  We develop techniques to probe the dynamics of quantum information, and
implement them experimentally on an IBM superconducting quantum processor. Our
protocols adapt shadow tomography for the study of time evolution channels
rather than of quantum states, and rely only on single-qubit operations and
measurements. We identify two unambiguous signatures of quantum information
scrambling, neither of which can be mimicked by dissipative processes, and
relate these to many-body teleportation. By realizing quantum chaotic dynamics
in experiment, we measure both signatures, and support our results with
numerical simulations of the quantum system. We additionally investigate
operator growth under this dynamics, and observe behaviour characteristic of
quantum chaos. As our methods require only a single quantum state at a time,
they can be readily applied on a wide variety of quantum simulators.
