We introduce a class of Markovian quantum master equations, able to describe the dissipative dynamics of a quantum system weakly coupled to one or several heat baths. The dissipative structure is driven by an entropic operator, the so called modular Hamiltonian, which makes it nonlinear. The generated Modular Dynamical Semigroup (MDS) is not, in general, a Quantum Dynamical Semigroup (QDS), whose dynamics is of the popular Lindblad type. The MDS has a robust thermodynamic structure, which guarantees for the positivity of the time evolved state, the correct steady state properties, the positivity of the entropy production, a positive Onsager matrix and Onsager symmetry relations (arising from Green-Kubo formulas). We show that the celebrated Davies generator, obtained through the Born and the secular approximations, generates a MDS. By unravelling the modular structure of the former, we provide a different and genuinely nonlinear MDS, not of QDS type, which is free from the severe spectral restrictions of the Davies generator, while still being supported by a weak coupling limit argument. With respect to the latter, the present work is a substantial extension of \cite{Ottinger2011_GEO,Ottinger2010_TLS_DHO}
We show that any Hermiticity and trace preserving continuous semigroup gamma_t in d dimensions is completely positive if and only if the semigroup gamma_t otimes gamma_t is positivity preserving.
A structure of generator of a quantum dynamical semigroup for the dynamics of a test particle interacting through collisions with the environment is considered, which has been obtained from a microphysical model. The related master-equation is shown to go over to a Fokker-Planck equation for the description of Brownian motion at quantum level in the long wavelength limit. The structure of this Fokker-Planck equation is expressed in this paper in terms of superoperators, giving explicit expressions for the coefficient of diffusion in momentum in correspondence with two cases of interest for the…
We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under fixed locality constraints. Beside recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law.
Dissipation can be used as a resource to control and simulate quantum systems. We discuss a modular model based on fast dissipation capable of performing universal quantum computation, and simulating arbitrary Lindbladian dynamics. The model consists of a network of elementary dissipation-generated modules and it is in principle scalable. In particular, we demonstrate the ability to dissipatively prepare all single qubit gates, and the CNOT gate; prerequisites for universal quantum computing. We also show a way to implement a type of quantum memory in a dissipative environment, whereby we can …
We consider a class of quantum dissipative semigroup on a von-Neumann algebra which admits a normal invariant state. We investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical Markov shift. In case the normal invariant state is also faithful, we also extend the notion of `quantum detailed balance' introduced by Frigerio-Gorini and prove that forward weak Markov process and backward weak Markov process are equivalent by an anti-unitary operator.
In the framework of open quantum systems, the propagation of polarized photons can be effectively described using quantum dynamical semigroups. These extended time-evolutions induce irreversibility and dissipation. Planned, high sensitive experiments, both in the laboratory and in space, will be able to put stringent bounds on these non-standard effects.
We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian controls and elucidate their geometry in terms of Lie semigroups. For standard dissipative interactions with the environment and different coherent controls, we particularly specify the tangent cones (Lie wedges) of the respective Lie semigroups of quantum channels. These cones are the counterpart of the infinitesimal generator of a single one-parameter semigroup. They comprise all directions the underlying open quantum system can be steered to and thus give insight into the geometry of controlled …
In the dynamics of open quantum systems, the interaction with the external environment usually leads to a contraction of the set of reachable states for the system as time increases, eventually shrinking to a single stationary point. In this contribution we describe to what extent it is possible to modify this asymptotic state by means of indirect control, that is by using an auxiliary system coupled to the target system in order to affect its dynamics, when there is a purely dissipative coupling between the two systems. We prove that, also in this restrictive case, it is possible to modify th…
We describe an array of quantum gates implementing Shor's algorithm for prime factorization in a quantum computer. The array includes a circuit for modular exponentiation with several subcomponents (such as controlled multipliers, adders, etc) which are described in terms of elementary Toffoli gates. We present a simple analysis of the impact of losses and decoherence on the performance of this quantum factoring circuit. For that purpose, we simulate a quantum computer which is running the program to factor N = 15 while interacting with a dissipative environment. As a consequence of this inter…
We consider the class of quantum stochastic evolutions ($SLH$-models) leading to a quantum dynamical semigroup over a fixed quantum mechanical system (taken to be finite-dimensional). We show that if the semigroup is dissipative, that is, the coupling operators are non-zero, then a dynamical decoupling scheme based on unitary rotations on the system space cannot suppress decoherence even in the limit where the period between pulses vanishes. We emphasize the role of the Fock space dilation used here to construct a quantum stochastic model, as there are often dilations of the same semigroup usi…
We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, whic…
We show that the generator of a GNS-symmetric quantum Markov semigroup can be written as the square of a derivation. This generalizes a result of Cipriani and Sauvageot for tracially symmetric semigroups. Compared to the tracially symmetric case, the derivations in the general case satisfy a twisted product rule, reflecting the non-triviality of their modular group. This twist is captured by the new concept of Tomita bimodules we introduce. If the quantum Markov semigroup satisfies a certain additional regularity condition, the associated Tomita bimodule can be realized inside the $L^2$ space …
In the weak-coupling limit approach to open quantum systems, the presence of the bath is eliminated and accounted for by a master equation that introduces dissipative contributions to the system reduced dynamics. Within this framework, there are no bath entropy contributions to the entropy balance. We show that, as a consequence, the entropy production fails to be positive for a class of physically legitimate (i.e., completely positive and trace preserving) non-Markovian dynamical maps. Moreover, in the absence of the semigroup property, if the reduced dynamics has a thermal asymptotic state, …
We analyze the dynamics of neutron beams in interferometry experiments using quantum dynamical semigroups. We show that these experiments could provide stringent limits on the non-standard, dissipative terms appearing in the extended evolution equations.
Open quantum systems evolving according to discrete-time dynamics are capable, unlike continuous-time counterparts, to converge to a stable equilibrium in finite time with zero error. We consider dissipative quantum circuits consisting of sequences of quantum channels subject to specified quasi-locality constraints, and determine conditions under which stabilization of a pure multipartite entangled state of interest may be exactly achieved in finite time. Special emphasis is devoted to characterizing scenarios where finite-time stabilization may be achieved robustly with respect to the order o…
Dissipation engineering is a powerful framework for quantum state preparation and autonomous error correction in few-qubit systems. In this work, we examine the scalability of this approach and give three criteria which any dissipative state stabilization protocol should satisfy to be truly scalable as the number of qubits grows. Besides the requirement that it can be constructed in a resource-efficient manner from simple-to-engineer building blocks, a scalable protocol must also exhibit favorable scaling of the stabilization time with the increase in system size. We present a family of protoc…
We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a t…
Markovian dynamics for a quantum/classical system and quantum trajectories
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano（意大利国家核物理研究所（INFN）米兰分部）；Istituto Nazionale di Alta Matematica (INDAM-GNAMPA)（意大利高等数学研究所（INDAM-GNAMPA））
Instantaneous Sobolev Regularization for Dissipative Bosonic Dynamics
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；University of Cambridge（剑桥大学）；Department of Mathematics, University of Tübingen（蒂宾根大学数学系）
Quantum dynamical semigroups provide a general framework for studying the evolution of open systems. Neutrino propagation both in vacuum and in matter can be analyzed using these techniques: they allow a consistent treatment of non-standard, dissipative effects that can alter the pattern of neutrino oscillations. In particular, initially massless neutrinos can give rise to a nonvanishing flavour transition probability, involving in addition the Majorana CP-violating mixing phase.
Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that decompose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These lo…
It is shown that piecewise deterministic dissipative quantum dynamics in a vector space with indefinite metric can lead to well defined, positive probabilities. The case of quantum jumps on the Poincar'e disk is studied in details, including results of numerical simulations of quantum fractals. Keywords: iterated function systems, dynamical semigroups, EEQT, indefinite metric, quantum jumps, quantum measurement, non-commuting observables, fractals, Poincare disk
We suggest to employ the dissipative nature of open quantum systems for the purpose of parameter estimation: The dynamics of open quantum systems is typically described by a quantum dynamical semigroup generator ${\cal L}$. The eigenvalues of ${\cal L}$ are complex, reflecting unitary as well as dissipative dynamics. For certain values of parameters defining ${\cal L}$, non-hermitian degeneracies emerge, i.e. exceptional points ($EP$). The dynamical signature of these $EP$s corresponds to a unique time evolution. This unique feature can be employed experimentally to locate the $EP$s and thereb…
Quantum memories can be regarded as quantum channels that transmit information through time without moving it through space. Aiming at a reliable storage of information we may thus not only encode at the beginning and decode at the end, but also intervene during the transmission - a possibility not captured by the ordinary capacities in Quantum Shannon Theory. In this work we introduce capacities that take this possibility into account and study them in particular for the transmission of quantum information via dynamical semigroups of Lindblad form. When the evolution is subdivided and supplem…
