  In quantum mechanics events can happen in no definite causal order: in
practice this can be verified by measuring a causal witness, in the same way
that an entanglement witness verifies entanglement. Indefinite causal order can
be observed in a quantum switch, where two operations act in a quantum
superposition of the two possible orders. Here we realise a photonic quantum
switch, where polarisation coherently controls the order of two operations,
$\hat{A}$ and $\hat{B}$, on the transverse spatial mode of the photons. Our
setup avoids the limitations of earlier implementations: the operations cannot
be distinguished by spatial or temporal position. We show that our quantum
switch has no definite causal order, by constructing a causal witness and
measuring its value to be 18 standard deviations beyond the definite-order
bound.
