Information theoretic treatment of tripartite systems and quantum channels

Abstract

A Holevo measure is used to discuss how much information about a given 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 \emph{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: e.g., if $\EC$ accurately transmits certain POVMs, the complementary channel $\FC$ 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 $c$.

21 pages. An earlier version of this paper attempted to prove our main uncertainty relation, Theorem 5, using the achievability of the Holevo quantity in a coding task, an approach that ultimately failed because it did not account for locking of classical correlations, e.g. see [DiVincenzo et al. PRL. 92, 067902 (2004)]. In the latest version, we use a very different approach to prove Theorem 5

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