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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.
The use of entanglement witness (EW) for non-full separability and the Bell operator for non-local hidden-variables (LHV) model are analyzed by relating them to the Hilbert-Schmidt (HS) decomposition of n-qubits states and these methods are applied e
The entanglement witness is an important and experimentally applicable tool for entanglement detection. In this paper, we provide a nonlinear improvement of any entanglement witness for $2otimes d$ quantum systems. Compared with any existing entangle
The practically useful criteria of separable states $rho=sum_{k}w_{k}rho_{k}$ in $d=2times2$ are discussed. The equality $G({bf a},{bf b})= 4[langle psi|P({bf a})otimes P({bf b})|psirangle-langle psi|P({bf a})otimes{bf 1}|psiranglelangle psi|{bf 1}ot
The generation and control of quantum correlations in high-dimensional systems is a major challenge in the present landscape of quantum technologies. Achieving such non-classical high-dimensional resources will potentially unlock enhanced capabilitie
We analyze entanglement and nonlocal properties of the convex set of symmetric $N$-qubits states which are diagonal in the Dicke basis. First, we demonstrate that within this set, positivity of partial transposition (PPT) is necessary and sufficient