  Differential privacy is a widely used notion of security that enables the
processing of sensitive information. In short, differentially private
algorithms map "neighbouring" inputs to close output distributions. Prior work
proposed several quantum extensions of differential privacy, each of them built
on substantially different notions of neighbouring quantum states. In this
paper, we propose a novel and general definition of neighbouring quantum
states. We demonstrate that this definition captures the underlying structure
of quantum encodings and can be used to provide exponentially tighter privacy
guarantees for quantum measurements. Our approach combines the addition of
classical and quantum noise and is motivated by the noisy nature of near-term
quantum devices. Moreover, we also investigate an alternative setting where we
are provided with multiple copies of the input state. In this case,
differential privacy can be ensured with little loss in accuracy combining
concentration of measure and noise-adding mechanisms. En route, we prove the
advanced joint convexity of the quantum hockey-stick divergence and we
demonstrate how this result can be applied to quantum differential privacy.
Finally, we complement our theoretical findings with an empirical estimation of
the certified adversarial robustness ensured by differentially private
measurements.
