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Hilbert-Schmidt (HS) decompositions are employed for analyzing systems ofn-qubits, and a qubit with a qudit. Negative eigenvalues, obtained bypartial-transpose (PT) plus local unitary transformations (PTU) for one qubitfrom the whole system, are used for indicating inseparability. A sufficientcriterion for full separability of the n-qubits and qubit-qudit systems isgiven. We use the singular value decomposition (SVD) for improving thecriterion for full separability. General properties of entanglement andseparability are analyzed for a system of a qubit and a qudit and n-qubitssystems, with emphasis on maximally disordered subsystems (MDS) (i.e., densitymatrices rho(MDS) for which tracing over any subsystem gives the unit densitymatrix). A sufficient condition that rho(MDS) is not separable is that it hasan eigenvalue larger than 1/d for a qubit and a qudit, and larger than1/2^(n-1) for n-qubits system. The PTU transformation does not change theeigenvalues of the n-qubits MDS density matrices for odd n. Thus thePeres-Horodecki criterion does not give any information about entanglement ofthese density matrices, but this criterion is useful for indicatinginseparability for even n. The changes of the entanglement and separabilityproperties of the GHZ state, the Braid entangled state and the W state bymixing them with white noise are analyzed by the use of the present methods.The entanglement and separability properties of the GHZ-diagonal densitymatrices, composed of mixture of 8 GHZ density matrices with probabilitiesp(i), is analyzed as function of these probabilities. In some cases we showthat the Peres-Horodecki criterion is both sufficient and necessary.

Separability and entanglement of n-qubits and a qubit with a qudit using Hilbert-Schmidt decompositions