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Register a new userConvex geometry of Markovian Lindblad dynamics and witnessing non-Markovianity
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Added by
Samyadeb Bhattacharya
Publication date
2018
fields
Physics
and research's language is
English
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We develop a theory of linear witnesses for detecting non-Markovianity, based on the geometric structure of the set of Choi states for all Markovian evolutions having Lindblad type generators. We show that the set of all such Markovian Choi states form a convex and compact set under the small time interval approximation. Invoking geometric Hahn-Banach theorem, we construct linear witnesses to separate a given non-Markovian Choi state from the set of Markovian Choi states. We present examples of such witnesses for dephasing channel and Pauli channel in case of qubits. We further investigate the geometric structure of the Markovian Choi states to find that they do not form a polytope. This presents a platform to consider non-linear improvement of non-Markovianity witnesses.
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We establish a convex resource theory of non-Markovianity under the constraint of small time intervals within the temporal evolution. We construct the free operations, free states and a generalized bona-fide measure of non-Markovianity. The framework satisfies the basic properties of a consistent resource theory. The proposed resource quantifier is lower bounded by the optimization free Rivas-Huelga-Plenio (RHP) measure of nonMarkovianity. We further define the robustness of non-Markovianity and show that it can directly be expressed as a function of the RHP measure of non-Markovianity. This enables a physical interpretation of the RHP measure.
The modeling of natural phenomena via a Markov process --- a process for which the future is independent of the past, given the present--- is ubiquitous in many fields of science. Within this context, it is of foremost importance to develop ways to check from the available empirical data if the underlying mechanism is indeed Markovian. A paradigmatic example is given by data processing inequalities, the violation of which is an unambiguous proof of the non-Markovianity of the process. Here, our aim is twofold. First we show the existence of a monogamy-like type of constraints, beyond data processing, respected by Markov chains. Second, to show a novel connection between the quantification of causality and the violation of both data processing and monogamy inequalities. Apart from its foundational relevance in the study of stochastic processes we also consider the applicability of our results in a typical quantum information setup, showing it can be useful to witness the non-Markovianity arising in a sequence of quantum non-projective measurements.
We identify a set of dynamical maps of open quantum system, and refer to them as $ epsilon $-Markovian maps. It is constituted of maps which, in a higher dimensional system-environment Hilbert space, possibly violate Born approximation but only a little. We characterize the $epsilon$-nonmarkovianity of a general dynamical map by the minimum distance of that map from the set of $epsilon$-Markovian maps. We analytically derive an inequality which gives a bound on the $ epsilon$-nonmarkovianity of the dynamical map, in terms of an entanglement-like resource generated between the system and its immediate environment. In the special case of a vanishing $epsilon$, this inequality gives a relation between the $epsilon$-nonmarkovianity of the reduced dynamical map on the system and the entanglement generated between the system and its immediate environment. We numerically investigate the behavior of the similar distant based measures of non-Markovianity for classes of amplitude damping and phase damping channels.
The correlated-projection technique has been successfully applied to derive a large class of highly non Markovian dynamics, the so called non Markovian generalized Lindblad type equations or Lindblad rate equations. In this article, general unravellings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unravelling can be interpreted in terms of measurements continuous in time, but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and we discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.
Non-Markovian effects can speed up the dynamics of quantum systems while the limits of the evolution time can be derived by quantifiers of quantum statistical speed. We introduce a witness for characterizing the non-Markovianity of quantum evolutions through the Hilbert-Schmidt speed (HSS), which is a special type of quantum statistical speed. This witness has the advantage of not requiring diagonalization of evolved density matrix. Its sensitivity is investigated by considering several paradigmatic instances of open quantum systems, such as one qubit subject to phase-covariant noise and Pauli channel, two independent qubits locally interacting with leaky cavities, V-type and $Lambda $-type three-level atom (qutrit) in a dissipative cavity. We show that the proposed HSS-based non-Markovianity witness detects memory effects in agreement with the well-established trace distance-based witness, being sensitive to system-environment information backflows.
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