Vanishing quantum discord is not necessary for completely positive maps

Abstract

The description of the dynamics of a system that may be correlated with its environment is only meaningful within the context of a specific framework. Different frameworks rely upon different assumptions about the initial system-environment state. We reexamine the connections between complete-positivity and quantum discord within two different sets of assumptions about the relevant family of initial states. We present an example of a system-environment state with non-vanishing quantum discord that leads to a completely-positive map. This invalidates an earlier claim on the necessity of vanishing quantum discord for completely-positive maps. In our final remarks we discuss the physical validity of each approach.

Colloquium: Non-Markovian dynamics in open quantum systems

“…In this framework the aim has been to determine under which conditions the system dynamics can be described by means of completely positive maps if initial correlations are present. Unfortunately, no generally acknowledged answer to this question has been found and the topic is still under discussion (Brodutch et al, 2013;Rodríguez-Rosario et al, 2010;Shabani and Lidar, 2009).…”

Complete Positivity, Markovianity, and the Quantum Data-Processing Inequality, in the Presence of Initial System-Environment Correlations

“…As it turns out, quantum operations can always be modeled as interactions of the input system with an environment, initially factorized from (and independent of) the input system, and discarded after the interaction took place [1][2][3][4][5]. Such a model, however, is not universally valid, but relies on an initial factorization condition.The question then naturally arises [6][7][8][9][10][11][12][13][14][15][16]: what happens when the initial factorization condition does not hold, namely, when system and environment are, already before the interaction is turned on, correlated? While this question arguably originated from practical motivations (e.g., the difficulty to experimentally enforce the initial factorization assumption), it soon moved to a more fundamental level, in an attempt to challenge the very physical arguments often put forth to promote CP dynamics as the only "physically reasonable" reduced dynamics.…”

“…As one would expect, by allowing the input system and its environment to start in a correlated state, it is possible that the reduced dynamics of the system are not CP anymore. The possibility of exploring phenomena outside the CP framework attracted considerable interest [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], in particular in connection with the possibility of circumventing thermodynamic or information-theoretic tenet like, e.g., the second law of thermodynamics (by anomalous heat flow [17,18]) or the data-processing inequality (by anomalous increase of distinguishability [20], by entanglement revivals via local operations [9,[21][22][23], or by violating the no-cloning theorem [19]). In the language of the theory of open quantum systems, all such violations are interpreted as signatures of the fact that the underlying global evolution is non-divisible [20][21][22][23][24], i.e., it cannot be decomposed into a chain of CP maps across successive time intervals.…”

“…[13], where it was proved that zero discord is sufficient for CP reduced dynamics, Ref. [14] later claimed the two conditions to be equivalent, but it turned out that the equivalence only holds for restricted spaces of initial states [15]. A rigorous framework for this problem has recently been provided in Ref.…”

An Introduction to Operational Quantum Dynamics

“…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.…”