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Many properties of partial transposition are unclear as yet. Here we carefully consider the number of the negative eigenvalues of ?T (?'s partial transposition) when ? is a two-partite state. There is strong evidence to show that the number of negative eigenvalues of ?T is N(N ? 1)/2 at most when ? is a state in Hilbert space CN ? CN. For the special case, the 2 ? 2 system, we use this result to give a partial proof of the conjecture |?T|T ? 0. We find that this conjecture is strongly connected with the entanglement of the state corresponding to the negative eigenvalue of ?T or the negative entropy of ?.

Natural thermal entanglement between two qubits with XXX Heisenberg interaction is studied. For the antiferromagnet, increasing coupling strength or decreasing temperature under critical point increases the entanglement. Based on the thermal entanglement as quantum channel, entanglement and information of an input entangled state are transferred via partial teleportation. We find that the entanglement transferred will be lost during the process, and for the entanglement fidelity the partial teleportation is superior to classical communication as concurrence of entangled channel beyond 1/4. We show that both correlation information in input entangled state and individual information of the teleported particle are linearly dissipated. With more entanglement in quantum channel, more entanglement and correlation information can be transferred.

It is well known that the positive partial transpose (PPT) criterion for determining separability is an operational separability criterion. However, in a high-dimensional (>6) situation, this criterion is not sufficient. How can we judge whether an entangled state in any high-dimensional quantum system is PPT or not? Here we propose a linear algebra method for checking the positivity of the partial transpose of a state and present a set of entangled mixed states with non-positive partial transpose.

We propose a scheme to generate a 3×3-dimensional maximally entangled state of two particles. Two three-level atoms interact with a strongly detuned cavity so that the cavity is only virtually excited and efficient decoherence time of the cavity is greatly prolonged. Compared to other protocols, this protocol is simpler and has a higher fidelity.