  As an important degree of freedom (DoF) in integrated photonic circuits, the
orthogonal transverse mode provides a promising and flexible way to increasing
communication capability, for both classical and quantum information
processing. To construct large-scale on-chip multimode multi-DoF quantum
systems, a transverse mode-encoded controlled-NOT (CNOT) gate is necessary.
Here, through design and integrate transverse mode-dependent directional
coupler and attenuators on a silicon photonic chip, we demonstrate the first
multimode implementation of a two-qubit quantum gate. With the aid of state
preparation and analysis parts, we show the ability of the gate to entangle two
separated transverse mode qubits with an average fidelity of $0.89\pm0.02$ and
the achievement of 10 standard deviations of violations in the quantum
nonlocality verification. In addition, a fidelity of $0.82\pm0.01$ was obtained
from quantum process tomography used to completely characterize the CNOT gate.
Our work paves the way for universal transverse mode-encoded quantum operations
and large-scale multimode multi-DoF quantum systems.
