  In the quantum world, a single particle can have various degrees of freedom
to encode quantum information. Controlling multiple degrees of freedom
simultaneously is necessary to describe a particle fully and, therefore, to use
it more efficiently. Here we introduce the transverse waveguide-mode degree of
freedom to quantum photonic integrated circuits, and demonstrate the coherent
conversion of a photonic quantum state between path, polarization and
transverse waveguide-mode degrees of freedom on a single chip. The preservation
of quantum coherence in these conversion processes is proven by single-photon
and two-photon quantum interference using a fibre beam splitter or on-chip beam
splitters. These results provide us with the ability to control and convert
multiple degrees of freedom of photons for quantum photonic integrated
circuit-based quantum information process.
