Gentle quantum leakage is proposed as a measure of information leakage to arbitrary eavesdroppers that aim to avoid detection. Gentle (also sometimes referred to as weak or non-demolition) measurements are used to encode the desire of the eavesdropper to evade detection. The gentle quantum leakage meets important axioms proposed for measures of information leakage including positivity, independence, and unitary invariance. Global depolarizing noise, an important family of physical noise in quantum devices, is shown to reduce gentle quantum leakage (and hence can be used as a mechanism to ensure privacy or security). A lower bound for the gentle quantum leakage based on asymmetric approximate cloning is presented. This lower bound relates information leakage to mutual incompatibility of quantum states. A numerical example, based on the encoding in the celebrated BB84 quantum key distribution algorithm, is used to demonstrate the results.