Programmable coherent linear quantum operations with high-dimensional optical spatial modes

A simple and flexible scheme for high-dimensional linear quantum operations on optical transverse spatial modes is demonstrated. The quantum Fourier transformation (QFT) and quantum state tomography (QST) via symmetric informationally complete positive operator-valued measures (SIC POVMs) are implemented with dimensionality of 15. The matrix fidelity of QFT is 0.85, while the statistical fidelity of SIC POVMs and fidelity of QST are ~0.97 and up to 0.853, respectively. We believe that our device has the potential for further exploration of high-dimensional spatial entanglement provided by spontaneous parametric down conversion in nonlinear crystals

Partially functional linear regression in high dimensions

In modern experiments, functional and nonfunctional data are often encountered simultaneously when observations are sampled from random processes and high-dimensional scalar covariates. It is difficult to apply existing methods for model selection and estimation. We propose a new class of partially functional linear models to characterize the regression between a scalar response and covariates of both functional and scalar types. The new approach provides a unified and flexible framework that simultaneously takes into account multiple functional and ultrahigh-dimensional scalar predictors, enables us to identify important features, and offers improved interpretability of the estimators. The underlying processes of the functional predictors are considered to be infinite-dimensional, and one of our contributions is to characterize the effects of regularization on the resulting estimators. We establish the consistency and oracle properties of the proposed method under mild conditions, demonstrate its performance with simulation studies, and illustrate its application using air pollution data

Universal linear optical operations on discrete phase-coherent spatial modes

Linear optical operations are fundamental and significant for both quantum mechanics and classical technologies. We demonstrate a non-cascaded approach to perform arbitrary unitary and non-unitary linear operations for N-dimensional phase-coherent spatial modes with meticulously designed phase gratings. As implemented on spatial light modulators (SLMs), the unitary transformation matrix has been realized with dimensionalities ranging from 7 to 24 and the corresponding fidelities are from 95.1% to 82.1%. For the non-unitary operators, a matrix is presented for the tomography of a 4-level quantum system with a fidelity of 94.9%. Thus, the linear operator has been successfully implemented with much higher dimensionality than that in previous reports. It should be mentioned that our method is not limited to SLMs and can be easily applied on other devices. Thus we believe that our proposal provides another option to perform linear operation with a simple, fixed, error-tolerant and scalable scheme