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arXiv:1610.04634 · Physical Review A

Tensor power of dynamical maps and positive versus completely positive divisibility

Fabio Benatti, Dariusz Chruściński, Sergey N. Filippov

Published Jan 11, 2017• 33 citations
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Abstract

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 ${\mathrm{\ensuremath{\Lambda}}}_{t}$.

DOI
10.1103/physreva.95.012112

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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 \ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​ is CP divisible if and only if the second tensor power \ensuremathΛt\ensuremath⨂\ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​\ensuremath⨂\ensuremathΛt​ is CP divisible as well. Moreover, the P divisibility of the map \ensuremathΛt\ensuremath⨂\ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​\ensuremath⨂\ensuremathΛt​ is equivalent to the CP divisibility of the map \ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​. Interestingly, the latter property is no longer true if we replace the P divisibility of \ensuremathΛt\ensuremath⨂\ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​\ensuremath⨂\ensuremathΛt​ by simple positivity and the CP divisibility of \ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​ by complete positivity. That is, unlike when \ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​ has a time-independent generator, positivity of \ensuremathΛt\ensuremath⨂\ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremath{\bigotimes}{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​\ensuremath⨂\ensuremathΛt​ does not imply complete positivity of \ensuremathΛt{\mathrm{\ensuremath{\Lambda}}}_{t}\ensuremathΛt​.
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 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.
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 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 a method to construct non-Markovian variants of completely positive (CP) dynamical maps, particularly, qubit Pauli channels. We identify non-Markovianity with the breakdown in CP divisibility of the map, i.e., appearance of a not-completely positive intermediate map. In particular, we consider the case of non-Markovian dephasing in detail. The eigenvalues of the Choi matrix of the intermediate map crossover at a point which corresponds to a singularity in the canonical decoherence rate of the corresponding master equation and thus to a momentary noninvertibility of the map. Thereafter, the rate becomes negative, indicating non-Markovianity. We quantify the non-Markovianity by two methods, one based on CP divisibility [Hall et al., Phys. Rev. A 89, 042120 (2014)], which does not require optimization but requires normalization to handle the singularity, and another method, based on distinguishability [Breuer et al. Phys. Rev. Lett. 103, 210401 (2009)], which requires optimization but is insensitive to the singularity.