  The quantum steering ellipsoid can be used to visualise two-qubit states, and
thus provides a generalisation of the Bloch picture for the single qubit.
Recently, a monogamy relation for the volumes of steering ellipsoids has been
derived for pure 3-qubit states and shown to be stronger than the celebrated
Coffman-Kundu-Wootters (CKW) inequality. We first demonstrate the close
connection between this volume monogamy relation and the classification of pure
3-qubit states under stochastic local operations and classical communication
(SLOCC). We then show that this monogamy relation does not hold for general
mixed 3-qubit states and derive a weaker monogamy relation that does hold for
such states. We also prove a volume monogamy relation for pure 4-qubit states,
and generalize our 3-qubit inequality to n qubits. Finally, we study the effect
of noise on the quantum steering ellipsoid and find that the volume of any
two-qubit state is non-increasing when the state is exposed to arbitrary local
noise. This implies that any volume monogamy relation for a given class of
multi-qubit states remains valid under the addition of local noise. We
investigate this quantitatively for the experimentally relevant example of
isotropic noise.
