Quantum information has been drawing a wealth of research in recent years, shedding light on questions in the heart of quantum mechanics¹, as well as advancing fields such as complexity theory², cryptography³, key distribution⁴, and chemistry⁵. These fundamental and applied aspects of quantum information rely on one crucial issue: the ability to characterize the quantum state from measurements, through a process called Quantum State Tomography (QST). However, QST is resource consuming, requiring a large number of measurements in multiple bases, derived from physical observables and corresponding to different experimental setups. Naturally, changing the setup results in also unwanted changes to the data, prolongs the measurement and hurts the integrity of the assumptions about the noise. We propose to overcome these problems by performing QST with a single observable. In many systems, a single observable is realized by a single setup, thus considerably reducing the experimental effort. Certainly, measurements of a single observable do not hold enough information to recover the quantum state. We overcome this by using concepts inspired by Compressed Sensing (CS)^6,7, using structure in the sought state, namely, that it is close to a pure state (and thus has low rank). Additionally, we increase the system dimension by adding a known ancilla to obtain more measurements, and introduce mixing between the degrees of freedom to maximize the benefit of each measurement. The dimensional increase is kept to a minimum by the structure of the state. We demonstrate this 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.

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