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arXiv:2407.16583 · Journal of Physics A Mathematical and Theoretical

Eventually entanglement breaking divisible quantum dynamics

Krzysztof Szczygielski, Dariusz Chruściński

Published Nov 15, 2024• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ad9368

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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.
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.
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We obtain a complete characterization of eventually entanglement breaking (EEB) unital channels in terms of some simple commutation properties of their eigenvectors. Specifically, the following properties are equivalent for a unital channel: (i) it is EEB; (ii) it is eventually positive partial transpose (PPT); (iii) it is eventually mixed twisted dephasing; and (iv) its eigenvectors corresponding to non-zero eigenvalues commute with all its peripheral eigenvectors. Similar results hold for one-parameter semigroups of unital quantum channels, where continuity forces such semigroups to become primitive. We employ a matrix integral technique inspired by a result of Watrous. We also characterize the extreme points of the convex set of unital EB channels and, using this characterization and the theory of equiangular tight frames (ETFs), identify several extreme points beyond the twisted dephasing channels.
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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.
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.