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A Brief History of the GKLS Equation
Dariusz Chruściński, Saverio Pascazio
TL;DR
The paper asks how the GKLS equation arose from overlapping historical and conceptual developments rather than as an isolated formula. It reconstructs the relevant hypotheses, derivations, and mathematical structures, including complete positivity and semigroup generators, and reports that related master-equation forms had appeared earlier. The authors propose the name GKLS equation for the shared formulation while acknowledging the account’s selective and partly arbitrary scope.
Problem
The paper addresses how to reconstruct the hypotheses, derivations, and conceptual developments that led to nearly simultaneous formulations of the GKLS equation.
Method
The authors compare historical events, technical findings, hypotheses, derivations, conceptual formulations, and scientific interactions rather than comparing only the final equations.
Results
The paper identifies a shared mathematical formulation whose complete-positivity structure characterizes the relevant semigroup generators, while noting that related master equations had worked before the GKLS papers.
Takeaways & Limitations
The authors propose naming the shared formulation the Gorini-Kossakowski-Lindblad-Sudarshan equation and emphasize that multiple historical factors contributed to its precise formulation.
Takeaways & Limitations
The account is not a review or tutorial, contains no novel results, assumes familiarity with dissipative quantum systems and notation, and necessarily involves arbitrariness and superficiality.
Abstract
from arXiv · showhide
We reconstruct the chain of events, intuitions and ideas that led to the formulation of the Gorini, Kossakowski, Lindblad and Sudarshan equation.
1 Introduction
The paper reconstructs how the GKLS equation emerged by comparing the hypotheses, derivations, concepts, and historical circumstances behind nearly simultaneous GKS and Lindblad papers. It acknowledges that the reconstruction is necessarily selective and partly arbitrary.
- Approach: The reconstruction draws on historical events and detailed technical findings to identify a significant level of similarity between the two cases.The paper frames similarity as meaningful only at an appropriately chosen level of detail.
- Historical setting: GKS and Lindblad published highly influential papers almost simultaneously in May and June 1976 after submitting them in March and April 1975.Their papers were probably developing during overlapping periods.
- Approach: The authors compare hypotheses, derivations, conceptual formulations, and surrounding influences rather than merely comparing final equations.They emphasize conferences, discussions, scientific literature, and personal interactions as factors in how ideas develop.
- Scope: The account accepts arbitrariness and superficiality because reconstructing similarities across historical events requires choosing a finite level of detail.The authors ask readers to allow for this limitation.
- Historical setting: Kossakowski’s 1972 axiomatic work, Gorini’s 1973 exposure to complete positivity, and the 1974 Gorini–Kossakowski–Sudarshan collaboration form key episodes in the chronology.The paper presents these events as facts to be considered in reconstructing the equation’s development.
- Scope: The authors state that the article is neither a review nor a tutorial, contains no novel results, and assumes familiarity with dissipative quantum systems and notation.It relies substantially on personal interactions with the four protagonists.
2 Formulation of the Problem
The formulation problem begins with replacing isolated-system evolution by a physically valid evolution for open quantum systems. The paper introduces quantum channels and Markovian master equations as the framework for this replacement.
- Closed systems: An isolated quantum system evolves through the Schrödinger equation and the corresponding von Neumann equation for its density matrix.The unitary evolution is generated by the Hamiltonian.
- Open systems: For an open system interacting with an environment, the isolated-system equation is no longer valid and must be replaced by another evolution law.The paper identifies environmental interaction as the defining departure from the closed-system case.
- Quantum channels: A quantum channel is a linear map that preserves trace and hermiticity and is completely positive.The channel maps quantum states between times, while its superoperator is denoted by Λ.
- Markovian dynamics: In the Markovian approximation, the channel description yields a differential master equation.This supplies the time-local evolution form used for dissipative dynamics.
- The generator problem: The GKLS problem is to characterize the generator L of a quantum dynamical semigroup acting on density operators.The semigroup maps positive trace-class states while preserving the relevant quantum-state structure.
- Motivation: Open quantum systems became increasingly important as quantum information and quantum technologies became central to quantum-mechanics foundations and applications.The paper places the mathematical formulation in this broader research context.
3 The Structure of the GKLS Generator
The GKLS structure characterizes generators of completely positive quantum dynamical semigroups under finite-dimensional or bounded-generator assumptions. GKS and Lindblad obtained equivalent forms in Schrödinger and Heisenberg pictures, with distinct uniqueness properties.
- Problem: The central problem is characterizing the generator L whose evolution defines a quantum dynamical semigroup.The semigroup is a one-parameter family of maps acting on quantum states, with an initial condition at t=0.
- Scope: GKS restricted its result to finite-dimensional Hilbert spaces, while Lindblad treated infinite dimensions under uniform continuity and bounded generators.Uniform continuity simplifies the analysis by implying that the generator is bounded.
- GKS form: In the Schrödinger picture, GKS gave a theorem expressing a matrix generator as the generator of a completely positive semigroup.The representation uses a positive coefficient matrix in the chosen operator basis.
- GKS form: The GKS form is determined by a Hamiltonian H and the Kossakowski matrix [c_kl].After fixing the basis, the traceless Hamiltonian is uniquely defined.
- Lindblad form: Lindblad independently obtained a Heisenberg-picture characterization of generators of completely positive semigroups.The corresponding Schrödinger-picture generator has the standard GKLS structure.
- Representation: The Lindblad representation is not unique in H and V_j, unlike the fixed-basis GKS representation's traceless Hamiltonian.This distinction concerns representation choices rather than the generated dynamics.
4 Master (Kinetic) Equations before GKLS
Master equations describing dissipation predated GKLS, with examples spanning damping, absorption, atomic decay, laser physics, and weak-coupling Markovian dynamics.
- Master equations were used to describe dissipative phenomena before the GKLS articles.
- 4.1 Landau’s approach to the damping problem: Landau’s 1927 equation described radiation-field damping using annihilation and creation operators with damping constant γ > 0.
- 4.2 Optical potential: Optical potentials modeled scattering and absorption, but their dynamics does not preserve trace because particles may be absorbed.
- 4.3 Lamb equation: Lamb’s two-level-atom equation generates legitimate evolution but is not trace-preserving; adding a suitable completely positive term converts it into a GKLS equation.
- Redfield’s Born–Markov equation preserves trace, but positivity of the evolving density matrix is not guaranteed.
- 4.5 Quantum optics: The Stuttgart laser equation had exactly GKLS form and successfully described laser–atom interaction, showing that this structure was used about ten years earlier.
- Davies generators derived through the weak-coupling limit are legitimate GKLS generators because the Kossakowski matrix is positive definite.
5 Complete Positivity and its Appearance in Physics
Complete positivity strengthened ordinary positivity into a central condition for quantum operations and dynamical maps, while mathematical and physical developments gradually brought the concept into physics.
- A completely positive map is k-positive for every k, extending positivity to all matrix amplifications.
- Stinespring’s theorem characterizes completely positive maps through a Hilbert-space representation involving a unital *-homomorphism and bounded operator.
- Complete positivity developed through mathematical work in the 1960s before becoming important in mathematical physics.
- Kraus introduced quantum operations requiring complete positivity and obtained the Kraus representation.
- In finite dimensions, Choi’s criterion reduces complete positivity to positivity of a matrix built from the maximally entangled projector.
- Complete positivity simplifies analysis because it provides a tractable condition for dynamical maps and master equations.
- The Marburg meeting exposed Gorini to lectures on positive maps and quantum operations, after which he brought complete positivity to Texas.
6 Positivity vs. Complete Positivity — the Great Simplification
The GKLS development distinguished positivity from complete positivity: the latter yields a simple generator characterization, whereas ordinary positivity permits broader and less tractable cases.
- Kossakowski initially studied generators of positive semigroups without knowing complete positivity.
- For a generator in GKLS form, the dynamical semigroup is completely positive if and only if its associated map Φ is completely positive.
- Kossakowski’s theorem gives necessary and sufficient trace conditions for a bounded operator to generate a dynamical semigroup.
- The general structure of generators producing merely positive semigroups remains unknown, although finite-dimensional cases have GKLS structure.
- For a two-level system with diagonal Kossakowski matrix c_kl = γ_kδ_kl, the GKLS condition is γ_k ≥ 0.
- Conditions weaker than γ_k ≥ 0 can preserve dynamical-map positivity without making the instantaneous map Φ positive.
- Belavin et al. anticipated positive-definite Kossakowski matrices as a natural multi-channel generalization, but GKLS established their link to complete positivity.
- Priority is delicate because Kossakowski wrote the right equation under positivity, while deriving the GKLS equation requires stronger premises.
7 The Mystery of Sudarshan-Mathews-Rau Paper
The Sudarshan–Mathews–Rau paper formulated a general dynamical law through conditions on the density matrix and a dynamical matrix. Its results anticipated the Kraus representation, although a gap linked positivity to a condition that modern theory distinguishes from complete positivity.
- Dynamical-law formulation: The authors sought necessary and sufficient conditions ensuring that the evolving density matrix remains a density matrix for t > t0.They expressed the evolution through a linear relation involving the density-matrix elements and imposed properties on an A matrix.
- Dynamical-law formulation: They introduced a dynamical matrix B to replace the complicated conditions on A with a more tractable representation.In modern language, B is the realignment of A.
- Representation theorem: B satisfies the key condition if and only if the evolution admits nonnegative parameters µα and n × n matrices Wα.This yields the operator-sum structure later recognized as the Kraus representation.
- Representation theorem: The paper’s representation is now identified as the Kraus representation of a quantum channel, derived nearly a decade before Kraus’s work.The accompanying normalization condition is expressed as Σα µαWα†Wα = I.
- Positivity and the gap: The derivation contains a gap because the relevant condition makes B blockpositive rather than necessarily positive.Modern language identifies B with the Choi matrix; complete positivity corresponds to the stronger condition, while the weaker condition corresponds to positivity of the map.
8 Recollection from G¨oran Lindblad
Lindblad recalled that complete positivity simplified the structure of quantum dynamical semigroups and motivated his 1974 conference announcement. Subsequent contact with Gorini established that their results substantially overlapped, differing mainly in mathematical machinery.
- Lindblad’s recollection: Lindblad recognized after completing his thesis that the complete-positivity condition made the structure of quantum dynamical semigroups much simpler.He planned to present these recent results at Ingarden’s conference.
- Lindblad’s recollection: He used the conference as an opportunity for a first announcement of his results.The recollection places this decision after Ingarden invited him to speak on entropy and related problems.
- Convergence with Gorini: After the December 1974 talk, Lindblad learned of similar results from Texas and later compared them with Gorini in Stockholm.Their January 1975 comparison found substantial overlap and essentially identical results, apart from the mathematical machinery.
9 Conclusions and Outlook
The paper attributes the precise GKLS evolution equations to several converging historical factors rather than a single episode. It concludes that the seminal articles share many similarities alongside some differences and proposes names for the equations and representation.
- Conclusions: Several concomitant factors contributed to the precise mathematical formulation of evolution equations for open quantum dynamical systems.The paper summarizes the chronology of these episodes in Table 1.
- Conclusions: The authors find many similarities and a few differences among the seminal articles they discuss.They leave the judgment about the significance of the comparison’s level of detail to the reader’s interpretation of Chomsky’s criterion.
- Terminology: They propose naming equation (4), together with (7), (9), and (10), the Gorini-Kossakowski-Lindblad-Sudarshan equation.They also propose calling equation (3) the Kraus-Stinespring-Sudarshan representation.