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Equivalence between completely-positive-divisibility and information flow

Bogna Bylicka, Markus Johansson, Antonio Acín

Published Mar 14, 2016• 0 citations
OpenAlex

Abstract

We prove the equivalence between two approaches to the characterisation of quantum Markovianity, divisibility and lack of information backflow. We show that a dynamical map is completely-positive-divisible if and only if a monotonic non-increase of distinguishability is observed for two equiprobable states of the evolving system and an ancilla. Moreover our proof is constructive for invertible dynamical maps: given any such a map that is not completely-positive-divisible, we give an explicit construction of two states that exhibit information backflow. Finally, while an ancilla is necessary for the equivalence to hold in general, we show that it is always possible to witness the non-Markovianity of maps without using any entanglement between system and ancilla.

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We present a general framework for the information backflow (IB) approach of Markovianity that not only includes a large number, if not all, of IB prescriptions proposed so far but also is equivalent to completely positive divisibility for invertible evolutions. Following the common approach of IB, where monotonic decay of some physical property or some information quantifier is seen as the definition of Markovianity, we propose in our framework a general description of what should be called a proper ``physicality quantifier'' to define Markovianity. We elucidate different properties of our framework and use them to argue that an infinite family of non-Markovianity measures can be constructed, which would capture varied strengths of non-Markovianity in the dynamics. Moreover, we show that generalized trace-distance measure in two dimensions serve as a sufficient criteria for IB Markovianity for a number of prescriptions suggested earlier in the literature.

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We prove the equivalence between two approaches to the characterisation of quantum Markovianity, divisibility and lack of information backflow. We show that a dynamical map is completely-positive-divisible if and only if a monotonic non-increase of distinguishability is observed for two equiprobable states of the evolving system and an ancilla. Moreover our proof is constructive for invertible dynamical maps: given any such a map that is not completely-positive-divisible, we give an explicit construction of two states that exhibit information backflow. Finally, while an ancilla is necessary for the equivalence to hold in general, we show that it is always possible to witness the non-Markovianity of maps without using any entanglement between system and ancilla.
We investigate the relation between two approaches to the characterisation of quantum Markovianity, divisibility and lack of information backflow. We show that a bijective dynamical map is completely-positive-divisible if and only if a monotonic non-increase of distinguishability is observed for two equiprobable states of the evolving system and an ancilla. Moreover our proof is constructive: given any such map that is not completely-positive-divisible, we give an explicit construction of two states that, when taken with the same a priori probability, exhibit information back-flow. Finally, while an ancilla is necessary for the equivalence to hold in general, we show that it is always possible to witness the non-Markovianity of bijective maps without using any entanglement between system and ancilla.
We investigate the relation between two approaches to the characterization of quantum Markovianity, divisibility and lack of information backflow. We show that a bijective dynamical map is completely positive divisible if and only if a monotonic nonincrease of distinguishability is observed for two equiprobable states of the evolving system and an ancilla. Moreover, our proof is constructive: given any such map that is not completely positive divisible, we give an explicit construction of two states that, when taken with the same a priori probability, exhibit information backflow. Finally, while an ancilla is necessary for the equivalence to hold in general, we show that it is always possible to witness the non-Markovianity of bijective maps without using any entanglement between the system and ancilla.
We analyze the relation between completely positive (CP) divisibility and the lack of information backflow for an arbitrary-not necessarily invertible-dynamical map. It is well known that CP divisibility always implies a lack of information backflow. Moreover, these two notions are equivalent for invertible maps. In this Letter, it is shown that for a map which is not invertible the lack of information backflow always implies the existence of a completely positive propagator which, however, needs not be trace preserving. Interestingly, for a wide class of image nonincreasing dynamical maps, this propagator becomes trace preserving as well, and hence, the lack of information backflow implies CP divisibility. This result sheds new light into the structure of the time-local generators giving rise to CP-divisible evolutions. We show that if the map is not invertible then positivity of dissipation/decoherence rates is no longer necessary for CP divisibility.
We study the relation between lack of information backflow and completely positive divisibility for noninvertible qubit dynamical maps. Recently, these two concepts were shown to be fully equivalent for the so-called image nonincreasing dynamical maps. Here we show that this equivalence is universal for any qubit dynamical map. A key ingredient in our proof is the observation that there does not exist a completely positive and trace-preserving projector onto a three-dimensional subspace spanned by qubit density operators. Our analysis is illustrated by several examples of qubit evolution, including dynamical maps which are not image nonincreasing.
We present a general framework for the information backflow (IB) approach of Markovianity that not only includes a large number, if not all, of IB prescriptions proposed so far but also is equivalent to completely positive divisibility for invertible evolutions. Following the common approach of IB, where monotonic decay of some physical property or some information quantifier is seen as the definition of Markovianity, we propose in our framework a general description of what should be called a proper ``physicality quantifier'' to define Markovianity. We elucidate different properties of our framework and use them to argue that an infinite family of non-Markovianity measures can be constructed, which would capture varied strengths of non-Markovianity in the dynamics. Moreover, we show that generalized trace-distance measure in two dimensions serve as a sufficient criteria for IB Markovianity for a number of prescriptions suggested earlier in the literature.