Decomposition of tracial positive maps and applications in quantum information

Every positive multilinear map between $C^{\ast }$-algebras is separately weak${}^{\ast }$-continuous. We show that the joint weak${}^{\ast }$-continuity is equivalent to the joint weak${}^{\ast }$-continuity of the multiplications of $C^{\ast }$-algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron–Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general $C^{\ast }$-algebras enjoys a decomposition $\mathrm{\Phi }={\phi }_2\circ {\phi }_1$, in which ${\phi }_1$ is a tracial positive linear map with the commutative range and ${\phi }_2$ is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map $\mathrm{\Phi }$ between $C^{\ast }$-algebra is a von Neumann algebra, then $\mathrm{\Phi }$ has a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics via arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented.