How Entanglement Reshapes the Geometry of Quantum Differential Privacy

TL;DR

This work extends quantum differential privacy by incorporating entanglement as a resource under entanglement-constrained inputs. It shows a sharp phase-transition in the optimal privacy leakage $\varepsilon^*(s)$ as entanglement entropy $s$ increases, governed by the non-convex geometry of the entanglement domain ${\mathbb H}_s$ and analyzed via Riemannian optimization on Schmidt-based parametrizations. The main results provide a closed-form expression for the maximal privacy energy $J_{\max}$ in the low-entanglement regime and a Gibbs-form optimum in the high-entanglement regime, with a computable lower bound for $J_{\min}$, collectively yielding a monotonically decreasing $\varepsilon^*(s)$ and the potential privatization of non-private mechanisms. Numerical experiments with block-depolarizing channels corroborate the phase transition and demonstrate that entanglement can turn non-private mechanisms into private ones, highlighting entanglement as a genuinely privacy-enhancing quantum resource with a rich geometric structure.

Abstract

Quantum differential privacy provides a rigorous framework for quantifying privacy guarantees in quantum information processing. While classical correlations are typically regarded as adversarial to privacy, the role of their quantum analogue, entanglement, is not well understood. In this work, we investigate how quantum entanglement fundamentally shapes quantum local differential privacy (QLDP). We consider a bipartite quantum system whose input state has a prescribed level of entanglement, characterized by a lower bound on the entanglement entropy. Each subsystem is then processed by a local quantum mechanism and measured using local operations only, ensuring that no additional entanglement is generated during the process. Our main result reveals a sharp phase-transition phenomenon in the relation between entanglement and QLDP: below a mechanism-dependent entropy threshold, the optimal privacy leakage level mirrors that of unentangled inputs; beyond this threshold, the privacy leakage level decreases with the entropy, which strictly improves privacy guarantees and can even turn some non-private mechanisms into private ones. The phase-transition phenomenon gives rise to a nonlinear dependence of the privacy leakage level on the entanglement entropy, even though the underlying quantum mechanisms and measurements are linear. We show that the transition is governed by the intrinsic non-convex geometry of the set of entanglement-constrained quantum states, which we parametrize as a smooth manifold and analyze via Riemannian optimization. Our findings demonstrate that entanglement serves as a genuine privacy-enhancing resource, offering a geometric and operational foundation for designing robust privacy-preserving quantum protocols.

Figures (3)

• Figure 1: Pipeline of Entanglement-constrained QLDP against local measurement adversary.
• Figure 2: Phase transition of the optimal QLDP leakage level with Entanglement entropy
• Figure 3: Finite QLDP leakage achieved via entanglement for a non-private mechanism.

Theorems & Definitions (15)

• Definition 1: Quantum Local Differential Privacy
• Theorem 1: Informal
• Definition 2: Local measurement adversary
• Definition 3: entanglement-constrained $\varepsilon$-local measurement QLDP
• Theorem 2: Optimal Privacy Budget
• Theorem 3: Riemannian Optimality Conditions for Privacy Energies
• proof
• Theorem 4: Phase-transition of Maximal Privacy Energy
• Theorem 5: Phase-transition of Relaxed Minimal Privacy Energy
• Proposition 1