Let $\varepsilon>0$. An $n$-tuple $(p_i)_{i=1}^n$ of probability vectors is called (classical) $\varepsilon$-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 $i,j=1,\ldots,n$. We denote by $\mathrm{C}_n(\varepsilon)$ the set of all $\varepsilon$-DP $n$-tuples, by $\mathrm{CQ}_n(\varepsilon)$ the set of all CQ $\varepsilon$-DP $n$-tuples, and by $\mathrm{EC}_n(\varepsilon)$ the set of all $n$-tuples $(\sum_k p_i(k)\sigma_k)_{i=1}^n$ with $(p_i)_{i=1}^n\in\mathrm{C}_n(\varepsilon)$ and density matrices $\sigma_k$. The set $\mathrm{EC}_n(\varepsilon)$ is a subset of $\mathrm{CQ}_n(\varepsilon)$, and is essentially classical in optimization. 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 give a sufficient condition for a CQ $\varepsilon$-DP $n$-tuple not to lie in $\mathrm{EC}_n(\varepsilon)$.

中文翻译：

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超越经典机制的经典量子差分私有机制

让 $\varepsilon>0$。概率向量的 $n$-元组 $(p_i)_{i=1}^n$ 被称为（经典）$\varepsilon$-差异私有（$\varepsilon$-DP）如果 $e^\varepsilon p_j- p_i$ 对于所有 $i,j=1,\ldots,n$ 都没有负项。密度矩阵的 $n$-元组 $(\rho_i)_{i=1}^n$ 称为经典量子 $\varepsilon$-差异私有 (CQ $\varepsilon$-DP) 如果 $e^\varepsilon \rho_j-\rho_i$ 对于所有 $i,j=1,\ldots,n$ 都是半正定的。我们用 $\mathrm{C}_n(\varepsilon)$ 表示所有 $\varepsilon$-DP $n$-元组的集合，用 $\mathrm{CQ}_n(\varepsilon)$ 表示所有 CQ $ 的集合\varepsilon$-DP $n$-元组，通过 $\mathrm{EC}_n(\varepsilon)$ 得到所有 $n$-元组的集合 $(\sum_k p_i(k)\sigma_k)_{i=1 }^n$ 与 $(p_i)_{i=1}^n\in\mathrm{C}_n(\varepsilon)$ 和密度矩阵 $\sigma_k$。集合 $\mathrm{EC}_n(\varepsilon)$ 是 $\mathrm{CQ}_n(\varepsilon)$ 的子集，本质上是优化的经典。在前面的研究中，已知$\mathrm{EC}_2(\varepsilon)=\mathrm{CQ}_2(\varepsilon)$。在本文中，我们证明了 $\mathrm{EC}_n(\varepsilon)\not=\mathrm{CQ}_n(\varepsilon)$ 对于每个 $n\ge3$，并给出了一个 CQ $\ varepsilon$-DP $n$-元组不要在 $\mathrm{EC}_n(\varepsilon)$ 中。