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In this semi-expository paper, we first explain key notions from currentquantum information theory and criteria for them in a coherent way. Theseinclude separability/entanglement, Schmidt numbers of bi-partite states andblock-positivity, together with various kinds of positive maps between matrixalgebras like entanglement breaking maps, $k$-superpositive maps, completelypositive maps, $k$-positive maps. We will begin with concrete examples ofelementary positive maps given by $x\mapsto s^*xs$, and use Choi matrices andduality to explain all the notions mentioned above. We also show that the Choimatrix can be defined free from coordinates. The above notions of positive mapsgive rise to mapping cones, whose dual cones are characterized in terms ofcompositions or tensor products of linear maps. Through the discussion, weexhibit an identity which connects tensor products and compositions of linearmaps between matrix algebras through the Choi matrices. Using this identity, weshow that the description of the dual cone with tensor products is possibleonly when the involving cones are mapping cones, and recover various knowncriteria with ampliation for the notions mentioned above. As anotherapplications of the identity, we construct various mapping cones arising fromampliation and factorization, and provide several equivalent statements to PPT(positive partial transpose) square conjecture in terms of tensor products.

Compositions and tensor products of linear maps between matrix algebras