# Classical-Quantum Differentially Private Mechanisms Beyond Classical Ones

Let ε>0. An n-tuple (p_i)_i=1^n of probability vectors is called (classical) ε-differentially private (ε-DP) if e^ε p_j-p_i has no negative entries for all i,j=1,…,n. An n-tuple (ρ_i)_i=1^n of density matrices is called classical-quantum ε-differentially private (CQ ε-DP) if e^ερ_j-ρ_i is positive semi-definite for all i,j=1,…,n. We denote by C_n(ε) the set of all ε-DP n-tuples, by CQ_n(ε) the set of all CQ ε-DP n-tuples, and by EC_n(ε) the set of all n-tuples (∑_k p_i(k)σ_k)_i=1^n with (p_i)_i=1^n∈C_n(ε) and density matrices σ_k. The set EC_n(ε) is a subset of CQ_n(ε), and is essentially classical in optimization. In a preceding study, it is known that EC_2(ε)=CQ_2(ε). In this paper, we show that EC_n(ε)≠CQ_n(ε) for every n≥3, and give a sufficient condition for a CQ ε-DP n-tuple not to lie in EC_n(ε).

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