  In this semi-expository paper, we first explain key notions from current
quantum information theory and criteria for them in a coherent way. These
include separability/entanglement, Schmidt numbers of bi-partite states and
block-positivity, together with various kinds of positive maps between matrix
algebras like entanglement breaking maps, $k$-superpositive maps, completely
positive maps, $k$-positive maps. We will begin with concrete examples of
elementary positive maps given by $x\mapsto s^*xs$, and use Choi matrices and
duality to explain all the notions mentioned above. We also show that the Choi
matrix can be defined free from coordinates. The above notions of positive maps
give rise to mapping cones, whose dual cones are characterized in terms of
compositions or tensor products of linear maps. Through the discussion, we
exhibit an identity which connects tensor products and compositions of linear
maps between matrix algebras through the Choi matrices. Using this identity, we
show that the description of the dual cone with tensor products is possible
only when the involving cones are mapping cones, and recover various known
criteria with ampliation for the notions mentioned above. As another
applications of the identity, we construct various mapping cones arising from
ampliation and factorization, and provide several equivalent statements to PPT
(positive partial transpose) square conjecture in terms of tensor products.
