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The secondary cosmic microwave background (CMB) $B$-modes stem from thepost-decoupling distortion of the polarization $E$-modes due to thegravitational lensing effect of large-scale structures. These lensing-induced$B$-modes constitute both a valuable probe of the dark matter distribution andan important contaminant for the extraction of the primary CMB $B$-modes frominflation. Planck provides accurate nearly all-sky measurements of both thepolarization $E$-modes and the integrated mass distribution via thereconstruction of the CMB lensing potential. By combining these two dataproducts, we have produced an all-sky template map of the lensing-induced$B$-modes using a real-space algorithm that minimizes the impact of sky masks.The cross-correlation of this template with an observed (primordial andsecondary) $B$-mode map can be used to measure the lensing $B$-mode powerspectrum at multipoles up to $2000$. In particular, when cross-correlating withthe $B$-mode contribution directly derived from the Planck polarization maps,we obtain lensing-induced $B$-mode power spectrum measurement at a significancelevel of $12\,\sigma$, which agrees with the theoretical expectation derivedfrom the Planck best-fit $\Lambda$CDM model. This unique nearly all-skysecondary $B$-mode template, which includes the lensing-induced informationfrom intermediate to small ($10\lesssim \ell\lesssim 1000$) angular scales, isdelivered as part of the Planck 2015 public data release. It will beparticularly useful for experiments searching for primordial $B$-modes, such asBICEP2/Keck Array or LiteBIRD, since it will enable an estimate to be made ofthe lensing-induced contribution to the measured total CMB $B$-modes.

Planck intermediate results. XLI. A map of lensing-induced B-modes