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

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