Separability and entanglement of n-qubit and a qubit and a qudit using Hilbert–Schmidt decompositions
Abstract
Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of n-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the n-qubit 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-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that ρ (MDS) is not separable is that it has an eigenvalue larger than 1∕d for a qubit and a qudit, and larger than 1∕2n−1 for n-qubit system. The PTU transformation does not change the eigenvalues of the n-qubit MDS density matrices for odd n. Thus, the Peres–Horodecki (PH) criterion does not give any information about entanglement of these density matrices. The PH criterion may be 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 8GHZ density matrices with probabilities pi(i=1,2,…,8), is analyzed as function of these probabilities. In some cases, we show that the PH criterion is both sufficient and necessary.
