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Let $\varepsilon>0$. An $n$-tuple $(p_i)_{i=1}^n$ of probability vectors iscalled $\varepsilon$-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 allCQ $\varepsilon$-DP $n$-tuples. By considering optimization problems underlocal differential privacy, we define the subset $\mathrm{EC}_n(\varepsilon)$of $\mathrm{CQ}_n(\varepsilon)$ that is essentially classical. Roughlyspeaking, 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 andtrace-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 showthat $\mathrm{EC}_n(\varepsilon)\not=\mathrm{CQ}_n(\varepsilon)$ for every$n\ge3$, and estimate the difference between $\mathrm{EC}_n(\varepsilon)$ and$\mathrm{CQ}_n(\varepsilon)$ in a certain manner.

Mathematical comparison of classical and quantum mechanisms in optimization under local differential privacy