A classical non-signalling (or causal) box is an operation on classical bipartite input with classical bipartite output such that no signal can be sent from a party to the other through the use of the box. The quantum counterpart of such boxes, i.e. completely positive trace-preserving maps on bipartite states, though studied in literature, have been investigated less intensively than classical boxes. We present here some results and remarks about such maps. In particular, we analyze: the relations among properties as causality, non-locality and entanglement; the connection between causal and entanglement breaking maps; the characterization of causal maps in terms of the classification of states with fixed reductions. We also provide new proofs of the fact that every non-product unitary transformation is not causal, as well as for the equivalence of the so-called semicausality and semilocalizability properties.

We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the difference between positive maps on M 2 ( ) and M 3 ( ). Our main tool stems from classical Grothendieck theorem on tensor product of Banach spaces and is an older and more general version of Choi-Jamiołkowski isomorphism between positive maps and block positive Choi matrices. It takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps. In this setting we show that in M 2 ( ) case a large class of nice positive maps can be generated from the small set of maps represented by self-adjoint unitaries, 2P x with x maximally entangled vector and p⊗½ with p rank 1 projector. We show why this construction fails in M 3 ( ) case. There are also similarities. In both M 2 ( ) and M 3 ( ) cases any unital positive map represented by selfadjoint unitary is unitarily equivalent to the transposition map. Consequently we obtain a large family of exposed maps. We also investigate a convex structure of the Choi map, the first example of non-decomposable map. As a result the nature of the Choi map will be explained. This gives an information on the origin of appearance of non-decomposable maps on M 3 ( ).

We provide mathematicaly rigorous justification of using term probability in connection to the so called non-signalling theories, known also as Popescu's and Rohrlich's box worlds. No only do we prove correctness of these models (in the sense that they describe composite system of two independent subsystems) but we obtain new properties of non-signalling boxes and expose new tools for further investigation. Moreover, it allows strightforward generalization to more complicated systems.