# Divisibility of qubit channels and dynamical maps

David Davalos1, Mario Ziman2,3, and Carlos Pineda1,4

1Instituto de Física, Universidad Nacional Autónoma de México, México, D.F., México
2Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia
3Faculty of Informatics, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic
4Faculty of Physics, University of Vienna, 1090 Vienna, Austria

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. [25]) 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.~[26]. 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.

Note: The originally published version was 1812.11437v2, after authors noticed a typo it was updated to 1812.11437v3.

Quantum channels are the basic mathematical objects to describe noisy manipulations of quantum information, irreversible quantum dynamics and open quantum systems. Thus, the study of their mathematical properties is crucial to understand how quantum systems evolve. Time evolutions of quantum systems, so-called quantum dynamical maps, are associated with continuous curves in the space of quantum channels,. Surprisingly, there are quantum channels that cannot be understood as concatenations of the others. From the perspective of continuous time the existence of such indivisible quantum processes is a bit perplexing, especially how the indivisibility is reflected by properties of the generating quantum dynamical maps.

We address the problem of which channels can be implemented as concatenation of other channels. We define several divisibility types, and investigate how they are related with the implementability of quantum channels using dynamical maps. To illustrate this, think in the following example. The flip of the direction of the magnetization of a spin might be understood as a discrete operation. But we can ask for the dynamical process that changes continuously the direction of the spin, and what mathematical properties the process must fulfill.

In the context of quantum information and open quantum systems, the dynamical maps are typically presented in the form of master equations generalizing the concept of Schroedinger equation determining the evolution of closed quantum systems. The most known and studied are the so-called Markovian quantum evolutions. The concept of Markovianity is closely related to the concept of channel divisibility. In particular, being a result of Markovian evolution implies the channel can be divided into an infinite number of quantum channels that can be chosen to be infinitesimal (i.e. they describe arbitrary small changes on the system).

Our contribution includes the analysis of tests deciding on the Markovianity features of quantum channels, their connections to different types of divisibility and entanglement. We also question the transitions between different types of Markovianity and divisibility. By analysing particular quantum dynamical maps we showed that all transitions are possible. We give an extensive characterization of the channels acting on the qubit system. Seizing the mathematical simplicity of the qubit channels that leave invariant the state with highest entropy, we present visually the structure and symmetries of channels enjoying each studied type of divisibility. We found that divisible but not infinitesimal divisible qubit channels are necessarily entanglement-breaking, i.e. they destroy all originally existing entanglement.

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### Cited by

[1] Sagnik Chakraborty and Dariusz Chruściński, "Information flow versus divisibility for qubit evolution", Physical Review A 99 4, 042105 (2019).

[2] Zbigniew Puchała, Łukasz Rudnicki, and Karol Życzkowski, "Pauli semigroups and unistochastic quantum channels", Physics Letters A 383 20, 2376 (2019).

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