  The framework of approximate differential privacy is considered, and
augmented by leveraging the notion of ``the total variation of a
(privacy-preserving) mechanism'' (denoted by $\eta$-TV). With this refinement,
an exact composition result is derived, and shown to be significantly tighter
than the optimal bounds for differential privacy (which do not consider the
total variation). Furthermore, it is shown that $(\varepsilon,\delta)$-DP with
$\eta$-TV is closed under subsampling. The induced total variation of commonly
used mechanisms are computed. Moreover, the notion of total variation of a
mechanism is studied in the local privacy setting and privacy-utility tradeoffs
are investigated. In particular, total variation distance and KL divergence are
considered as utility functions and studied through the lens of contraction
coefficients. Finally, the results are compared and connected to the locally
differentially private setting.
