  We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard
exclusive $\rho ^0$ meson muoproduction at COMPASS using 160~GeV/$c$ polarised
$ \mu ^{+}$ and $ \mu ^{-}$ beams impinging on a liquid hydrogen target. The
measurement covers the kinematic range 5.0~GeV/$c^2$ $< W <$ 17.0~GeV/$c^2$,
1.0 (GeV/$c$)$^2$ $< Q^2 <$ 10.0 (GeV/$c$)$^2$ and 0.01 (GeV/$c$)$^2$ $<
p_{\rm{T}}^2 <$ 0.5 (GeV/$c$)$^2$. Here, $W$ denotes the mass of the final
hadronic system, $Q^2$ the virtuality of the exchanged photon, and $p_{\rm{T}}$
the transverse momentum of the $\rho ^0$ meson with respect to the
virtual-photon direction. The measured non-zero SDMEs for the transitions of
transversely polarised virtual photons to longitudinally polarised vector
mesons ($\gamma^*_T \to V^{ }_L$) indicate a violation of $s$-channel helicity
conservation. Additionally, we observe a dominant contribution of
natural-parity-exchange transitions and a very small contribution of
unnatural-parity-exchange transitions, which is compatible with zero within
experimental uncertainties. The results provide important input for modelling
Generalised Parton Distributions (GPDs). In particular, they may allow one to
evaluate in a model-dependent way the role of parton helicity-flip GPDs in
exclusive $\rho ^0$ production.
