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 a 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 fundamental dependence on the measurement devices 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 task of de-mixing a sparse sum of low-rank quantum states. Building on
techniques from model-based compressed sensing, 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 our algorithm requires a close-to-optimal number
measurement settings for solving the blind tomography task. Complementing these
conceptual and mathematical insights, we numerically demonstrate that blind
quantum tomography is possible by exploiting low-rank assumptions in a
practical setting inspired by an implementation of trapped ions using
constrained alternating optimization.Comment: 22 pages, 8 Figure
