is positive semi-definite for all
$i,j=1,\ldots,n$. Denote by $\mathrm{C}_n(\varepsilon)$ the set of all
$\varepsilon$-DP $n$-tuples, and by $\mathrm{CQ}_n(\varepsilon)$ the set of all
CQ $\varepsilon$-DP $n$-tuples. By considering optimization problems under
local differential privacy, we define the subset $\mathrm{EC}_n(\varepsilon)$
of $\mathrm{CQ}_n(\varepsilon)$ that is essentially classical. Roughly
speaking, an element in $\mathrm{EC}_n(\varepsilon)$ is the image of
$(p_i)_{i=1}^n\in\mathrm{C}_n(\varepsilon)$ by a completely positive and
trace-preserving linear map (CPTP map). In a preceding study, it is known that
$\mathrm{EC}_2(\varepsilon)=\mathrm{CQ}_2(\varepsilon)$. In this paper, we show
that $\mathrm{EC}_n(\varepsilon)\not=\mathrm{CQ}_n(\varepsilon)$ for every
$n\ge3$, and estimate the difference between $\mathrm{EC}_n(\varepsilon)$ and
$\mathrm{CQ}_n(\varepsilon)$ in a certain manner.
