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.

翻译：量子差分隐私为量化量子信息处理中的隐私保障提供了严格框架。虽然经典相关性通常被视为对隐私不利，但其量子对应物——纠缠的作用尚未得到充分理解。在本研究中，我们探究量子纠缠如何从根本上塑造量子局部差分隐私（QLDP）。我们考虑一个二分量子系统，其输入态具有预设的纠缠水平，以纠缠熵的下界为特征。每个子系统随后通过局部量子机制处理，并仅使用局部操作进行测量，确保过程中不产生额外纠缠。我们的主要结果揭示了纠缠与QLDP关系中的尖锐相变现象：当低于机制依赖的熵阈值时，最优隐私泄露水平与无纠缠输入的情况一致；超过该阈值，隐私泄露水平随熵值增加而下降，这严格提升了隐私保障，甚至能将某些非隐私机制转化为隐私机制。尽管底层量子机制和测量是线性的，相变现象导致隐私泄露水平对纠缠熵产生非线性依赖。我们证明该相变由纠缠约束量子态集合的内在非凸几何性质所主导，我们将其参数化为光滑流形并通过黎曼优化进行分析。我们的研究结果表明，纠缠作为一种真正的隐私增强资源，为设计鲁棒的隐私保护量子协议提供了几何与操作基础。