We investigate the dynamics of two-level systems using the quasidistribution
Q function projected onto the space of collective measurements. Particular attention is given to evolution under Hamiltonians that are invariant under particle permutations, for pure states that are either invariant or noninvariant under such permutations. This requires constructing an approximate analog of the star product between quasidistribution functions: the projected
Q function of a product of two operators is expressed approximately in terms of the individual operators'
Q and
P functions. We rewrite the Liouville equation to determine approximate state dynamics and compare the result with exact evolution using the Hellinger distance. Overall, the method performs well for permutation-invariant pure states and Hamiltonians, whereas deviations arise for arbitrary pure states and noninvariant Hamiltonians.
The Bose-Hubbard Hamiltonian can be simplified to have only two lattice sites, in which case the system being described is referred to as a dimer. Due to its structure, the hopping term of the dimer Hamiltonian enjoys invariance in a family of subspaces indexed by a whole number
k, each subspace corresponding to a system of only
k particles. We have invented an inductive argument using the bosonic canonical commutation relations to find the eigenvalues and eigenvectors of the dimer hopping Hamiltonian in its
k-particle subspaces. In particular, this Hamiltonian, when restricted to one of the
k-particle subspaces, is exactly the spin projection operator along the
x-axis, where the number of particles
k in the dimer system yields the projection matrix for spin quantum number
s=k/2. Thus, a new method for computing the eigenvalues and eigenvectors of the
x-axis spin projector has been unearthed. We use the explicit construction to study the dynamics of coherent states induced by the square of the dimer hopping hamiltonian. We find that it generates Schrödinger cat states in the two-site setting.
We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a `relativistic' transformation group analogous to that of the Lorentz group SL(2,C)_R ~ SO(3,1) for the qubit case. This is the group SL(3,C)_R, presented as a group of real 9x9 matrices acting linearly on the 9-dimensional space of projective coordinates for the qutrit density matrix. The counterpart, for qutrits, of the invariant 4x4 Minkowski metric of the qubit case, proves to be a certain 9x9x9 totally symmetric three-fold tensor generalizing the Gell-Mann d coefficient. Using this structure, we provide a count of the corresponding LSL polynomial invariants using group character methods. Finally, we give an explicit construction of the lowest degree quantity (the cubic invariant) and its expansion in terms of SU(3)xSU(3) invariants, and we indicate how to construct higher degree analogues. These quantities are proven to yield entanglement monotones. This work generalizes and partly extends the paper of King, Welsh and Jarvis (Journal of Physics A, 40(33):10083, 2007) on the mixed two qubit system, which is reviewed in an appendix.