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arXiv:2004.09264 · New Journal of Physics

Construction of propagators for divisible dynamical maps

Ujan Chakraborty, Dariusz Chruściński

Published Dec 16, 2020• 9 citations
arXivHTMLPDF

Abstract

Abstract Divisible dynamical maps play an important role in characterizing Markovianity on the level of quantum evolution. Divisible maps provide an important generalization of Markovian semigroups. Usually one analyzes either completely positive or just positive divisibility meaning that the corresponding propagators are defined in terms of completely positive or positive maps, respectively. For maps which are invertible at any moment of time the very existence of propagator is already guaranteed and hence the only issue is (complete) positivity and trace-preservation. However, for maps which are not invertible the problem is much more involved since even the existence of a propagator is not guaranteed. In this paper we propose a simple method to construct propagators of dynamical maps using the concept of generalized inverse. We analyze both time-continuous and time-discrete maps. Since the generalized inverse is not uniquely defined the same applies for the corresponding propagator. In simple examples of qubit evolution we analyze it turns out that the additional requirement of complete positivity possibly makes the propagator unique.

DOI
10.1088/1367-2630/abd43b

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Abstract Divisible dynamical maps play an important role in characterizing Markovianity on the level of quantum evolution. Divisible maps provide an important generalization of Markovian semigroups. Usually one analyzes either completely positive or just positive divisibility meaning that the corresponding propagators are defined in terms of completely positive or positive maps, respectively. For maps which are invertible at any moment of time the very existence of propagator is already guaranteed and hence the only issue is (complete) positivity and trace-preservation. However, for maps which are not invertible the problem is much more involved since even the existence of a propagator is not guaranteed. In this paper we propose a simple method to construct propagators of dynamical maps using the concept of generalized inverse. We analyze both time-continuous and time-discrete maps. Since the generalized inverse is not uniquely defined the same applies for the corresponding propagator. In simple examples of qubit evolution we analyze it turns out that the additional requirement of complete positivity possibly makes the propagator unique.
Abstract We propose and explore a notion of decomposably divisible (D-divisible) differentiable quantum evolution families on matrix algebras. This is achieved by replacing the complete positivity requirement, imposed on the propagator, by more general condition of decomposability. It is shown that such D-divisible dynamical maps satisfy a generalized version of Master equation and are totally characterized by their time-local generators. Necessary and sufficient conditions for D-divisibility are found. Additionally, decomposable trace preserving semigroups are examined.
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.
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.
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.
In this paper we study quantum dynamical mappings called also quantum processes.The set of values of such mapping is a one-parameter family of completely positive trace-preserving linear operators defined on a finite-dimensional Hilbert space.In quantum information theory such operators are referred to as quantum channels.An important concept for quantum dynamical mappings is their divisibility.There are different types of this concept.The present paper deals with so-called completely positive divisible quantum processes.For two such processes, which are bijective and satisfy a commutativity condition, we construct a compound quantum process.It is shown that this compound quantum process is also completely positive divisible.Endowing a set of quantum channels with the norm topology, we consider continuous quantum processes and continuous completely positive evolutions.The latter are defined as two-parameter families of quantum channels satisfying additional properties.We prove that a continuous bijective completely positive divisible quantum process generates a continuous completely positive evolution.In order to illustrate the considered concepts and the results on them, we provide examples of quantum dynamical mappings with values in the set of qubit channels.In particular, a completely positive divisible compound quantum process is constructed for two bijective commuting quantum processes.Geometric and physical interpretations of this compound quantum process are given.
Abstract It is shown that a large class of quantum dynamical maps on complex matrix algebras governed by time-local Master Equations tend to become entanglement breaking in the course of time. Such situation seems to be generic for quantum evolution and in particular, completely positive dynamical semigroups with a unique faithful stationary state enjoy this property. Inspired by this observation, we propose a new concept of eventually entanglement breaking divisible (eEB-divisible) dynamics. A dynamical map is eEB-divisible if any propagator becomes entanglement breaking in finite time. It turns out that eEB-divisibility is quite general and holds for a large class of quantum evolutions.