Optimal quantum locally differentially private mechanisms in the high-privacy regime

Abstract: We optimize the trade-off between privacy and utility in the high-privacy regime. We adopt local differential privacy (LDP) and its quantum extension, quantum local differential privacy (QLDP), for privacy protection, and investigate utility functions including the Holevo information (which reduces to the mutual information in the classical case) and the error exponents in symmetric and asymmetric hypothesis testing. These utility functions have classical and quantum optimal values, which are denoted by $C$ and $Q$, respectively, in this abstract for simplicity. In this paper, we provide optimal LDP and QLDP mechanisms achieving the classical and quantum optimal values in the high-privacy regime, and prove that the asymptotic ratio $Q/C$ in this regime takes the same value regardless of the utility function. Our results reveal quantum advantages (more precisely, $Q/C\ge3/2$) for the above utility functions when the protected private data are $n$-ary with $n\ge3$.

• The paper establishes a unified asymptotic analysis showing a quantum advantage ratio strictly greater than one for non-binary alphabets (n ≥ 3).
• It employs a second-order Taylor expansion and Fréchet derivatives to derive tight analytical bounds for privacy-utility trade-offs in various tasks including hypothesis testing and mutual information.
• Explicit constructions using binary mechanisms and EITFF-based isoclinic frameworks are proposed, paving the way for quantum-enhanced, privacy-preserving data analysis.

Optimal Quantum Locally Differentially Private Mechanisms in the High-Privacy Regime

Problem Formulation and Motivation

The paper rigorously investigates the optimization of privacy-utility trade-off in the context of quantum local differential privacy (QLDP) and its classical counterpart, local differential privacy (LDP). Unlike global DP, LDP mechanisms provide privacy guarantees without reliance on trusted aggregators, making them crucial for decentralized privacy-preserving data analysis and distributed learning.

The quantum extension, QLDP, utilizes quantum states in place of classical randomized outputs, enriching the space of privacy mechanisms. The principal question is whether quantum mechanisms offer tangible advantages over classical ones in utility, under the same privacy constraints, particularly in the high-privacy ($\epsilon \to 0$) regime. Previous works have shown no quantum advantage in binary cases ($n=2$), while the situation for $n \geq 3$ remains less understood except for limited utility functions and parameter ranges.

Utility Functions and Optimization Framework

The paper considers a broad class of utility functions:

• Information-theoretic quantities such as mutual information and Holevo information.
• Error exponents in symmetric and asymmetric hypothesis testing.
• Functions representable as sums of symmetric sublinear functions of output conditional distributions, covering pairwise $f$-divergences and general measures relevant to hypothesis testing and estimation.

For classical mechanisms, the optimization problem is to maximize utility $\Phi_C(q)$ subject to $q$ being $\epsilon$-LDP. For quantum mechanisms, the analogous optimization is maximizing $\Phi_Q(\vec{\rho})$ with $\vec{\rho}$ being $\epsilon$-QLDP. Quantum extensions $n=2$0 are required to be unitary-invariant and consistent with $n=2$1 on diagonal states.

The paper leverages the second-order Taylor expansion in Fréchet derivatives near the high-privacy regime, providing analytical tractability and tight asymptotic bounds.

Main Results: Optimality and Quantum Advantage

Unified Asymptotic Quantum Advantage

The primary theorem establishes the asymptotic ratio of quantum to classical optimal values for utility in the high-privacy regime:

$n=2$2

for utility functions whose quadratic term in expansion matches a normalized Petz monotone metric (e.g., BKM, SLD, RLD metrics), with $n=2$3 given by symmetric sublinear forms and $n=2$4 (2605.27278). This ratio is always at least $n=2$5 for $n=2$6, rigorously demonstrating quantum advantage across a broad utility spectrum, including hypothesis testing exponents and Holevo/mutual information.

Strong Numerical Claims

The asymptotic quantum-to-classical ratio $n=2$7 is:

• Independent of the particular utility function (as long as specified conditions are met).
• Strictly greater than one for non-binary alphabets ($n=2$8).
• For $n=2$9, this ratio remains valid and optimal, solving previously open cases in concrete processing scenarios.

For mutual/Holevo information:

• Classical optimum in high-privacy: $n \geq 3$0.
• Quantum optimum: $n \geq 3$1.

For symmetric/asymmetric hypothesis testing (error exponents):

• Classical symmetric: $n \geq 3$2.
• Quantum symmetric: $n \geq 3$3.
• Classical asymmetric: $n \geq 3$4.
• Quantum asymmetric: $n \geq 3$5.

Quantum mechanisms strictly improve utility for $n \geq 3$6; optimality is proved analytically, and exact privacy parameter thresholds for the advantage are determined.

Construction of Optimal Mechanisms

• Classical optimal mechanisms: Binary mechanisms (partitioning input alphabet into two blocks) provably achieve the classical optimum in high-privacy [Kairouz et al.].
• Quantum optimal mechanisms: "Isoclinic" mechanisms based on equi-isoclinic tight fusion frames (EITFFs), which generalize projective measurements to optimal rank projections, achieve quantum optimum. For $n \geq 3$7 (even dimensions), their existence is tied to the Radon-Hurwitz numbers; explicit constructions provided for $n \geq 3$8.

Implications and Theoretical Insights

The paper's results offer a unified explanation for quantum advantage in locally private mechanisms, analytically connecting optimality to abstract information geometry (Petz monotone metrics, quadratic expansions). The quantum optimal mechanism outperforms classical counterparts for privacy-sensitive information processing tasks, with ratios independent of utility specifics.

The reduction of classical mechanisms to binary partitions and the use of EITFFs in quantum constructions settle longstanding questions regarding exact optimality and minimum-dimension realizations in privacy mechanisms.

Quantitative privacy thresholds for quantum advantage are rigorously evaluated; for $n \geq 3$9 and $f$0 in $f$1, quantum advantage is always exhibited for both symmetric and asymmetric hypothesis testing. The privacy regime for quantum superiority expands with increasing $f$2.

Practical and Future Directions

For practical privacy-preserving machine learning, the quantum advantage in utility implies that quantum channels can enable higher statistical accuracy or distinguishability under equivalent privacy constraints, especially in high-privacy (small $f$3) settings. Potential applications include quantum-enhanced locally private federated learning, quantum-secure statistical estimation, and hypothesis testing on distributed quantum datasets.

Open questions include:

• Minimum dimension required for quantum-optimal mechanisms outside $f$4 (non-EITFF cases).
• Extension of optimality results to moderate-privacy regimes ($f$5 not small).
• Density conjecture for realization of isoclinic frames, which may affect possible values of quantum utility in larger alphabets.
• Verification of tightness in quantum-to-classical ratios for intermediate privacy settings.

Conclusion

This paper establishes optimal mechanisms for quantum local differential privacy in the high-privacy regime, demonstrates universal quantum advantage for a broad class of utility functions, and settles several previously unexplored cases in information-theoretic privacy optimization. It provides explicit constructions and analytic bounds that clarify the landscape of privacy-utility trade-off in the quantum setting. These results pave the way for deeper integration of quantum mechanisms in decentralized privacy-preserving data analysis and for further study of optimality in the moderate-privacy regime (2605.27278).