  The problem of using observed correlations to infer causal relations is
relevant to a wide variety of scientific disciplines. Yet given correlations
between just two classical variables, it is impossible to determine whether
they arose from a causal influence of one on the other or a common cause
influencing both, unless one can implement a randomized intervention. We here
consider the problem of causal inference for quantum variables. We introduce
causal tomography, which unifies and generalizes conventional quantum
tomography schemes to provide a complete solution to the causal inference
problem using a quantum analogue of a randomized trial. We furthermore show
that, in contrast to the classical case, observed quantum correlations alone
can sometimes provide a solution. We implement a quantum-optical experiment
that allows us to control the causal relation between two optical modes, and
two measurement schemes -- one with and one without randomization -- that
extract this relation from the observed correlations. Our results show that
entanglement and coherence, known to be central to quantum information
processing, also provide a quantum advantage for causal inference.
