Compositions and tensor products of linear maps between matrix algebras

TL;DR

This paper explores the relationships between positive maps, tensor products, and mapping cones in quantum information theory, providing new identities and criteria for understanding entanglement and related properties in matrix algebras.

Contribution

It introduces an identity linking tensor products and compositions of linear maps, enabling new characterizations of dual cones and criteria for quantum entanglement.

Findings

Derived an identity connecting tensor products and compositions of linear maps.

Characterized dual cones in terms of mapping cones and tensor products.

Provided new criteria related to the PPT square conjecture.

Abstract

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.