  The set of all qubit states that can be steered to by measurements on a
correlated qubit is predicted to form an ellipsoid---called the quantum
steering ellipsoid---in the Bloch ball. This ellipsoid provides a simple visual
characterisation of the initial 2-qubit state, and various aspects of
entanglement are reflected in its geometric properties. We experimentally
verify these properties via measurements on many different
polarisation-entangled photonic qubit states. Moreover, for pure 3-qubit
states, the volumes of the two quantum steering ellipsoids generated by
measurements on the first qubit are predicted to satisfy a tight monogamy
relation, which is strictly stronger than the well-known monogamy of
entanglement for concurrence. We experimentally verify these predictions, using
polarisation and path entanglement. We also show experimentally that this
monogamy relation can be violated by a mixed entangled state, which
nevertheless satisfies a weaker monogamy relation.
