A natural operational paradigm for distributed quantum and classical
information processing involves local operations coordinated by multiple rounds
of public communication. In this paper we consider the minimum number of
communication rounds needed to perform the locality-constrained task of
entanglement transformation and the analogous classical task of secrecy
manipulation. Specifically we address whether bipartite mixed entanglement can
always be converted into pure entanglement or whether unsecure classical
correlations can always be transformed into secret shared randomness using
local operations and a bounded number of communication exchanges. Our main
contribution in this paper is an explicit construction of quantum and classical
state transformations which, for any given r, can be achieved using r
rounds of classical communication exchanges but no fewer. Our results reveal
that highly complex communication protocols are indeed necessary to fully
harness the information-theoretic resources contained in general quantum and
classical states. The major technical contribution of this manuscript lies in
proving lower bounds for the required number of communication exchanges using
the notion of common information and various lemmas built upon it. We propose a
classical analog to the Schmidt rank of a bipartite quantum state which we call
the secrecy rank, and we show that it is a monotone under stochastic local
classical operations.Comment: Submitted to QIP 2017. Proof strategies have been streamlined and
differ from the submitted versio
