  Quantum information has been drawing a wealth of research in recent years,
shedding light on questions at the heart of quantum mechanics, as well as
advancing fields such as complexity theory, cryptography, key distribution, and
chemistry. These fundamental and applied aspects of quantum information rely on
a crucial issue: the ability to characterize a quantum state from measurements,
through a process called Quantum State Tomography (QST). However, QST requires
a large number of measurements, each derived from a different physical
observable corresponding to a different experimental setup. Unfortunately,
changing the setup results in unwanted changes to the data, prolongs the
measurement and impairs the assumptions that are always made about the
stationarity of the noise. Here, we propose to overcome these drawbacks by
performing QST with a single observable. A single observable can often be
realized by a single setup, thus considerably reducing the experimental effort.
In general, measurements of a single observable do not hold enough information
to recover the quantum state. We overcome this lack of information by relying
on concepts inspired by Compressed Sensing (CS), exploiting the fact that the
sought state - in many applications of quantum information - is close to a pure
state (and thus has low rank). Additionally, we increase the system dimension
by adding an ancilla that couples to information evolving in the system,
thereby providing more measurements, enabling the recovery of the original
quantum state from a single-observable measurements. We demonstrate our
approach on multi-photon states by recovering structured quantum states from a
single observable, in a single experimental setup. We further show how this
approach can be used to recover quantum states without number-resolving
detectors.
