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arXiv:2402.04415 · Physical Review A

Non-Markovian quantum dynamics from symmetric measurements

Katarzyna Siudzińska

Published Jul 15, 2024• 5 citations
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Abstract

We use symmetric measurement operators to construct quantum channels that provide a further generalization of generalized Pauli channels. The resulting maps are bistochastic but, in general, no longer mixed unitary. We analyze their important properties, such as complete positivity and the ability to break quantum entanglement. In the main part, we consider the corresponding open quantum systems dynamics with time-local generators. From the divisibility properties of dynamical maps, we derive sufficient Markovianity and non-Markovianity conditions. As instructive examples, we present the generators of P-divisible generalized Pauli dynamical maps that allow for more negativity in the decoherence rates.

DOI
10.1103/physreva.110.012440

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We use symmetric measurement operators to construct quantum channels that provide a further generalization of generalized Pauli channels. The resulting maps are bistochastic but, in general, no longer mixed unitary. We analyze their important properties, such as complete positivity and the ability to break quantum entanglement. In the main part, we consider the corresponding open quantum systems dynamics with time-local generators. From the divisibility properties of dynamical maps, we derive sufficient Markovianity and non-Markovianity conditions. As instructive examples, we present the generators of P-divisible generalized Pauli dynamical maps that allow for more negativity in the decoherence rates.
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.
Abstract We analyze the connections between the non-Markovianity degree of the most general phase-damping qubit maps and their legitimate mixtures. Using the results for image non-increasing dynamical maps, we formulate the necessary and sufficient conditions for the Pauli maps to satisfy specific divisibility criteria. Next, we examine how the non-Markovianity properties for (in general noninvertible) Pauli dynamical maps influence the properties of their convex combinations. Our results are illustrated with instructive examples. For P-divisible maps, we propose a legitimate time-local generator whose all decoherence rates are temporarily infinite.
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.
We introduce a necessary and sufficient criterion for the non-Markovianity of Gaussian quantum dynamical maps based on the violation of divisibility. The criterion is derived by defining a general vectorial representation of the covariance matrix which is then exploited to determine the condition for the complete positivity of partial maps associated with arbitrary time intervals. Such construction does not rely on the Choi-Jamiolkowski representation and does not require optimization over states.
We discuss fundamental features of the local expression for the entropy production rate of the nonequilibrium quantum dynamics of open systems and its relations to memory effects and the spectrum of the generator of the dynamics. Defining the entropy production rate as negative rate of change of the relative entropy with respect to an instantaneous fixed point, it is shown that positivity of the entropy production rate for all possible initial states implies that the real parts of the eigenvalues of the time-local generator for the quantum master equation are always negative. It is further demonstrated that Markovian dynamics, identified as P-divisibility of the quantum dynamical map, implies positivity of entropy production rate, thus providing a kind of generalized second law in the nonequilibrium regime. We also prove by means of the counterexample of a phase covariant quantum master equation that the converse of this statement is not true, i.e., there are non-Markovian dynamics for which the entropy production rate is always positive. Thus, we conclude that the emergence of negative entropy production rates is a sufficient but not necessary condition for non-Markovianity of the quantum dynamics. Finally, we also consider a recently introduced map-based notion of entropy production and show the equivalence between its positivity and Markovianity for general finite-dimensional systems.