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arXiv:2610.07986

Divisibility of Non-Invertible Quantum Dynamics in the Heisenberg Picture

Federico Settimo, Dariusz Chruściński

quant-phSubmitted Oct 6, 2026• Announced Oct 7, 2026• 0 citations
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Abstract

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.

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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.
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.
Divisibility of dynamical maps is a central notion in the study of quantum non-Markovianity, providing a natural framework to characterize memory effects via time-local master equations. In this work, we generalize the notion of divisibility of quantum dynamical maps from the Schrödinger to the Heisenberg picture. While the two pictures are equivalent at the level of physical predictions, we show that the divisibility properties of the corresponding dual maps are, in general, not equivalent. This inequivalence originates from the distinction between left and right generators of time-local master equations, which interchange roles under duality. We demonstrate that Schrödinger and Heisenberg divisibility are distinct concepts by constructing explicit dynamics divisible only in one picture. Furthermore, we introduce a quantifier for the violation of Heisenberg P-divisibility, analogous to the trace-distance-based measure of non-Markovianity, and provide it with an operational interpretation in terms of the guessing probability between effects. Our results show that Heisenberg divisibility is an independent witness of memory effects and highlight the need to consider both pictures when characterizing non-Markovian quantum dynamics.
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.
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.
We analyze the divisibility properties of generalized Pauli dynamical maps. Using the results for image non-increasing dynamical maps, we formulate the conditions for the underlying evolution to satisfy specific non-Markovianity criteria. For the qubit evolution, we find the necessary and sufficient conditions for P and CP-divisibility of the associated, in general noninvertible, Pauli dynamical maps. Finally, we analyze the divisibility degree for mixtures of noninvertible phase-damping channels. For P-divisible maps, we propose a legitimate time-local generator with all temporarily infinite decoherence rates.