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

Let varepsilon>0. An n-tuple \(p_i\)_i=1ⁿ 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=1ⁿ 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 C_n\(varepsilon\) the set of all varepsilon-DP n-tuples, and by CQ_n\(varepsilon\) the set of all CQ varepsilon-DP n-tuples. By considering optimization problems under local differential privacy, we define the subset EC_n\(varepsilon\) of CQ_n\(varepsilon\) that is essentially classical. Roughly speaking, an element in EC_n\(varepsilon\) is the image of \(p_i\)_i=1^ninC_n\(varepsilon\) by a completely positive and trace-preserving linear map (CPTP map). In a preceding study, it is known that EC₂\(varepsilon\)=CQ₂\(varepsilon\). In this paper, we show that EC_n\(varepsilon\)not=CQ_n\(varepsilon\) for every nge3, and estimate the difference between EC_n\(varepsilon\) and CQ_n\(varepsilon\) in a certain manner.