Information-theoretic treatment of tripartite systems and quantum channels

A Holevo measure is used to discuss how much information about a given positive operator valued measure (POVM) on system $a$ is present in another system $b$, and how this influences the presence or absence of information about a different POVM on $a$ in a third system $c$. The main goal is to extend information theorems for mutually unbiased bases or general bases to arbitrary POVMs, and especially to generalize “all-or-nothing” theorems about information located in tripartite systems to the case of partial information, in the form of quantitative inequalities. Some of the inequalities can be viewed as entropic uncertainty relations that apply in the presence of quantum side information, as in recent work by Berta et al. [Nature Physics 6, 659 (2010)]. All of the results also apply to quantum channels: For example, if $\mathcal{E}$ accurately transmits certain POVMs, the complementary channel $\mathcal{F}$ will necessarily be noisy for certain other POVMs. While the inequalities are valid for mixed states of tripartite systems, restricting to pure states leads to the basis invariance of the difference between the information about $a$ contained in $b$ and $c$.

Figure 1

How $|\Omega \rangle$ is produced by applying the isometry $V$ to an entangled state $|\Phi \rangle$.