Frontier between separability and quantum entanglement in a many spin system

We discuss the critical point $x_c$ separating the quantum entangled and separable states in two series of N spins S in the simple mixed state characterized by the matrix operator $ρ=x|\tildeϕ><\tildeϕ| + \frac{1-x}{D^N} I_{D^N}$ where $x \in [0,1]$, $D =2S+1$, ${\bf I}_{D^N}$ is the $D^N \times D^N$ unity matrix and $|\tilde ϕ>$ is a special entangled state. The cases x=0 and x=1 correspond respectively to fully random spins and to a fully entangled state. In the first of these series we consider special states $|\tildeϕ>$ invariant under charge conjugation, that generalizes the N=2 spin S=1/2 Einstein-Podolsky-Rosen state, and in the second one we consider generalizations of the Weber density matrices. The evaluation of the critical point $x_c$ was done through bounds coming from the partial transposition method of Peres and the conditional nonextensive entropy criterion. Our results suggest the conjecture that whenever the bounds coming from both methods coincide the result of $x_c$ is the exact one. The results we present are relevant for the discussion of quantum computing, teleportation and cryptography.

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