Decomposition of tracial positive maps and applications in quantum information

Abstract

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 $\Phi ={\phi }_2\circ {\phi }_1$, 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 $\Phi$ between $C^{\ast }$-algebra is a von Neumann algebra, then $\Phi$ 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.

Cited by in corpus (2)

• Banach bimodule-valued positive maps: Inequalities and induced representations
• Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces