论文标题
在当地差异隐私下优化中经典和量子机制的数学比较
Mathematical comparison of classical and quantum mechanisms in optimization under local differential privacy
论文作者
论文摘要
令$ \ varepsilon> 0 $。如果$ n $ -tuple $（p_i）_ {i = 1}^n $称为$ \ varepsilon $ -differtially私有（$ \ varepsilon $ -dp），如果$ e^\ e^\ varepsilon p_jj-p_i $没有负面的$ i，j = 1，j = 1，n $ i，n $ i，n $。一个$ n $ -tuple $（ρ_i）_ {i = 1}^n $的密度矩阵称为经典 - Quantum $ \ varepsilon $ -divientally私有（CQ $ \ VAREPSILON $ -DP）用$ \ MATHRM {C} _n（\ VAREPSILON）$表示所有$ \ VAREPSILON $ -DP $ n $ -tuples的集合，以及$ \ Mathrm {cq} _n（\ varepsilon）$的所有CQ $ \ VAREPSILON $ -DPPEPER。通过考虑局部微分隐私下的优化问题，我们定义了$ \ mathrm {cq} _n（\ varepsilon）$的子集$ \ mathrm {ec} _n（\ varepsilon）$，本质上是古典的。粗略地说，$ \ mathrm {ec} _n（\ varepsilon）$中的一个元素是$（p_i）_ {i = 1}^n \ in \ in \ mathrm {c} _n（\ varepsilon）$的图像。在上一项研究中，众所周知，$ \ mathrm {ec} _2（\ varepsilon）= \ mathrm {cq} _2（\ varepsilon）$。在本文中，我们表明每个$ n \ ge3 $ $ \ mathrm {ec} _n（\ varepsilon）\ not = \ not = \ mathrm {cq} _n（\ varepsilon）$ for $ n \ ge3 $，并估算$ \ \ \ m artrm {ec} _n（ec} _n（\ varepsilon）$之间的差额$ \ mathrm {cq} _n（\ varepsilon）$以某种方式。
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 $(ρ_i)_{i=1}^n$ of density matrices is called classical-quantum $\varepsilon$-differentially private (CQ $\varepsilon$-DP) if $e^\varepsilonρ_j-ρ_i$ 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.
