In this paper, we study the diagrammatic categorification of q-boson algebra and also q-fermion algebra. We construct a graphical category corresponding to q-boson algebra. q-Fock states correspond to some kind of 1-morphisms, and the graded dimension of the graded vector space of 2-morphisms is exactly the inner product of the corresponding q-Fock states. We also find that this graphical category can be used to categorify q-fermion algebra.

A deformed boson algebra is naturally introduced from studying quantum mechanics on noncommutative phase space in which both positions and momenta are noncommuting each other. Based on this algebra, corresponding intrinsic noncommutative coherent and squeezed state representations are constructed, and variances of single-and twomode quadrature operators on these states are evaluated. The result indicates that in order to maintain Heisenberg's uncertainty relations, a restriction between the noncommutative parameters is required.

We study the quantum entropy and entanglement of the 2D isotropic harmonic oscillators in noncommutative phase space. We propose a definition of quantum Rényi entropy by the Wigner functions in noncommutative phase space. Using the Rényi entropy, we calculate the entanglement entropy of the ground state of the harmonic oscillators. We find that the 2D isotropic harmonic oscillators can be entangled in noncommutative phase space. This is an entanglement-like effect caused by the noncommutativity of the phase space. We also study the Tsallis entropy of the harmonic oscillators in noncommutative phase space.

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.