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A short introduction to the Lindblad Master Equation
Daniel Manzano
TL;DR
Open quantum theory needs tractable equations for subsystems interacting with environments, especially when the interaction is Markovian. The paper develops the Lindblad Master Equation through two derivations, explains its properties, and reviews methods for solving it. It concludes that the equation provides the general generator of Markovian quantum dynamics while reduced density matrices connect the formalism to open-system observables.
Problem
The paper addresses how to characterize the most general Markovian transformation between density matrices and obtain a continuous master equation describing its time evolution.
Method
The paper presents two derivations of the Lindblad Master Equation, develops the relevant mathematical and quantum-mechanical framework, and describes integration and diagonalisation methods for solving it.
Results
The Lindblad Master Equation is obtained as the generator of Markovian completely positive trace-preserving maps, with methods provided for calculating time-dependent and steady-state density matrices.
Takeaways & Limitations
The paper supplies a self-contained route from reduced open-system dynamics to Lindblad-equation derivation, properties, and practical resolution methods.
Abstract
from arXiv · showhide
The theory of open quantum system is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) Master Equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.
I. INTRODUCTION
Open quantum theory addresses how to derive tractable dynamics for a subsystem interacting with an environment. The paper focuses on the Lindblad equation as the most general generator of Markovian quantum dynamics and introduces the concepts and methods needed to use it.
- Motivation: Open quantum theory separates a system of interest from its environment to derive reduced equations of motion.The reduced equations should be easier to solve than the full dynamics, so derivations usually require approximations.
- Scope: The Lindblad equation generates the most general quantum dynamics for systems connected to several baths through a Markovian interaction.It is also called the Gorini-Kossakowski-Sudarshan-Lindblad equation.
- Paper organization: The paper provides basic knowledge about the Lindblad Master Equation, including its mathematical requirements, quantum-mechanical prerequisites, and the Fock-Liouville-space framework.The presentation is organized to support readers with limited previous knowledge of the field.
- Paper organization: The paper derives the Lindblad Master Equation from two approaches, discusses its properties, and presents several resolution methods.A decaying two-level system illustrates the sections, and Mathematica notebooks solve the proposed problems.
III. (VERY SHORT) INTRODUCTION TO QUANTUM MECHANICS
This chapter reviews the quantum-mechanical concepts required for the Lindblad Master Equation, including states, measurements, unitary evolution, density matrices, and reduced dynamics. It emphasizes that tracing out the environment yields the reduced state used to describe an open subsystem.
- Quantum states: The chapter introduces quantum states as vectors in a complex Hilbert space for isolated systems and notes that all unit vectors in a finite Hilbert space represent possible physical states.The paper assumes familiarity with Hilbert spaces and basic infinitesimal calculus rather than serving as a full quantum-mechanics course.
- Density matrices: Density matrices describe mixed states as positive, unit-trace operators, with diagonal elements representing populations and off-diagonal elements representing coherences.For pure states, Tr[ρ^2] = 1; mixed states have Tr[ρ^2] < 1.
- Measurements: Measurements are represented by Hermitian observables, whose eigenvalues give possible outcomes and whose eigenvectors determine the post-measurement state.The chapter also gives probability and expectation-value formulas for pure and mixed states.
- Time evolution: Closed-system time evolution is governed by the Schrödinger equation and can equivalently be expressed as unitary evolution generated by the Hamiltonian.The Hamiltonian is the system’s energy operator, and the paper uses units with hbar = 1.
- Time evolution: Hamiltonian dynamics preserves the purity and therefore the mixing rate of a quantum state.The paper derives this result using the cyclic property of the trace.
- Open-system dynamics: Tracing over environmental degrees of freedom produces the reduced density matrix, from which all measurement statistics of the subsystem can be calculated.This partial-trace operation is presented as the main idea of open quantum systems.
IV. THE FOCK-LIOUVILLE HILBERT SPACE. THE LIOUVILLE SUPEROPERATOR
The Fock-Liouville space converts density matrices into vectors in a Hilbert space, allowing the Liouvillian and evolution operator to be represented as matrices.
- Fock-Liouville space treats density matrices as vectors by defining a scalar product on the matrix space.The representation is written as ρ → |ρ⟩⟩.
- The Liouville super-operator acts on the Hilbert space of density matrices and enables matrix representations of evolution.
- A two-level system is used to illustrate time evolution in the Fock-Liouville representation.
- For a mixed state, time evolution follows the von-Neumann equation, with the Liouvillian expressed as a matrix.
- In a computational density-matrix basis, each Liouvillian row is obtained from the operation −i[H,ρ].The resulting evolution is a matrix equation for |ρ⟩⟩.
A. Completely positive maps
This section defines completely positive trace-preserving maps as the physically valid transformations of density matrices and motivates their role in Markovian dynamics.
- The target is the most general Markovian transformation of density matrices and its corresponding dynamical equation.
- A physical map must transform density matrices into density matrices, requiring trace preservation and complete positivity.
- Positivity preserves positive operators, but complete positivity additionally requires V ⊗ 1_n to remain positive for every n.
- Matrix transposition provides a positive map that is not completely positive, demonstrated using a Bell state and a non-positive output.
- Imposing complete positivity and trace preservation yields a unique master-equation generator for Markovian CPT maps.
B. Derivation of the Lindblad Equation from microscopic dynamics
The microscopic derivation reduces closed system–environment dynamics to a system equation through tracing, weak-coupling and decorrelation assumptions, Markovianization, and the rotating wave approximation.
- The total Hilbert space is divided into the system of interest and environment, and the reduced state is obtained by tracing over environmental degrees of freedom.The total Hamiltonian is split into system, environment, and interaction parts.
- The derivation assumes weak system–environment coupling and initially separable system and environment states.It also approximates the system and environment as non-correlated throughout evolution.
- Extending the memory-kernel integral to infinity produces a local-in-time Markovian equation identified as the Redfield equation.The equation remains dependent on the system’s initial-state preparation and is therefore initially non-Markovian.
- The Redfield equation does not guarantee positivity, so the rotating wave approximation is introduced to ensure complete positivity.
- The environmental effect enters through coefficients that are decomposed into Hamiltonian and non-Hamiltonian parts.The Hamiltonian contribution is the Lamb shift, which renormalizes system energy levels.
- Diagonalizing the positive coefficient matrix yields the Lindblad master equation, whose operators are called jump operators.In the simplest case, only one relevant frequency remains.
C. Derivation of the Lindblad Equation as a CPT generator
A second derivation starts from completely positive trace-preserving maps and constructs their time-independent generator, arriving at the Lindblad master equation.
- The quantum-information approach asks for the most general Markovian map from density matrices onto density matrices.
- Choi’s theorem characterizes completely positive maps, while the Choi–Kraus theorem adds the trace-preserving condition for CPT maps.
- Complete positivity is tested by extending the system with an auxiliary space and applying the map to one part of a maximally entangled state.
- Kraus operators represent CPT maps and may depend on time provided they satisfy the trace-preserving relation at every time.
- The remaining task is to obtain a continuous differential equation and a time-independent generator L satisfying V(t) = e^Lt.
- Expanding Kraus operators in an orthonormal operator basis and separating Hermitian components leads, after imposing trace preservation, to the Lindblad master equation.
D. Properties of the Lindblad Master Equation
The paper highlights structural properties of Lindblad dynamics and applies the equation to a decaying two-level atom. These properties include purity monotonicity under Hermitian jumps and invariance under transformations of jump operators.
- For Hermitian jump operators, Lindblad dynamics makes the purity derivative non-positive.The paper supplies a proof in Appendix A.
- The Lindblad Master Equation is invariant under unitary transformations of the jump operators.
- These invariances allow traceless jump operators to be chosen without loss of generality.
- It is also invariant under inhomogeneous jump-operator transformations involving complex coefficients and a real parameter.
- The two-level-atom example incorporates photon-emission decay through interaction with the surrounding vacuum.
A. Integration
The integration approach numerically evolves the density matrix using differential equations, with fourth-order Runge–Kutta presented as a common method. Long-time evolution becomes costly and can introduce trace-preservation problems in larger systems.
- The density-matrix evolution requires solving as many equations as the density matrix has variables, with dimensionality increasing exponentially with system size.The example requires four variables, while larger systems require dimension-reduction techniques.
- Fourth-order Runge–Kutta numerically integrates the equations of motion to calculate the density matrix at arbitrary times.
- Long-time evolution reaches the steady state as t →∞ but becomes too slow for systems beyond a few qubits.
- Finite-difference integration errors can make the density-matrix trace drift from one and produce non-physical states.
- Trace-preserving methods such as Crank-Nicholson avoid this issue but require more computational power than Runge–Kutta.
- Figure 4 compares populations under pure incoherent dynamics with populations under combined coherent and incoherent dynamics.The blue curves represent ρ11 and the orange curves represent ρ00.
B. Diagonalisation
Diagonalising the Liouvillian provides time-dependent states and steady states through its eigenvectors, avoiding interval-by-interval integration. The zero-eigenvalue eigenvector identifies the steady state, which is generally unique for finite systems.
- Diagonalising the Liouvillian yields both time-dependent and steady-state density matrices.For short times, direct differential-equation integration may be more efficient because Liouvillian diagonalisation becomes costly at high dimensionality.
- Because the Liouvillian is non-Hermitian, its left and right eigenvectors may differ.
- The diagonalisation method calculates the state at time t without integrating over the interval [0, t].
- The eigenvector associated with the zero eigenvalue gives the steady state because it is the only mode surviving as t →∞.
- For finite systems, Evans’ Theorem guarantees at least one zero Liouvillian eigenvalue, while symmetries can produce multiple zero eigenvalues.The paper treats a unique fixed point as the generic case.
- Figure 5 displays the Liouvillian spectrum for Γ = 0.2, n = 1, Ω= 0, and E = 1.
- In the two-level decay example, one zero eigenvalue implies that every initial density matrix evolves to the same steady state.The steady-state density matrix is obtained by selecting and normalizing the corresponding right eigenvector.
Appendix A: Proof of d dtTr
Appendix A proves the non-increasing purity property for Lindblad dynamics with Hermitian jump operators. The proof differentiates the trace expression, uses the Lindblad equation and trace identities, and reduces the result to elementary inequalities.
- The argument relies on Hermitian jump operators and a time-dependent eigenbasis remains valid because the inequality must hold at every time.
- The proof begins by interchanging the trace and derivative, using linearity of the trace.
- Substituting the Lindblad equation and applying cyclicity of the trace reduces the target inequality after the first term vanishes.
- Diagonalising the Hermitian density matrix expresses it through real eigenvalues and corresponding eigenvectors.The eigenvalues are ordered as Λ0 ≥Λ1 ≥· · · ≥Λd.
- The jump operators are expanded in the density-matrix eigenbasis, introducing coefficients used to reorganize the inequality.
- The proof reduces the purity inequality to a set of inequalities established using nonnegative coefficients and the triangular inequality.