A Brief History of the GKLS Equation

We reconstruct the chain of events, intuitions and ideas that led to the formulation of the Gorini, Kossakowski, Lindblad and Sudarshan equation.Comment: Based on a talk given by D.C. at the 48th Symposium on Mathematical Physics "Gorini-Kossakowski-Lindblad-Sudarshan Master Equation - 40 Years After" (Toru\'n, June 10-12, 2016). To be published in the special volume of OSI

Kick and fix: the roots of quantum control

When two operators $A$ and $B$ do not commute, the calculation of the exponential operator $e^{A+B}$ is a difficult and crucial problem. The applications are vast and diversified: to name but a few examples, quantum evolutions, product formulas, quantum control, Zeno effect. The latter are of great interest in quantum applications and quantum technologies. We present here a historical survey of results and techniques, and discuss differences and similarities. We also highlight the link with the strong coupling regime, via the adiabatic theorem, and contend that the "pulsed" and "continuous" formulations differ only in the order by which two limits are taken, and are but two faces of the same coin.Comment: 6 page

Universal spectra of random Lindblad operators

To understand typical dynamics of an open quantum system in continuous time, we introduce an ensemble of random Lindblad operators, which generate Markovian completely positive evolution in the space of density matrices. Spectral properties of these operators, including the shape of the spectrum in the complex plane, are evaluated by using methods of free probabilities and explained with non-Hermitian random matrix models. We also demonstrate universality of the spectral features. The notion of ensemble of random generators of Markovian qauntum evolution constitutes a step towards categorization of dissipative quantum chaos.Comment: 6 pages, 4 figures + supplemental materia

Remembering George Sudarshan

Selected Papers from the 16th International Conference on Squeezed States and Uncertainty Relations (ICSSUR 2019), 17-21 June 2019, Universidad Complutense de Madrid, Spain.In these brief notes we want to render homage to the memory of E.C.G. Sudarshan, adding it to the many contributions devoted to preserve his memory from a personal point of view.This research was funded by the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015/0554). A.I. and F.C. would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD++, S2018/TCS-A4342. G.M. would like to thank the support provided by the Santander/UC3M Excellence Chair Programme 2019/2020

Application of Shemesh theorem to quantum channels

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators $\mathcal{A}(A_1,\ldots A_K)$. By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra $\mathcal{A}(A_1,\ldots A_K)$ one can predict the asymptotic dynamics for a class of operations

Generalized Adiabatic Theorem and Strong-Coupling Limits

We generalize Kato's adiabatic theorem to nonunitary dynamics with an isospectral generator. This enables us to unify two strong-coupling limits: one driven by fast oscillations under a Hamiltonian, and the other driven by strong damping under a Lindbladian. We discuss the case where both mechanisms are present and provide nonperturbative error bounds. We also analyze the links with the quantum Zeno effect and dynamics

Nature and origin of the operators entering the master equation of an open quantum system

By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the Parametric Representation with Environmental Coherent States) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini-Kossakowski-Sudarshan-Lindblad equation, but holds regardless of markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We find that, instead of a single set of Lindblad operators, in the general (non-markovian) case there one set of Lindblad-like operators for each and every point of a symplectic manifold associated to the environment. This intricacy disappears under some assumptions (which are related to markovianity and the classical limit of the environment), under which it is possible to recover the usual master equation formalism. Finally, we find such Lindblad-like operators for two different models of a qubit in a bosonic environment, and show that in the classical limit of the environment their renown master equations are recovered.Comment: 19 pages, 1 figur

Dissipative self-interference and robustness of continuous error-correction to miscalibration

We derive an effective equation of motion within the steady-state subspace of a large family of Markovian open systems (i.e., Lindbladians) due to perturbations of their Hamiltonians and system-bath couplings. Under mild and realistic conditions, competing dissipative processes destructively interfere without the need for fine-tuning and produce no dissipation within the steady-state subspace. In quantum error-correction, these effects imply that continuously error-correcting Lindbladians are robust to calibration errors, including miscalibrations consisting of operators undetectable by the code. A similar interference is present in more general systems if one implements a particular Hamiltonian drive, resulting in a coherent cancellation of dissipation. On the opposite extreme, we provide a simple implementation of universal Lindbladian simulation