  Measurements of joint-polarisation states of $W$ and $Z$ gauge bosons in
$W^{\pm}Z$ production are presented. The data set used corresponds to an
integrated luminosity of $139$ fb$^{-1}$ of proton-proton collisions at a
center-of-mass energy of $13$ TeV recorded by the ATLAS detector at the CERN
Large Hadron Collider. The $W^{\pm}Z$ candidate events are reconstructed using
leptonic decay modes of the gauge bosons into electrons and muons. The
simultaneous pair-production of longitudinally polarised vector bosons is
measured for the first time with a significance of $7.1$ standard deviations.
The measured joint helicity fractions integrated over the fiducial region are
$f_{\mathrm{00}} = 0.067 \pm 0.010$, $f_{\mathrm{0T}} = 0.110 \pm 0.029$,
$f_{\mathrm{T0}} = 0.179 \pm 0.023$ and $f_{\mathrm{TT}} = 0.644 \pm 0.032$, in
agreement with the next-to-leading-order Standard Model predictions. Individual
helicity fractions of the $W$ and $Z$ bosons are also measured and found to be
consistent with joint helicity fractions within the expected amount of
correlations. Both the joint and individual helicity fractions are also
measured separately in $W^+Z$ and $W^-Z$ events. Inclusive and differential
cross sections for several kinematic observables sensitive to polarisation are
presented.
