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arXiv:1710.06771 · Physical Review Letters

Divisibility and Information Flow Notions of Quantum Markovianity for Noninvertible Dynamical Maps

Dariusz Chruściński, ́Angel Rivas, Erling Størmer

Published Aug 24, 2018• 74 citations
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Abstract

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.

DOI
10.1103/physrevlett.121.080407

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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 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.
The are several nonequivalent notions of Markovian quantum evolution. In this paper we show that the one based on the so-called CP divisibility of the corresponding dynamical map enjoys the following stability property: the dynamical map ${\mathrm{\ensuremath{\Lambda}}}_{t}$ is CP divisible if and only if the second tensor power ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ is CP divisible as well. Moreover, the P divisibility of the map ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ is equivalent to the CP divisibility of the map ${\mathrm{\ensuremath{\Lambda}}}_{t}$. Interestingly, the latter property is no longer true if we replace the P divisibility of ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ by simple positivity and the CP divisibility of ${\mathrm{\ensuremath{\Lambda}}}_{t}$ by complete positivity. That is, unlike when ${\mathrm{\ensuremath{\Lambda}}}_{t}$ has a time-independent generator, positivity of ${\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}$ does not imply complete positivity of ${\mathrm{\ensuremath{\Lambda}}}_{t}$.
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 study divisibility of finite-dimensional quantum dynamics in the Heisenberg picture when the dynamical maps need not be invertible. Heisenberg divisible dynamics determines unique reduced propagators on the accessible operator systems, and Heisenberg CP-divisibility is shown to be equivalent to an operator-norm monotonicity. Although each reduced propagator admits a unital completely positive extension, independently chosen extensions need not satisfy a composition law. Generalized inverses restore this law algebraically, but do not, in general, preserve complete positivity. We characterize the resulting obstruction and give a sufficient construction based on unital completely positive projections onto the accessible operator systems. It is shown that for qubit dynamics Heisenberg CP-divisibility always guarantees a completely positive generalized-inverse propagator. Finally, we analyze time-local generators at singular times and identify the associated gauge freedom on dynamically inaccessible directions.