quant-phNov 3, 2025

Quantum Information Ordering and Differential Privacy

Authors: Naqueeb Ahmad Warsi, Ayanava Dasgupta, Masahito Hayashi

Organizations: Indian Statistical Institute, Kolkata 700108, India. · Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan.

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Abstract

We study quantum differential privacy (QDP) by defining a notion of the order of informativeness between two pairs of quantum states. In particular, we show that if the hypothesis testing divergence of the one pair dominates over that of the other pair, then this dominance holds for every ff-divergence. This approach completely characterizes (ε,δ)(\varepsilon,δ)-QDP mechanisms by identifying the most informative (ε,δ)(\varepsilon,δ)-DP quantum state pairs. We apply this to study precise limits for privatized hypothesis testing and privatized quantum parameter estimation, including tight upper-bounds on the quantum Fisher information under QDP. Finally, we establish near-optimal contraction bounds for differentially private quantum channels with respect to the Hockey-Stick divergence.

Figures & tables

ResultsClassical DomainQuantum Domain
ε\varepsilon -DP(ε,δ)(\varepsilon,\delta) -DPε\varepsilon -DP(ε,δ)(\varepsilon,\delta) -DP
Explicit weakest / most informative private object[ 3 , Theorem 5][ 3 , Theorem 18][ 14 , Theorem 2]Lemma 3 of this manuscript
Extremal-pair dominance bounds for divergences[ 15 , Lemma D.8] Corollary 3 of this manuscriptTheorem 2 of this manuscript
Privatized hypothesis testing[ 3 , Theorem 18] *[ 3 , Theorem 18] *[ 14 , Section IIB]Theorem 2 and Corollary 5 of this manuscript
Privatized parameter estimation[ 3 , Theorem 18] *[ 3 , Theorem 18] *[ 14 , Section IIA]Theorem 3 of this manuscript
Contraction coefficient of trace distance[ 3 , Corollary 11] [ 5 , Theorem 2][ 5 , Theorem 5]
TABLE I : Relationship between results obtained in this work and related studies
Fig. 1 : Graphical representation of R(ε,δ)\mathcal{R}(\varepsilon,\delta) : characteristic region of (ε,δ)(\varepsilon,\delta) -QDP (in shaded region), where we define the boundaries (a) , (b) , (c) and (d) to be β=1−δ−eεα\beta=1-\delta-e^{\varepsilon}\alpha , β=e−ε(1−δ−α)\beta=e^{-\varepsilon}(1-\delta-\alpha) , β=1−e−ε(α−δ)\beta=1-e^{-\varepsilon}(\alpha-\delta) and β=eε(1−α)+δ\beta=e^{\varepsilon}(1-\alpha)+\delta respectively and R(ε,δ)\mathcal{R}(\varepsilon,\delta) has two symmetric error points (1−δ1+eε,1−δ1+eε)(\frac{1-\delta}{1+e^{\varepsilon}},\frac{1-\delta}{1+e^{\varepsilon}}) and (eε+δ1+eε,eε+δ1+eε)(\frac{e^{\varepsilon}+\delta}{1+e^{\varepsilon}},\frac{e^{\varepsilon}+\delta}{1+e^{\varepsilon}}) as its extremal points, where the former and the latter points are known to be the worst and the best-case symmetric error points respectively from the perspective of privacy.
Fig. 2 : Graphical representation of R(ρ(ε,δ)′,σ(ε,δ)′)\mathcal{R}(\rho^{\prime}_{(\varepsilon,\delta)},\sigma^{\prime}_{(\varepsilon,\delta)}) (in the whole shaded region) which is strictly larger than R(ε,δ)\mathcal{R}(\varepsilon,\delta) (in inner shaded region), where the two extremal points of R(ρ(ε,δ)′,σ(ε,δ)′)\mathcal{R}(\rho^{\prime}_{(\varepsilon,\delta)},\sigma^{\prime}_{(\varepsilon,\delta)}) outside R(ε,δ)\mathcal{R}(\varepsilon,\delta) are (1−δ1+eε−δ,1−δ1+eε−δ)\left(\frac{1-\delta}{1+e^{\varepsilon}}-\delta,\frac{1-\delta}{1+e^{\varepsilon}}-\delta\right) and (eε+δ1+eε+δ,eε+δ1+eε+δ)\left(\frac{e^{\varepsilon}+\delta}{1+e^{\varepsilon}}+\delta,\frac{e^{\varepsilon}+\delta}{1+e^{\varepsilon}}+\delta\right) .
Fig. 3 : Graphical representation of R(ρ(ε,δ)′,σ(ε,δ)′)\mathcal{R}(\rho^{\prime}_{(\varepsilon,\delta)},\sigma^{\prime}_{(\varepsilon,\delta)}) (in the whole shaded region) which is strictly larger than R(ε,δ)\mathcal{R}(\varepsilon,\delta) (in inner shaded region), where the two extremal points of R(ρ(ε,δ)′,σ(ε,δ)′)\mathcal{R}(\rho^{\prime}_{(\varepsilon,\delta)},\sigma^{\prime}_{(\varepsilon,\delta)}) outside R(ε,δ)\mathcal{R}(\varepsilon,\delta) are (1−δ1+eε−δ,1−δ1+eε−δ)\left(\frac{1-\delta}{1+e^{\varepsilon}}-\delta,\frac{1-\delta}{1+e^{\varepsilon}}-\delta\right) and (eε+δ1+eε+δ,eε+δ1+eε+δ)\left(\frac{e^{\varepsilon}+\delta}{1+e^{\varepsilon}}+\delta,\frac{e^{\varepsilon}+\delta}{1+e^{\varepsilon}}+\delta\right) .

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