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.

翻译：提取量子态的层析信息是开发高精度量子器件过程中的关键任务。现有方案通常需要事先经过高精度校准的测量装置。讽刺的是，测量校准的精度从根本上受限于量子态制备的精度，从而形成恶性循环。本文证明该循环可被打破，且对测量装置校准的依赖性可显著降低。我们证明，利用目标量子态天然的低秩结构，足以构建高度可扩展的"盲"层析方案，并配合经典高效的后处理算法。通过利用校准的稀疏结构，我们进一步提升了方案的效率。这通过将盲量子层析问题松弛为稀疏低秩矩阵混合的分离问题来实现。我们证明，当测量模型满足受限等距性质时，所提算法能同时恢复低秩量子态与校准参数。对于一般测量情形，我们证明其所需的测量配置数量接近最优。除概念与数学洞见外，我们通过数值实验证明，在受离子阱实验启发的实际场景中，鲁棒盲量子层析是可行的。