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Owing in large part to the advent of integrated biphoton frequency combs(BFCs), recent years have witnessed increased attention to quantum informationprocessing in the frequency domain for its inherent high dimensionality andentanglement compatible with fiber-optic networks. Quantum state tomography(QST) of such states, however, has required complex and precise engineering ofactive frequency mixing operations, which are difficult to scale. To addressthese limitations, we propose a novel solution that employs a pulse shaper andelectro-optic phase modulator (EOM) to perform random operations instead ofmixing in a prescribed manner. We successfully verify the entanglement andreconstruct the full density matrix of BFCs generated from an on-chipSi$_{3}$N$_{4}$ microring resonator(MRR) in up to an $8\times8$-dimensionaltwo-qudit Hilbert space, the highest dimension to date for frequency bins. Moregenerally, our employed Bayesian statistical model can be tailored to a varietyof quantum systems with restricted measurement capabilities, forming anopportunistic tomographic framework that utilizes all available data in anoptimal way.

Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements