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arXiv:2105.12644 · Physical Review A

Dissipative evolution of quantum Gaussian states

Tomasz Linowski, Alexander Evgen’evich Teretenkov, Łukasz Rudnicki

Published Nov 10, 2022• 21 citations
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Abstract

Recent works on quantum resource theories of non-Gaussianity, which are based upon the type of tools available in contemporary experimental settings, put Gaussian states and their convex combinations on equal footing. Motivated by this, in this paper, we derive a model of dissipative time evolution based on unitary Lindblad operators which, while it does not preserve the set of Gaussian states, preserves the set of their convex combinations, i.e., so-called quantum Gaussian states. As we demonstrate, the considered evolution proves useful both as a description for random scattering and as a tool in dissipator engineering.

DOI
10.1103/physreva.106.052206

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Recent works on quantum resource theories of non-Gaussianity, which are based upon the type of tools available in contemporary experimental settings, put Gaussian states and their convex combinations on equal footing. Motivated by this, in this paper, we derive a model of dissipative time evolution based on unitary Lindblad operators which, while it does not preserve the set of Gaussian states, preserves the set of their convex combinations, i.e., so-called quantum Gaussian states. As we demonstrate, the considered evolution proves useful both as a description for random scattering and as a tool in dissipator engineering.
The notion of classicality of quantum evolution of light is an object of both conceptual and practical importance. The main goal of this work is to derive the exact conditions for the classicality of quantum Gaussian evolution, i.e., the evolution of Gaussian states of light and their convex combinations, a model which is of great significance in quantum optics and information. According to our findings, quantum Gaussian evolution should be considered classical if the Hamiltonian and Lindblad operators generating it correspond to passive optical transformations. This is illustrated with several explicit examples, ranging from Gaussian thermal operations to entanglement-maximizing dissipative engineering. Our results are obtained using the recently introduced mesoscopic formalism of the reduced state of the field, which was originally devised as as a description of macroscopic quantum fields. Here, to make the framework suitable for our goal, we redevelop it as a tool for probing classicality, which constitutes our second main contribution.
The covariance matrix contains the complete information about the second-order expectation values of the mode quadratures (position and momentum operators) of the system. Due to its prominence in studies of continuous variable systems, most significantly Gaussian states, special emphasis is put on time evolution models that result in self-contained equations for the covariance matrix. So far, despite not being explicitly implied by this requirement, virtually all such models assume a so-called quadratic, or second-order case, in which the generator of the evolution is at most second-order in the mode quadratures. Here, we provide an explicit model of covariance matrix evolution of infinite order. Furthermore, we derive the solution, including stationary states, for a large subclass of proposed evolutions. Our findings challenge the contemporary understanding of covariance matrix dynamics and may give rise to new methods and improvements in quantum technologies employing continuous variable systems.
Abstract This paper studies a class of single-mode bosonic models of open quantum systems in which the dynamics of the creation and annihilation operators remains linear despite the strongly non-Gaussian nature of the induced dissipation. It is shown that, in contrast to isolated bosonic systems—where linear dynamics uniquely corresponds to quadratic Hamiltonians and Gaussian states—linearity of the operator dynamics in open systems does not guarantee Gaussianity of the state evolution. We construct quantum channels and Gorini–Kossakowski–Sudarshan–Lindblad generators for which the dynamics of the creation and annihilation operators is linear and closed, while Gaussian states are driven beyond the convex hull of Gaussian states during the evolution. In particular, the emergence of negativity of the Wigner function at a finite evolution time is demonstrated.
The Lindblad equation is commonly used to model dissipation and decoherence in open quantum systems. Yet many of the properties of Lindblad dynamics are still little understood. This thesis explores various aspects of the Lindblad equation and its connections with non-Hermitian quantum theories. The evolution speed of Lindblad dynamics is investigated and an explicit expression for the speed is derived. The radial component, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. The semiclassical time evolution of Gaussian states under Lindblad dynamics is then considered and a new type of phase-space dynamics is derived for the centre of a Gaussian Wigner function. By viewing the phase-space Lindblad equation as a Schroedinger equation with a non-Hermitian Hamiltonian, a further set of semiclassical equations are derived. These can describe the interference terms in Wigner functions. Further, a PT-symmetric two-level system driven through two consecutive exceptional points at finite speed is considered and an analytical expression for the probability of a non-adiabatic transition is derived. It is shown that the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations. Finally, a quantum implementation of this PT-symmetric system using ultracold atoms in an optical lattice with single-particle losses is investigated.
Quantum Non-Gaussian states are considered as a useful resource for many tasks in quantum information processing, from quantum metrology and quantum sensing to quantum communication and quantum key distribution. Another useful tool that is growing attention is the newly constructed quantum switch. Its applications in many tasks in quantum information have been proved to be outperforming many existing schemes in quantum communication and quantum thermometry. In this contribution, we are addressing this later to be very useful to engineer highly non-Gaussian states from Gaussian operations whose order is controlled by degrees of freedom of a control qubit. The non-convexity of the set of Gaussian states and the set of Gaussian operations guarantees the emergence of non-Gaussianity after postselection on the control qubit deterministically, in contrast to existing protocols in the literature. The non-classicality of the resulting states is discussed accordingly.
We present a quantum algorithm to simulate general finite dimensional Lindblad master equations without the requirement of engineering the system-environment interactions. The proposed method is able to simulate both Markovian and non-Markovian quantum dynamics. It consists in the quantum computation of the dissipative corrections to the unitary evolution of the system of interest, via the reconstruction of the response functions associated with the Lindblad operators. Our approach is equally applicable to dynamics generated by effectively non-Hermitian Hamiltonians. We confirm the quality of our method providing specific error bounds that quantify its accuracy.