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Could any one give me an example of a state whose density matrix is positive semidefinnite but partial transpose is not positive semidefinnite?

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If you want an example of a quantity of entanglement, it's always a good idea to check the qubit maximally entangled Bell states first. Here, this immediately gives you the answer.

A proof that this state does what you want is that in the qubit case (i.e. four dimensional Hilbert space of the joint system qubit x qubit) the Peres criterion states that separable states are exactly the states that have a positive partial transpose. Since any state is positive, this implies that any entangled state gives you an example of a positive matrix with nonpositive partial transpose.

Of course, you can also just write down the state and its partial transpose.

Other important examples are some of the Werner states, see e.g. Can isotropic states have bound entanglement?

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  • $\begingroup$ It's worth mentioning that in general (i.e. non 2x2 or 2x3 systems), PPT does not imply separability. In fact, there are entangled states that are PPT, i.e. bound entangled states. Of course, if the state has negative partial transpose, then for sure it's entangled. $\endgroup$ – vsoftco Oct 20 '16 at 14:14

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