SOT-separability For Positive Operators On Tensor Product Of Hilbert Spaces

In this paper,the definition of the general separable operators on the tensor product space of two Hilbert spaces with infinite-dimension is introduced,and more attention is paid to the SOT-separable positive operators.Two sufficient and necessary criteria of detecting the SOT-separability are established,that is,the SOT-inseparability witness criterion and the positive elementary operator criterion.The relations between different SOT-inseparability witnesses is discussed and the necessary and sufficient conditions for two witnesses to be comparable,to be equivalent,to be optimal,to detect some common SOT-inseparable positive operators,to detect no common SOT-inseparable positive operators are given respectively.As an application to the quantum information theory,a new class of separable quantum states,that is,the semi-SSPPT states,are constructed.In addition,the WOT-separability,UWT-separability and UT-separability of positive operators are also discussed.The criteria for SOT-separable operators can be used to detect the WOT-separ ability and UWT-separ ability of positive operators,while the positive partial transpose(PPT)criterion for UT-separ ability is established.A class of UT-separable operators called SSPPT operators are constructed.