Lindblad Equation

Purpose

The Gorini–Kossakowski–Sudarshan–Lindblad equation is the general generator of a finite-dimensional, norm-continuous, time-homogeneous completely positive trace-preserving quantum dynamical semigroup. In a common convention,

$$\frac{d\rho}{dt} = - \frac{i}{\hbar} [H,\rho] + \sum_k \gamma_k \left( L_k\rho L_k^\dagger - \frac12 \{L_k^\dagger L_k,\rho\} \right), \qquad \gamma_k\ge0.$$

Define

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12 \{L^\dagger L,\rho\}.$$

Then

$$\dot\rho = \mathcal L(\rho) = - \frac{i}{\hbar}[H,\rho] + \sum_k \gamma_k\mathcal D[L_k]\rho.$$

The rates may be absorbed into the operators:

$$J_k=\sqrt{\gamma_k}\,L_k.$$

Never use both an explicit $\gamma_k$ and an already rate-weighted $J_k$ in the same dissipator.

The structural derivation belongs to Lindblad–GKSL Equation and Lindblad Theorem. This card is the calculation and convention reference.

At a glance

Task	Formula or condition
Dissipator	$\mathcal D[L]\rho=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}$
Time-independent solution	$\rho(t)=e^{t\mathcal L}\rho(0)$
Observable generator	$\mathcal L^\dagger(A)=(i/\hbar)[H,A]+\sum_k\gamma_k\mathcal D^\dagger[L_k]A$
Adjoint dissipator	$\mathcal D^\dagger[L]A=L^\dagger A L-\tfrac12\{L^\dagger L,A\}$
Expectation dynamics	$d\langle A\rangle/dt=\langle\mathcal L^\dagger(A)\rangle+\langle\partial_tA\rangle$
Steady state	$\mathcal L(\rho_{\mathrm{ss}})=0$, $\rho_{\mathrm{ss}}\ge0$, $\operatorname{Tr}\rho_{\mathrm{ss}}=1$
Jump probability in $dt$	$dp_k=\gamma_kdt\,\operatorname{Tr}(L_k^\dagger L_k\rho)$
Effective no-jump Hamiltonian	$H_{\mathrm{eff}}=H-(i\hbar/2)\sum_k\gamma_kL_k^\dagger L_k$
Pure-state purity slope	$\dot P=-2\sum_k\gamma_k(\langle L_k^\dagger L_k\rangle-\lvert\langle L_k\rangle\rvert^2)$

The jump rows refer to one quantum-trajectory unraveling. Different monitoring schemes can unravel the same unconditional generator differently.

Symbols and dimensions

Symbol	Meaning	Typical dimensions
$\rho$	Density operator	dimensionless
$H$	Effective Hamiltonian, including any Lamb shift	energy
$\mathcal L$	Liouvillian or GKSL generator	inverse time
$L_k$	Dimensionless Lindblad operator in the explicit-rate convention	dimensionless
$\gamma_k$	Nonnegative canonical rate	inverse time
$J_k=\sqrt{\gamma_k}L_k$	Rate-weighted jump operator	inverse square root of time

Authors often call $J_k$ itself a Lindblad operator and omit $\gamma_k$. Check dimensions before comparing equations.

The label “dissipator” does not imply energy loss. A Hermitian $L$ commuting with $H$ can produce pure dephasing while leaving energy populations fixed.

Trace and Hermiticity checks

For one dissipator,

$$\begin{aligned} \operatorname{Tr}\mathcal D[L]\rho &= \operatorname{Tr}(L\rho L^\dagger) - \frac12 \operatorname{Tr}(L^\dagger L\rho) \\ &\quad- \frac12 \operatorname{Tr}(\rho L^\dagger L). \end{aligned}$$

Cyclicity makes all three traces equal, so

$$\operatorname{Tr}\mathcal D[L]\rho=0.$$

The Hamiltonian commutator also has zero trace. Therefore

$$\frac{d}{dt}\operatorname{Tr}\rho=0.$$

If $\rho^\dagger=\rho$ and $H^\dagger=H$, then

$$\mathcal L(\rho)^\dagger = \mathcal L(\rho),$$

so Hermiticity is preserved. Positivity at finite time is the subtler condition; the GKSL structure with nonnegative canonical rates promotes the infinitesimal equation to completely positive dynamics.

Short-time Kraus step

For a small interval $dt$, define

$$K_0 = I - \frac{i}{\hbar}Hdt - \frac12 \sum_k \gamma_kL_k^\dagger L_k\,dt$$

and

$$K_k = \sqrt{\gamma_kdt}\,L_k.$$

Then

$$\rho(t+dt) = K_0\rho(t)K_0^\dagger + \sum_k K_k\rho(t)K_k^\dagger + O(dt^2),$$

and

$$K_0^\dagger K_0 + \sum_k K_k^\dagger K_k = I+O(dt^2).$$

Expanding the map through first order gives the GKSL equation. This is a useful memory aid for the anticommutator and rate square roots. It is not, by itself, a proof of the generator theorem.

For finite $dt$, do not treat these first-order Kraus operators as an exactly trace-preserving integrator. Use a converged ODE method, a matrix exponential, an exact finite-time channel when available, or a completely positive structure-preserving scheme.

Semigroup statement

A time-homogeneous quantum dynamical semigroup satisfies

$$\Phi_{t+s} = \Phi_t\circ\Phi_s, \qquad \Phi_0=I, \qquad t,s\ge0.$$

If

$$\Phi_t=e^{t\mathcal L}$$

is norm continuous, completely positive, and trace preserving in finite dimension, then its generator has GKSL form. Conversely, a time-independent GKSL generator with nonnegative rates produces a CPTP semigroup.

The theorem determines the allowed generator structure. It does not prove that a proposed laboratory model is Markovian, that its chosen $L_k$ are microscopically correct, or that its fitted rates are valid over all times.

Kossakowski-matrix form

Choose a Hilbert–Schmidt orthonormal traceless operator basis $\{F_a\}$ for a $d$-dimensional system. The dissipative part can be written

$$\mathcal L_{\mathrm{diss}}(\rho) = \sum_{a,b} c_{ab} \left( F_a\rho F_b^\dagger - \frac12 \{F_b^\dagger F_a,\rho\} \right).$$

Complete positivity of the semigroup requires the Kossakowski matrix to be positive semidefinite:

$$C=(c_{ab})\ge0.$$

Diagonalizing

$$C=U^\dagger \operatorname{diag}(\gamma_1,\gamma_2,\ldots) U$$

with $\gamma_k\ge0$ produces the diagonal Lindblad-operator form. This is why the physically relevant signs are the eigenvalues of the canonical rate matrix, not arbitrary coefficients in a nonorthogonal operator expansion.

Time-dependent generators

A time-local equation has

$$\dot\rho(t) = \mathcal L_t(\rho(t)).$$

Its solution is

$$\rho(t) = \mathcal T \exp\left[ \int_0^t \mathcal L_s\,ds \right] \rho(0).$$

If $\mathcal L_t$ has canonical GKSL form with nonnegative rates at every time, the evolution is CP-divisible under the usual regularity assumptions. If a canonical rate becomes negative, the instantaneous generator is not of CP-divisible GKSL form. The overall map from the initial time may nevertheless remain CPTP; complete positivity must then be checked from the full propagator, not inferred from one instantaneous coefficient.

“Time local” and “Markovian semigroup” are therefore not synonyms. A time-dependent Lindblad-form equation is more general than a homogeneous semigroup, while memory-kernel and initially correlated dynamics can lie outside both.

Adjoint evolution

The adjoint generator is defined by

$$\operatorname{Tr} \left[ A\mathcal L(\rho) \right] = \operatorname{Tr} \left[ \mathcal L^\dagger(A)\rho \right].$$

It is

$$\mathcal L^\dagger(A) = \frac{i}{\hbar}[H,A] + \sum_k \gamma_k \left( L_k^\dagger A L_k - \frac12 \{L_k^\dagger L_k,A\} \right).$$

For an observable with explicit time dependence,

$$\frac{d}{dt}\langle A\rangle = \left\langle \mathcal L^\dagger(A) \right\rangle + \left\langle \frac{\partial A}{\partial t} \right\rangle.$$

Trace preservation appears as

$$\mathcal L^\dagger(I)=0.$$

The adjoint form is usually the shortest route to equations for populations, moments, oscillator amplitudes, and conserved observables.

Steady states and relaxation

A steady state must satisfy all three conditions

$$\mathcal L(\rho_{\mathrm{ss}})=0, \qquad \rho_{\mathrm{ss}}\ge0, \qquad \operatorname{Tr}\rho_{\mathrm{ss}}=1.$$

Solving the homogeneous linear equation alone can return nonphysical null vectors, so positivity and normalization are separate checks.

A finite-dimensional trace-preserving semigroup has at least one stationary state under standard compactness assumptions, but it need not be unique. A unique full-rank stationary state does not by itself guarantee a particular thermal form; detailed balance or a microscopic thermal derivation is needed.

For a relaxing finite-dimensional generator, nonzero Liouvillian eigenvalues have nonpositive real parts. A spectral relaxation scale is often defined by

$$\Delta_{\mathcal L} = - \max_{\lambda_j\ne0} \operatorname{Re}\lambda_j,$$

provided the zero mode is isolated and the relevant modes are included. Jordan blocks can add polynomial prefactors to exponential decay. See Steady States and Relaxation for fixed-point structure and gaps.

Vectorized Liouvillian

With column-stacking vectorization,

$$\operatorname{vec}(AXB) = (B^{\mathsf T}\otimes A) \operatorname{vec}(X).$$

The master equation becomes

$$\frac{d}{dt} \lvert\rho\rangle\rangle = \mathbb L \lvert\rho\rangle\rangle.$$

The Hamiltonian part is

$$\mathbb L_H = - \frac{i}{\hbar} \left( I\otimes H - H^{\mathsf T}\otimes I \right),$$

and one dissipator contributes

$$\begin{aligned} \mathbb D[L] ={}& L^*\otimes L - \frac12 I\otimes L^\dagger L \\ &- \frac12 (L^\dagger L)^{\mathsf T}\otimes I. \end{aligned}$$

Thus

$$\mathbb L = \mathbb L_H + \sum_k \gamma_k\mathbb D[L_k].$$

This representation is useful for matrix exponentials, eigenmodes, and steady states, but its Kronecker ordering depends on the vectorization convention. Test it on $\operatorname{vec}(AXB)$ before trusting a software implementation.

Purity diagnostic

Let

$$P=\operatorname{Tr}\rho^2.$$

Then

$$\frac{dP}{dt} = 2\operatorname{Tr} \left[ \rho\mathcal L(\rho) \right].$$

The Hamiltonian commutator contributes zero. For a pure state $\rho=\lvert\psi\rangle\langle\psi\rvert$,

$$\frac{dP}{dt} = - 2 \sum_k \gamma_k \left( \langle L_k^\dagger L_k\rangle - \lvert\langle L_k\rangle\rvert^2 \right) \le0.$$

The inequality follows from Cauchy–Schwarz. A pure state initially loses no purity only when it is a simultaneous zero-variance state of every active $L_k$.

For mixed states, a nonunital channel can increase purity. Amplitude damping, for example, can drive a mixed qubit toward a pure ground state. Do not claim that every Lindblad term monotonically increases entropy or decreases purity.

Quantum-jump form

With rate-weighted jumps

$$J_k=\sqrt{\gamma_k}\,L_k,$$

define

$$H_{\mathrm{eff}} = H - \frac{i\hbar}{2} \sum_k J_k^\dagger J_k.$$

Over $dt$, the jump probability is

$$dp_k = dt\, \operatorname{Tr} \left( J_k^\dagger J_k\rho \right).$$

Conditioned no-jump evolution uses $H_{\mathrm{eff}}$ and is not norm preserving; its norm loss equals the total jump probability to first order. Adding the normalized stochastic branches and averaging over records recovers the unconditional master equation.

The same generator can have inequivalent-looking unravelings corresponding to photon counting, homodyne detection, heterodyne detection, or unitary mixing of jump channels. A Lindblad operator is not automatically an actually observed event.

Example: pure dephasing

For a qubit, choose

$$L=Z, \qquad \gamma=\frac{\Gamma_\phi}{2}.$$

Because $Z^\dagger Z=I$,

$$\dot\rho = \frac{\Gamma_\phi}{2} \left( Z\rho Z-\rho \right).$$

For

$$\rho = \begin{pmatrix} \rho_{00}&\rho_{01}\\ \rho_{10}&\rho_{11} \end{pmatrix},$$

the populations are constant and

$$\dot\rho_{01} = -\Gamma_\phi\rho_{01}.$$

Thus

$$\rho_{01}(t) = e^{-\Gamma_\phi t} \rho_{01}(0)$$

in the interaction picture. If a source instead writes $\Gamma_\phi\mathcal D[Z]$, its off-diagonal decay rate is $2\Gamma_\phi$. This factor-of-two convention must be checked explicitly.

Example: zero-temperature amplitude damping

Take

$$L=\sigma_-=\lvert g\rangle\langle e\rvert, \qquad \gamma=\Gamma_1.$$

In the interaction picture,

$$\dot\rho = \Gamma_1 \mathcal D[\sigma_-]\rho.$$

The matrix elements obey

$$\begin{aligned} \dot\rho_{ee} &= -\Gamma_1\rho_{ee}, \\ \dot\rho_{gg} &= \Gamma_1\rho_{ee}, \\ \dot\rho_{eg} &= - \frac{\Gamma_1}{2} \rho_{eg}. \end{aligned}$$

Therefore

$$\rho_{ee}(t) = e^{-\Gamma_1t}\rho_{ee}(0)$$

and

$$\rho_{eg}(t) = e^{-\Gamma_1t/2}\rho_{eg}(0).$$

The finite-time channel has decay probability

$$p(t)=1-e^{-\Gamma_1t}.$$

Population relaxation alone contributes

$$\frac{1}{T_2} = \frac{1}{2T_1}, \qquad T_1=\Gamma_1^{-1}.$$

Additional pure dephasing gives

$$\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi}.$$

See Amplitude Damping Master Equation for finite temperature and oscillator loss.

Example: thermal qubit

With downward and upward jumps,

$$\dot\rho = \Gamma_\downarrow \mathcal D[\sigma_-]\rho + \Gamma_\uparrow \mathcal D[\sigma_+]\rho.$$

The excited population satisfies

$$\dot\rho_{ee} = - \Gamma_\downarrow\rho_{ee} + \Gamma_\uparrow(1-\rho_{ee}),$$

so

$$\rho_{ee}^{\mathrm{ss}} = \frac{ \Gamma_\uparrow }{ \Gamma_\uparrow+\Gamma_\downarrow }.$$

A thermal Gibbs state follows only when the rates obey the appropriate detailed-balance relation, such as

$$\frac{\Gamma_\uparrow}{\Gamma_\downarrow} = e^{-\beta\hbar\omega_0}$$

for a two-level spacing $\hbar\omega_0$ under the standard weak-coupling thermal assumptions.

Representation freedom

The displayed $H$, rates, and Lindblad operators are not unique.

• Positive rates can be absorbed into $L_k$ as square roots.
• Rate-weighted jump operators can be mixed by a unitary matrix without changing the dissipative sum.
• Identity components of jump operators can be shifted into a compensating Hamiltonian term.
• Degenerate eigenvalues of the Kossakowski matrix allow basis rotations among canonical noise operators.
• Different environment monitoring schemes give different trajectories for the same unconditional $\mathcal L$.

Compare generators as superoperators, not operator lists label by label. A microscopic interpretation requires additional information about the system–bath coupling and the monitored environment observable.

Microscopic validity

The GKSL theorem is structural. A microscopic weak-coupling derivation often also assumes:

• a fixed, initially factorized system–bath state;
• weak system–bath coupling;
• a stationary bath with rapidly decaying correlations;
• coarse graining beyond the bath memory time;
• a Markov approximation;
• a secular or rotating-wave approximation separating Bohr frequencies.

These approximations determine $H$, Lamb shifts, operators, and rates. They can fail for strong coupling, structured reservoirs, short times, nearly degenerate transitions, initial correlations, or appreciable information backflow. Dropping nonsecular terms can be inaccurate, while keeping them naively can produce a generator that is not completely positive.

In infinite-dimensional systems, unbounded $H$ and $L_k$ introduce domain and generator-closure questions absent from the finite-dimensional theorem.

Calculation workflow

1. State the picture, basis, effective Hamiltonian, and whether rates are explicit or absorbed.
2. Check $H^\dagger=H$, dimensions, and nonnegative eigenvalues of the canonical rate matrix.
3. Verify trace preservation analytically using $\mathcal L^\dagger(I)=0$.
4. Derive matrix-element or adjoint-observable equations and confirm known rate factors.
5. Find steady states from $\mathcal L(\rho)=0$, then impose Hermiticity, positivity, and unit trace.
6. If vectorizing, verify the $\operatorname{vec}(AXB)$ convention on a small test matrix.
7. Check finite-time positivity, trace, and Hermiticity numerically for representative states.
8. Vary time step, basis cutoff, and solver tolerance; compare against an exact finite-time channel when available.
9. Audit the microscopic timescale and coupling assumptions separately from the GKSL algebra.

Common mistakes

• Dropping the anticommutator term.
• Counting a rate twice after absorbing its square root into a jump operator.
• Confusing population, amplitude, and coherence decay rates.
• Assuming every time-local equation with temporarily negative rates is a CPTP semigroup.
• Treating nonnegative coefficients in a noncanonical operator basis as the complete-positivity test instead of checking the Kossakowski matrix.
• Assuming a steady state is unique or thermal without further conditions.
• Claiming that every dissipator lowers energy, entropy, or purity.
• Reading Lindblad operators as unique observed jumps.
• Using first-order Kraus steps as exact finite-time channels.
• Reversing Kronecker factors in a vectorized Liouvillian.
• Believing the GKSL theorem validates the microscopic Markov approximation.
• Ignoring domains and truncation effects in oscillator or field models.

Exercises

1. Prove trace preservation for one dissipator and recover its short-time Kraus operators.

Solution

Cyclicity gives

$$\operatorname{Tr}(L\rho L^\dagger) = \operatorname{Tr}(L^\dagger L\rho) = \operatorname{Tr}(\rho L^\dagger L).$$

Hence

$$\operatorname{Tr}\mathcal D[L]\rho=0.$$

For rate $\gamma$, choose

$$K_1=\sqrt{\gamma dt}\,L$$

and

$$K_0 = I - \frac{i}{\hbar}Hdt - \frac{\gamma}{2} L^\dagger L\,dt.$$

Expanding $K_0\rho K_0^\dagger+K_1\rho K_1^\dagger$ through first order gives

$$\rho + dt \left[ - \frac{i}{\hbar}[H,\rho] + \gamma\mathcal D[L]\rho \right].$$

2. For

$$\dot\rho = c \left( Z\rho Z-\rho \right),$$

find the population and coherence equations. What value of $c$ gives coherence-decay rate $\Gamma_\phi$?

Solution

Conjugation by $Z$ leaves the diagonal entries unchanged and changes the sign of each off-diagonal entry. Therefore

$$\dot\rho_{00} = \dot\rho_{11} = 0$$

and

$$\dot\rho_{01} = -2c\rho_{01}.$$

To obtain

$$\dot\rho_{01} = -\Gamma_\phi\rho_{01},$$

choose

$$c=\frac{\Gamma_\phi}{2}.$$

3. For zero-temperature amplitude damping, derive the population and coherence decay rates and the corresponding finite-time Kraus probability.

Solution

With $L=\sigma_-$,

$$\dot\rho_{ee} = -\Gamma_1\rho_{ee}, \qquad \dot\rho_{eg} = - \frac{\Gamma_1}{2}\rho_{eg}.$$

Thus

$$\rho_{ee}(t) = e^{-\Gamma_1t}\rho_{ee}(0)$$

and

$$\rho_{eg}(t) = e^{-\Gamma_1t/2}\rho_{eg}(0).$$

The survival probability of an initial excitation is $e^{-\Gamma_1t}$, so the finite-time amplitude-damping channel has

$$p(t)=1-e^{-\Gamma_1t}.$$

4. Derive the initial purity slope for a pure state under one dissipator with rate $\gamma$.

Solution

For $P=\operatorname{Tr}\rho^2$,

$$\dot P = 2\operatorname{Tr} \left[ \rho\dot\rho \right].$$

The commutator has zero contribution. If $\rho=\lvert\psi\rangle\langle\psi\rvert$, then $\rho^2=\rho$ and

$$\operatorname{Tr} \left( \rho L\rho L^\dagger \right) = \lvert\langle L\rangle\rvert^2,$$

while

$$\operatorname{Tr} \left( \rho L^\dagger L\rho \right) = \langle L^\dagger L\rangle.$$

Therefore

$$\dot P = - 2\gamma \left( \langle L^\dagger L\rangle - \lvert\langle L\rangle\rvert^2 \right) \le0.$$

Canonical links

• Lindblad–GKSL Equation
• Lindblad Theorem
• Lindblad Operators
• Quantum Dynamical Semigroups
• Steady States and Relaxation
• Pure Dephasing Master Equation
• Amplitude Damping Master Equation
• Solving Lindblad Equations

References

• V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely Positive Dynamical Semigroups of N-Level Systems,” Journal of Mathematical Physics 17, 821–825 (1976).
• G. Lindblad, “On the Generators of Quantum Dynamical Semigroups,” Communications in Mathematical Physics 48, 119–130 (1976).
• E. B. Davies, Quantum Theory of Open Systems, Academic Press, 1976.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002, Chs. 3–4.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer, 2012.