The Gorini-Kossakowski-Sudarshan-Lindblad generation theorem,and a generalization to non-stationary evolutions

• URL: http://arxiv.org/abs/2507.11766v2
• Date: Mon, 04 Aug 2025 19:42:28 GMT
• Title: The Gorini-Kossakowski-Sudarshan-Lindblad generation theorem,and a generalization to non-stationary evolutions
• Authors: Paul E. Lammert,
• Abstract summary: Gorini-Kossakowski-Sudarshan-Lindblad generation theorem says precisely which superoperators can appear on its right-hand side.<n>The finite-dimensional case is handled using a form of Jamiolkowski transform.<n>The infinite-dimensional case is handled by means of a sequence of finite-dimensional approximations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Lindblad equation embodies a fundamental paradigm of the quantum theory of open systems, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generation theorem says precisely which superoperators can appear on its right-hand side. These are the generators of completely positive trace-preserving (or nonincreasing) semigroups. A complete exposition of this theorem is given. The finite-dimensional case is handled using a form of Jamio\l{}kowski transform. The treatment requires no previous knowledge of complete positivity and obtains the Choi-Kraus presentation along the way. The (separable) infinite-dimensional case is handled by means of a sequence of finite-dimensional approximations, using the finite-dimensional case as a crucial tool. An extension to time-dependent generator is given.The condition for CP evolution is just that for semigroups applied at each instant, and the Lindblad decomposition can be chosen continuous in time.

Related papers

• Recursive perturbation approach to time-convolutionless master equations: Explicit construction of generalized Lindblad generators for arbitrary open systems [44.99833362998488]
  We develop a perturbative expansion for the time-convolutionless (TCL) generator of an open quantum system in a generalized Lindblad form.<n>This formulation provides a systematic approach to derive the generator at arbitrary order while preserving a Lindblad-like structure.<n>To validate the method and show its effectiveness in addressing non-Markovian dynamics and strong-coupling effects, we compute the generator explicitly up to fourth order.
  arXiv  Detail & Related papers  (2025-06-04T15:51:26Z)
• A unified approach to quantum de Finetti theorems and SoS rounding via geometric quantization [0.0]
  We study a connection between a Hermitian version of the SoS hierarchy, related to the quantum de Finetti theorem. We show that previously known HSoS rounding algorithms can be recast as quantizing an objective function.
  arXiv  Detail & Related papers  (2024-11-06T17:09:28Z)
• Lindbladian reverse engineering for general non-equilibrium steady states: A scalable null-space approach [49.1574468325115]
  We introduce a method for reconstructing the corresponding Lindbaldian master equation given any target NESS. The kernel (null-space) of the correlation matrix corresponds to Lindbladian solutions. We illustrate the method in different systems, ranging from bosonic Gaussian to dissipative-driven collective spins.
  arXiv  Detail & Related papers  (2024-08-09T19:00:18Z)
• On the Liouville-von Neumann equation for unbounded Hamiltonians [44.99833362998488]
  We present and discuss a characterization of the domain of the Liouville superoperator originally due to M. Courbage. We also provide, with explicit proofs, some domains of essential self-adjointness (cores) of the Liouvillian.
  arXiv  Detail & Related papers  (2024-08-09T10:13:35Z)
• Interacting chiral fermions on the lattice with matrix product operator norms [37.69303106863453]
  We develop a Hamiltonian formalism for simulating interacting chiral fermions on the lattice. The fermion doubling problem is circumvented by constructing a Fock space endowed with a semi-definite norm. We demonstrate that the scaling limit of the free model recovers the chiral fermion field.
  arXiv  Detail & Related papers  (2024-05-16T17:46:12Z)
• Decoherence is an echo of Anderson localization in open quantum systems [0.0]
  We study the time evolution of single-particle quantum states described by a Lindblad master equation with local terms. We establish a finite-volume-type criterion for the decay of the off-diagonal matrix elements in the position basis of the time-evolved or steady states.
  arXiv  Detail & Related papers  (2023-10-15T16:38:31Z)
• Connecting classical finite exchangeability to quantum theory [45.76759085727843]
  Exchangeability is a fundamental concept in probability theory and statistics.<n>It allows to model situations where the order of observations does not matter.<n>It is well known that both theorems do not hold for finitely exchangeable sequences.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• D-divisible quantum evolution families [0.0]
  We propose and explore a notion of decomposably divisible (D-divisible) differentiable quantum evolution families on matrix algebras. Necessary and sufficient conditions for D-divisibility are found.
  arXiv  Detail & Related papers  (2023-03-07T22:06:14Z)
• Continuous percolation in a Hilbert space for a large system of qubits [58.720142291102135]
  The percolation transition is defined through the appearance of the infinite cluster. We show that the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient. Our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.
  arXiv  Detail & Related papers  (2022-10-15T13:53:21Z)
• On the use of total state decompositions for the study of reduced dynamics [0.0]
  We show that a decomposition of any possibly correlated bipartite state allows one to fix reduced-system evolution via a finite set of time-dependent CPTP maps. In particular, we show that such a decomposition always exists, also for infinite dimensional Hilbert spaces, and that the number of resulting CPTP maps is bounded by the Schmidt rank of the initial global state. For two simple qubit models, we identify the positivity domain defined by the initial states that are mapped into proper states at any time of the evolution fixed by the CPTP semigroups.
  arXiv  Detail & Related papers  (2022-09-06T08:33:42Z)
• The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) Equation for Two-Dimensional Systems [62.997667081978825]
  Open quantum systems can obey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space dimension of $2$.
  arXiv  Detail & Related papers  (2022-04-16T07:03:54Z)
• Mean hitting time formula for positive maps [0.0]
  We present an analogous construction for the setting of irreducible, positive, trace preserving maps. The problem at hand is motivated by questions on quantum information theory.
  arXiv  Detail & Related papers  (2022-03-21T01:25:25Z)
• Euclidean Time Approach to Entanglement Entropy on Lattices and Fuzzy Spaces [0.0]
  In a recent letter, we developed a novel Euclidean time approach to compute entanglement entropy on lattices and fuzzy spaces based on Green's function. The present work is devoted in part to the explicit proof of the Green's matrix function formula which was quoted and used in the previous letter, and on the other part to some applications of this formalism.
  arXiv  Detail & Related papers  (2022-01-10T19:03:06Z)
• Exact Recovery in the General Hypergraph Stochastic Block Model [92.28929858529679]
  This paper investigates fundamental limits of exact recovery in the general d-uniform hypergraph block model (d-HSBM) We show that there exists a sharp threshold such that exact recovery is achievable above the threshold and impossible below it.
  arXiv  Detail & Related papers  (2021-05-11T03:39:08Z)
• Eternal Adiabaticity [0.0]
  We apply adiabatic theorem for the strong-coupling limit in finite-dimensional quantum systems. We prove the equivalence of the Schrieffer-Wolff and des Cloiseaux approaches in the unitary case. We show that ideal effective generators for open systems do not exist in general.
  arXiv  Detail & Related papers  (2020-11-09T19:41:56Z)
• Nonlinear extension of the quantum dynamical semigroup [0.0]
  We consider deterministic nonlinear time evolutions satisfying so called convex quasi-linearity condition. We show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasi-linear operations also possesses the semigroup property.
  arXiv  Detail & Related papers  (2020-03-20T10:04:51Z)
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder [68.8204255655161]
  We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder. We retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting.
  arXiv  Detail & Related papers  (2020-03-16T11:37:23Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.