High Quantum Local Differential Privacy Breaks Entanglement

Differential privacy (DP) is a standard mathematical framework for guaranteeing privacy of sensitive data. In quantum information processing, an important open question is how privacy constraints interact with quantum resources such as entanglement. Since the utility of many protocols—and often the presence of a quantum advantage—relies on entanglement, it is crucial to understand when a privacy requirement for a quantum channel is compatible with the channel's ability to preserve entanglement.

This work studies that question for **quantum local differential privacy (QLDP)**. The central object is a quantum channel (completely positive, trace-preserving map) acting on a $d$-dimensional input, and the goal is to characterize when an $\epsilon$-QLDP channel must destroy entanglement. The authors prove a sharp high-privacy threshold, an approximate version for $(\epsilon ,\delta )$-QLDP, a composition result for private channels with entangled inputs and global measurements, and applications to private quantum learning theory.

The paper's main results are as follows.

1. **Exact high-privacy threshold.** Every $\epsilon$-QLDP channel with a $d$-dimensional input is entanglement-breaking whenever

This gives a sharp privacy regime in which local differential privacy forces the channel to destroy entanglement.

2. **Approximate version.** For $(\epsilon ,\delta )$-QLDP channels in the same high-privacy regime, the channel is close in diamond norm to an entanglement-breaking channel. Thus even allowing a small privacy failure probability $\delta$ does not permit entanglement preservation beyond the threshold.

3. **Composition result.** The authors prove a composition result for a collection of private quantum channels having entangled inputs and global measurements in the high-privacy regime. This extends the entanglement-breaking behavior to composed settings.

4. **Private quantum learning theory.** Any learning protocol using arbitrary quantum memory on copies of the output of an entanglement-breaking channel can be simulated by a protocol that measures the corresponding unprocessed input copies one at a time while storing only classical information.

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