  In recent years, centralized differential privacy has been successfully
extended to quantum computing and information processing to safeguard privacy
and prevent leaks in neighboring relationships of quantum states. This paper
introduces a framework known as quantum local differential privacy (QLDP) and
initializes the algorithmic study of QLDP. QLDP utilizes a parameter $\epsilon$
to manage privacy leaks and ensure the privacy of individual quantum states.
The optimization of the QLDP value $\epsilon$, denoted as $\epsilon^*$, for any
quantum mechanism is addressed as an optimization problem. The introduction of
quantum noise is shown to provide privacy protections similar to classical
scenarios, with quantum depolarizing noise identified as the optimal unital
privatization mechanism within the QLDP framework. Unital mechanisms represent
a diverse set of quantum mechanisms that encompass frequently employed quantum
noise types. Quantum depolarizing noise optimizes both fidelity and trace
distance utilities, which are crucial metrics in the field of quantum
computation and information, and can be viewed as a quantum counterpart to
classical randomized response methods. Additionally, a composition theorem is
presented for the application of QLDP framework in distributed (spatially
separated) quantum systems, ensuring the validity (additivity of QLDP value)
irrespective of the states' independence, classical correlation, or
entanglement (quantum correlation). The study further explores the trade-off
between utility and privacy across different quantum noise mechanisms,
including unital and non-unital quantum noise mechanisms, through both
analytical and numerically experimental approaches. Meanwhile, this highlights
the optimization of quantum depolarizing noise in QLDP framework.
