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arXiv:2309.10806 · Physical Review A

Relating completely positive divisibility of dynamical maps with compatibility of channels

Arindam Mitra, Debashis Saha, Samyadeb Bhattacharya, A. S. Majumdar

Published Jun 13, 2024• 2 citations
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Abstract

The role of completely positive--indivisibility (CP-indivisibility) and incompatibility as valuable resources for various information-theoretic tasks is widely acknowledged. This study delves into the intricate relationship between CP-divisibility and channel compatibility. Our investigation focuses on the behavior of incompatibility robustness of quantum channels for a pair of generic dynamical maps. We show that the incompatibility robustness of channels is monotonically nonincreasing for a pair of generic CP-divisible dynamical maps. Further, our explicit study of the behavior of incompatibility robustness with time for some specific dynamical maps reveals nonmonotonic behavior in the CP-indivisible regime. Additionally, we propose a measure of CP-indivisibility based on the incompatibility robustness of quantum channels. Our investigation provides valuable insights into the nature of quantum dynamical maps and their relevance in information-theoretic applications.

DOI
10.1103/physreva.109.062213

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The role of completely positive--indivisibility (CP-indivisibility) and incompatibility as valuable resources for various information-theoretic tasks is widely acknowledged. This study delves into the intricate relationship between CP-divisibility and channel compatibility. Our investigation focuses on the behavior of incompatibility robustness of quantum channels for a pair of generic dynamical maps. We show that the incompatibility robustness of channels is monotonically nonincreasing for a pair of generic CP-divisible dynamical maps. Further, our explicit study of the behavior of incompatibility robustness with time for some specific dynamical maps reveals nonmonotonic behavior in the CP-indivisible regime. Additionally, we propose a measure of CP-indivisibility based on the incompatibility robustness of quantum channels. Our investigation provides valuable insights into the nature of quantum dynamical maps and their relevance in information-theoretic applications.
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.
Open quantum dynamics can be categorized in several ways, including according to their divisibility. For any quantum channel acting for any finite time period, we propose measures of P-indivisibility and CP-indivisibility, where P and CP stand for positivity and complete positivity respectively. Subsequently, we also propose two quantities to measure the resourcefulness - with respect to an arbitrary quantum resource - of any quantum channel within any finite time interval. Moreover, we find a bridge between these two classes of metrics, viz. those quantifying divisibilities of quantum channels and those gauging their resourcefulness, by identifying two separate relations between elements of one class with those of the other. Lastly, we verify the two relations using quantum non-Markovianity as a channel resource.
The divisibility of dynamical maps is visualized by trajectories in the parameter space and analyzed within the framework of collision models. We introduce ultimate completely positive (CP) divisible processes, which lose CP divisibility under infinitesimal perturbations, and characterize Pauli dynamical semigroups exhibiting such a property. We construct collision models with factorized environment particles, which realize additivity and multiplicativity of generators of CP divisible maps. A mixture of dynamical maps is obtained with the help of correlated environment. The mixture of ultimate CP divisible processes is shown to result in a class of eternal CP indivisible evolutions. We explicitly find collision models leading to weakly and essentially non-Markovian Pauli dynamical maps.
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 ${\mathrm{\ensuremath{\Lambda}}}_{t}$.
The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the simulability of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al. \cite{cirac}) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets and show several equivalences for qubit channels. To this end we study the conditions of L-divisibility for finite dimensional channels, especially the cases with negative eigenvalues, extending and completing the results of Ref.~\cite{Wolf2008}. Furthermore we show that transitions between every two of the defined divisibility sets are allowed. We explore particular examples of dynamical maps to compare these concepts. Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.
We introduce map-dependent quantum characteristic functions constructed from the normalized Choi operator of quantum dynamical maps. We prove a Bochner--Choi positivity theorem establishing that the positive-type condition of the associated Gram matrix is equivalent to complete positivity of the underlying quantum channel. Applying the construction to intermediate dynamical maps, we obtain a characterization of CP-divisibility in terms of positivity of two-time characteristic functions. Numerical examples for amplitude damping and pure dephasing models demonstrate that negativity of the Gram matrix coincides with the breakdown of CP-divisibility and the emergence of information backflow. The proposed framework provides a new bridge between characteristic-function methods in quantum statistics and structural properties of quantum dynamical maps.