  Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of
n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by
partial-transpose (PT) plus local unitary transformations (PTU) for one qubit
from the whole system, are used for indicating inseparability. A sufficient
criterion for full separability of the n-qubits and qubit-qudit systems is
given. We use the singular value decomposition (SVD) for improving the
criterion for full separability. General properties of entanglement and
separability are analyzed for a system of a qubit and a qudit and n-qubits
systems, with emphasis on maximally disordered subsystems (MDS) (i.e., density
matrices rho(MDS) for which tracing over any subsystem gives the unit density
matrix). A sufficient condition that rho(MDS) is not separable is that it has
an eigenvalue larger than 1/d for a qubit and a qudit, and larger than
1/2^(n-1) for n-qubits system. The PTU transformation does not change the
eigenvalues of the n-qubits MDS density matrices for odd n. Thus the
Peres-Horodecki criterion does not give any information about entanglement of
these density matrices, but this criterion is useful for indicating
inseparability for even n. The changes of the entanglement and separability
properties of the GHZ state, the Braid entangled state and the W state by
mixing them with white noise are analyzed by the use of the present methods.
The entanglement and separability properties of the GHZ-diagonal density
matrices, composed of mixture of 8 GHZ density matrices with probabilities
p(i), is analyzed as function of these probabilities. In some cases we show
that the Peres-Horodecki criterion is both sufficient and necessary.
