Completely positive maps and classical correlations

Abstract

We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)]. We prove that initial states that have only classical correlations lead to completely positive reduced dynamics. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We outline the implications of our results to quantum process tomography experiments.

Necessary and sufficient condition for saturating the upper bound of quantum discord

“…Quantum discord was viewed as a figure of merit for characterizing the nonclassical resources in the deterministic quantum computation with one-qubit [6,7]. It was also discussed that zero-discord of the initial system-environment states is a necessary and sufficient condition for completely positivity of reduced dynamical maps [8,9]. At the same time, a necessary and sufficient condition for nonzero-discord was also given for any dimensional bipartite states [10].…”

Negativity and quantum discord in Davies environments

“…This classification is non-trivial, as e.g. classically correlated states always lead to completely positive maps while states with quantum correlations may give rise to non-completely positive maps [6]. For our set-up here here, classically correlated states exhibit zero quantum discord D(t) = 0, while states with quantum correlations exhibit a non-zero discord D(t) ≥ 0.…”

“…Let us remark that, generally, this quantum discord (20) presents neither a unique nor the most optimal quantifier for quantum correlation [5]. However, for the case of bipartite systems one can summarize that the states can be divided into two groups [6]; namely entangled (quantum correlated) and separable states. In turn, the separable states can either be classically correlated or quantum correlated (but not entangled).…”

An Introduction to Operational Quantum Dynamics

“…Proving that the dynamics of an open system for a particular initial state is CP amounts to showing that it can be written in terms of a Kraus decomposition [46,4,82,74], and the existence of a minimal Kraus decomposition can be employed to show the existence of generalised Stinespring dilations [19,20].…”

“…Research along these lines led to the claim that "vanishing quantum discord is necessary and sufficient for completely positive maps" [86] which received a great deal of attention, but then was subsequently proven to be incorrect [12], leading to an erratum [87]. In [82], it was shown that if the initial se state has vanishing quantum discord, then a CP map can be ascribed to the dynamics of s. Consequently, by projectively measuring the system part of any initial state ρ 0 se -which will always produce a discord zero state -one can associate a CP map from the measurement outcome at the initial time to the quantum state at the final time. The problem with this approach is that the CP maps depend on the choice of measurement, which does not depend on the pre-measurement state of the system.…”