Mathematical comparison of classical and quantum mechanisms in optimization under local differential privacy
Yuuya Yoshida
TL;DR
The paper investigates the boundary between classical and quantum mechanisms in optimization under local differential privacy. It formalizes classical $ε$-DP and classical-quantum $ε$-DP, defines the essentially classical subset $\mathrm{EC}_n(ε)$, and proves that $\mathrm{EC}_n(ε)\neq\mathrm{CQ}_n(ε)$ for all $n\ge3$, establishing a quantum advantage in optimization tasks under CQ-$ε$-DP. It provides a concrete objective based on the RLD Fisher information that yields equal classical and essentially classical optima, but strict superiority for CQ optima when $n\ge3$, together with explicit bounds via auxiliary functions. The work also constructs explicit CQ-$ε$-DP $n$-tuples not lying in $\mathrm{EC}_n(ε)$, demonstrating the practical gap between these sets, and discusses implications for dimension handling and future refinements of the bounds. Overall, the results quantify the separation between classical- and quantum-driven DP mechanisms and illuminate when quantum resources provide a genuine advantage in private data optimization.
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 $(ρ_i)_{i=1}^n$ 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.