  Extracting tomographic information about quantum states is a crucial task in
the quest towards devising high-precision quantum devices. Current schemes
typically require measurement devices for tomography that are a priori
calibrated to high precision. Ironically, the accuracy of the measurement
calibration is fundamentally limited by the accuracy of state preparation,
establishing a vicious cycle. Here, we prove that this cycle can be broken and
the dependence on the measurement device's calibration significantly relaxed.
We show that exploiting the natural low-rank structure of quantum states of
interest suffices to arrive at a highly scalable `blind' tomography scheme with
a classically efficient post-processing algorithm. We further improve the
efficiency of our scheme by making use of the sparse structure of the
calibrations. This is achieved by relaxing the blind quantum tomography problem
to the de-mixing of a sparse sum of low-rank matrices. We prove that the
proposed algorithm recovers a low-rank quantum state and the calibration
provided that the measurement model exhibits a restricted isometry property.
For generic measurements, we show that it requires a close-to-optimal number of
measurement settings. Complementing these conceptual and mathematical insights,
we numerically demonstrate that robust blind quantum tomography is possible in
a practical setting inspired by an implementation of trapped ions.
