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

• URL: http://arxiv.org/abs/2011.09960v2
• Date: Sun, 19 Dec 2021 00:15:01 GMT
• Title: Mathematical comparison of classical and quantum mechanisms in optimization under local differential privacy
• Authors: Yuuya Yoshida
• Abstract summary: We show that $mathrmEC_n(varepsilon)not=mathrmCQ_n(varepsilon)$ for every $nge3$, and estimate the difference between $mathrmEC_n(varepsilon)$ and $mathrmCQ_n(varepsilon)$ in a certain manner.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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