Lindblad / GKSL Master Equation

A closed quantum system evolves by a unitary rotation and nothing is lost. Open a window to an environment — a bath of photons, phonons, or stray two-level defects — and the reduced state $\rho$ leaks energy and coherence into degrees of freedom you no longer track, so its evolution is no longer unitary. When the bath forgets its own past faster than the system changes, that leakage becomes memoryless, or Markovian, and the whole dynamics collapses into a single fixed linear map on operators, the Lindbladian $\mathcal{L}$ , acting through $\dot{\rho}=\mathcal{L}\rho$ . Integrating once gives a one-parameter family $\Lambda_t=e^{\mathcal{L}t}$ : a quantum dynamical semigroup. The central theorem — proved for finite dimension by Gorini, Kossakowski and Sudarshan and for bounded generators by Lindblad, both in 1976 — says exactly which $\mathcal{L}$ are allowed, and the answer is strikingly rigid. Any legal generator splits into a commutator with a Hermitian Hamiltonian and a dissipator built from jump operators $L_k$ , each carrying a non-negative rate. Bend that structure even slightly — let one effective rate go negative — and the evolution eventually drags some density operator outside the cone of valid states, manufacturing negative probabilities. The GKSL form is therefore not a convenient parametrization but the precise boundary of physical law: it is what "completely positive and trace-preserving at every instant" looks like as a differential equation. Specialize the jumps and the standard channels fall out — $\sigma_-$ gives $T_1$ relaxation, $\sigma_z$ gives dephasing, isotropic Pauli jumps give depolarization — which is why this one equation underlies quantum optics, NMR, and every superconducting-qubit error budget written today.

Key takeaways

The Lindblad (Gorini–Kossakowski–Sudarshan–Lindblad, GKSL) equation is the most general generator of a continuous semigroup of completely positive, trace-preserving (CPTP) maps — of Markovian open-system dynamics.
Canonical form: $\dot{\rho} = -\tfrac{i}{\hbar}[H,\rho] + \sum_k(L_k\rho L_k^\dagger - \tfrac12\{L_k^\dagger L_k,\rho\})$ — a reversible Hamiltonian part plus an irreversible dissipator of jump operators $L_k$ .
Complete positivity is exactly the positive-semidefiniteness of the Kossakowski matrix $a_{mn}$ ; diagonalizing gives jumps with non-negative rates $\gamma_k\ge0$ .
It follows microscopically from a system–bath model via the Born, Markov and secular approximations; dropping the secular step gives the Redfield equation, more accurate briefly but able to violate positivity.
Specializing the jumps reproduces every canonical channel and the coherence relation $1/T_2 = 1/(2T_1) + 1/T_\phi$ , so $T_2\le2T_1$ .
Key idea: Markovian open dynamics is completely positive and trace-preserving at every instant if and only if its generator has the GKSL form — a Hamiltonian commutator plus a dissipator $\sum_k\gamma_k(L_k\rho L_k^\dagger-\tfrac12\{L_k^\dagger L_k,\rho\})$ with all $\gamma_k\ge 0$ , equivalently a positive-semidefinite Kossakowski matrix.

The core insight

Markovian open dynamics is completely positive and trace-preserving at every instant if and only if its generator has the GKSL form: a Hamiltonian commutator plus a dissipator $\sum_k\gamma_k(L_k\rho L_k^\dagger-\tfrac12\{L_k^\dagger L_k,\rho\})$ with all rates $\gamma_k\ge 0$ — equivalently, a positive-semidefinite Kossakowski matrix.

Where this lives today

The Lindblad equation is the daily arithmetic of quantum-hardware engineering. Every superconducting or trapped-ion processor is characterized by its $T_1$ and $T_2$ times, and those numbers are the jump rates $\gamma$ and $\gamma_\phi$ of a two-jump Lindbladian; simulators such as QuTiP and qiskit-dynamics integrate equation (1) with $\sigma_-$ and $\sigma_z$ jumps to predict gate fidelities, leakage, and readout errors, feeding the fault-tolerance thresholds that set how many physical qubits a logical qubit costs. In quantum optics the same equation governs cavity photon loss ( $L=\sqrt{\kappa}\,a$ ) and atomic decay. The frontier has flipped dissipation from foe to tool. In reservoir engineering, one designs jumps whose unique null state $\mathcal{L}\rho_{ss}=0$ is a desired entangled state, so the environment drives the system there and holds it — the trapped-ion experiments of Barreiro et al. (2011) stabilized Bell states this way, central to autonomous quantum error correction. Its most striking descendant is the dissipatively stabilized cat qubit, where an engineered two-photon-loss Lindbladian pins a superconducting cavity to a cat-state manifold exponentially protected against bit flips — the operating principle of the bosonic-code processors now being built.

The physical picture

Picture the density operator as a point inside a convex body: the Hermitian, unit-trace, positive operators. For a qubit that body is the Bloch ball and the point is the Bloch vector $\vec{r}$ . Unitary evolution rotates the ball rigidly — it moves the point but never shrinks the body, because a closed system loses no purity. The dissipator does what a rotation cannot: it deforms the ball inward. Coherences shrink, populations drift toward a fixed distribution, and the point spirals to a steady state $\rho_{ss}$ inside. The GKSL structure is exactly the family of inward "velocity fields" that never push the point out through the surface into negative eigenvalues — that geometric no-escape condition is complete positivity.

Figure

Dissipation contracts the Bloch ball while the Hamiltonian rotates it. Under isotropic depolarization the Bloch vector shrinks uniformly, r(t) = e^(−γt) r(0), spiralling to the maximally mixed centre; amplitude damping instead pulls toward the ground-state pole. The coherence time obeys T₂ ≤ 2T₁

Each jump operator $L_k$ names one elementary event the environment can trigger — emit a photon ( $\sigma_-$ ), deliver a phase kick ( $\sigma_z$ ), flip a spin. The "sandwich" term $L_k\rho L_k^\dagger$ injects the outcome of that jump; the anticommutator $-\tfrac12\{L_k^\dagger L_k,\rho\}$ is the no-jump back-action that renormalizes the state for the fact that, most of the time, no jump happened. Average over the random jump record — the unraveling — and the smooth Lindblad flow reappears; sample individual records and you have the quantum-trajectory method. One slogan holds the two halves: commutator = reversible rotation, anticommutator plus sandwich = irreversible relaxation, and their imbalance drives the system to its fixed point. That fixed point and the rates of approach are what an experimentalist measures. For a qubit they are two times: $T_1$ , the energy-relaxation time, and $T_2$ , the coherence time. The Lindblad equation ties them with an inequality no device escapes — coherence never outlasts twice the relaxation time — and the second figure is exactly that measurement.

Figure

Qubit relaxation from the Lindblad equation with superconducting values T₁ = 100 μs and T₂ = 120 μs. The excited population decays as exp(−t/T₁) in blue, the coherence faster as exp(−t/T₂) in amber. Inverting 1/T₂ = 1/(2T₁) + 1/T_φ gives T_φ = 300 μs; the dashed curve exp(−t/2T₁) is the T₂ = 2T₁ ceiling reached only when pure dephasing vanishes

The core equations

The whole subject hangs on one equation and one positivity condition; everything else — the semigroup law, trace preservation, the adjoint picture, the qubit times — is a consequence or specialization of it. The canonical GKSL generator is a reversible commutator plus an irreversible dissipator: "rotate the state, then let a set of independent noise channels each nibble at it."

\dot{\rho} = -\frac{i}{\hbar}[H,\rho] + \sum_{k}\Big(L_k\rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k,\rho\}\Big) \qquad (1)

where: $\rho$ — reduced density operator, $\rho\ge0$ , $\mathrm{Tr}\,\rho=1$ ; $H$ — effective Hermitian Hamiltonian, possibly including a bath-induced Lamb shift; $L_k$ — jump operators, one per channel, rate absorbed as $L_k=\sqrt{\gamma_k}\,A_k$ ; $\{A,B\}=AB+BA$ . Reading: the commutator alone gives unitary flow; each $L_k$ pairs a "jump in" with a "no-jump" back-action so probability is conserved. Written in a fixed operator basis rather than pre-chosen jumps, the same generator exposes the object that decides legality — the Kossakowski matrix, the form in which the GKSL theorem is stated.

\dot{\rho} = -\frac{i}{\hbar}[H,\rho] + \sum_{m,n=1}^{N^2-1} a_{mn}\Big(F_m\rho F_n^\dagger - \frac12\{F_n^\dagger F_m,\rho\}\Big) \qquad (2)

where: $\{F_m\}$ — orthonormal traceless operator basis, $\mathrm{Tr}(F_m^\dagger F_n)=\delta_{mn}$ ; $a_{mn}$ — Hermitian $(N^2-1)\times(N^2-1)$ Kossakowski (GKS) matrix; $N=\dim\mathcal{H}$ . Reading: the map is completely positive for all $t\ge0$ iff $a\succeq0$ . Diagonalizing $a=u\,\mathrm{diag}(\gamma_k)\,u^\dagger$ and setting $L_k=\sqrt{\gamma_k}\sum_m u_{mk}F_m$ turns (2) back into (1) with non-negative rates — so " $a\succeq0$ " and "all $\gamma_k\ge0$ " are one statement. Integrating once produces the finite-time channels. They form a semigroup, not a group — no inverse, the signature of irreversibility.

\Lambda_t = e^{\mathcal{L}t},\qquad \Lambda_t\circ\Lambda_s = \Lambda_{t+s},\qquad \Lambda_0=\mathrm{id} \qquad (3)

where: $\Lambda_t$ — CPTP propagator from $0$ to $t$ ; $\mathcal{L}$ — Lindbladian superoperator, $\mathcal{L}=\tfrac{\mathrm d}{\mathrm dt}\Lambda_t\big|_{0}$ ; $\circ$ — composition. Reading: each $\Lambda_t$ is a legal channel with a Kraus representation, and the semigroup law is the precise content of "memoryless." Trace preservation is built in term by term, which is why $\rho$ stays a valid probability object at every instant.

\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{Tr}\,\rho = \mathrm{Tr}(\mathcal{L}\rho) = 0 \qquad (4)

where: $\mathrm{Tr}$ — trace over the system Hilbert space. Reading: cyclicity kills the commutator's trace, and inside each channel $\mathrm{Tr}(L_k\rho L_k^\dagger)=\mathrm{Tr}(L_k^\dagger L_k\rho)$ cancels exactly against the anticommutator half — probability is conserved channel by channel, not just on average. Observables evolve under the adjoint generator, the Heisenberg-picture partner of (1). Where $\mathcal{L}$ is trace-preserving, its adjoint $\mathcal{L}^\dagger$ is unital: it fixes the identity.

\mathcal{L}^\dagger(A) = \frac{i}{\hbar}[H,A] + \sum_k\Big(L_k^\dagger A L_k - \frac12\{L_k^\dagger L_k,A\}\Big),\qquad \mathcal{L}^\dagger(\mathbb{1})=0 \qquad (5)

where: $A$ — a system observable, $\dot A=\mathcal{L}^\dagger A$ ; $\mathcal{L}^\dagger$ — adjoint with respect to the Hilbert–Schmidt pairing $\mathrm{Tr}(A\,\mathcal{L}\rho)=\mathrm{Tr}((\mathcal{L}^\dagger A)\rho)$ . Reading: the Hamiltonian term flips sign between the two pictures; the sandwich keeps $L_k^\dagger A L_k$ . Setting $A=\mathbb{1}$ recovers trace preservation as the dual of unitality. Finally, the workhorse special case: a single qubit with a lowering jump — the bridge from the abstract generator to a measured $T_1$ .

L=\sqrt{\gamma}\,\sigma_-,\quad \sigma_-=|0\rangle\langle1|,\quad \rho_{11}(t)=e^{-\gamma t}\rho_{11}(0),\quad \mathcal{L}\rho_{ss}=0\ \Rightarrow\ \rho_{ss}=|0\rangle\langle0| \qquad (6)

where: $\sigma_-$ — lowering operator; $\gamma=1/T_1$ — spontaneous emission rate; $\rho_{ss}$ — stationary state (null eigenvector of $\mathcal{L}$ ). Reading: the excited population decays at rate $\gamma$ , the coherence at $\gamma/2$ , and the unique fixed point is the ground state — dissipation has erased all memory of the initial state.

Derivation AFrom a system coupled to a bath to the GKSL form

The goal is to show that the rigid GKSL structure of equation (1) is not postulated but forced: starting only from a system and a large reservoir evolving unitarily together, three controlled approximations — Born (weak coupling), Markov (short bath memory), and secular (rotating-wave) — reduce the intractable joint dynamics to a closed, completely positive equation for the reduced state alone, with jump operators fixed by the system's energy levels and rates by the bath's spectrum. We work in the interaction picture with $H_{tot}=H_S+H_B+H_{int}$ and coupling $H_{int}=\sum_\alpha A_\alpha\otimes B_\alpha$ .

Step 1. Assume weak coupling and a large bath, factorizing the joint state as $\rho_{tot}(t)\approx\rho(t)\otimes\rho_B$ with $\rho_B$ a fixed stationary bath state, $\mathrm{Tr}_B(B_\alpha\rho_B)=0$ . Why: a macroscopic reservoir is barely perturbed, so its state is frozen and the back-reaction is kept only to second order in $H_{int}$ — the Born approximation.

Step 2. Expand the von Neumann equation to second order and trace out the bath, giving the time-nonlocal equation $\dot{\rho}(t)=-\tfrac{1}{\hbar^2}\int_0^t\mathrm ds\,\mathrm{Tr}_B[H_{int}(t),[H_{int}(s),\rho(s)\otimes\rho_B]]$ . Why: the double commutator of second-order perturbation theory, traced over the bath, packages all environmental influence into two-point correlators $\langle B_\alpha(t)B_\beta(s)\rangle$ .

Step 3. Invoke the Markov approximation: replace $\rho(s)\to\rho(t)$ and extend the integral to $\int_0^\infty$ . Why: the bath correlations decay on a memory time $\tau_B$ far shorter than the relaxation time, so the kernel is sharply peaked at $s=t$ — a bath that has forgotten its past makes the equation local in time.

Step 4. Decompose each coupling operator into eigenoperators of $H_S$ , $A_\alpha(\omega)=\sum_{\epsilon'-\epsilon=\omega}\Pi(\epsilon)A_\alpha\Pi(\epsilon')$ , so each carries a clean phase $e^{-i\omega t}$ . Why: sorting by Bohr frequency lets the next step separate slow from fast terms — each $A_\alpha(\omega)$ is a candidate jump that removes energy $\hbar\omega$ .

Step 5. Apply the secular (rotating-wave) approximation: drop cross terms with $\omega\neq\omega'$ , whose phases $e^{i(\omega'-\omega)t}$ average to zero. What survives is $\dot{\rho}=-\tfrac{i}{\hbar}[H_S+H_{LS},\rho]+\sum_{\omega,\alpha\beta}\gamma_{\alpha\beta}(\omega)\big(A_\beta(\omega)\rho A_\alpha^\dagger(\omega)-\tfrac12\{A_\alpha^\dagger(\omega)A_\beta(\omega),\rho\}\big)$ , with $\gamma_{\alpha\beta}(\omega)$ the Fourier transform of the bath correlation function. Why: keeping only co-rotating terms is what makes the rate matrix a genuine power spectrum — the step that buys complete positivity.

Step 6. Identify $\gamma_{\alpha\beta}(\omega)$ as the Kossakowski matrix of (2). By Bochner's theorem it is positive-semidefinite (the Fourier transform of a positive-definite correlation function); diagonalizing gives the jumps and non-negative rates, while its imaginary part is the Lamb shift $H_{LS}$ , which commutes with $H_S$ . Why: the microscopics do not merely permit the GKSL form — they deliver a manifestly positive rate matrix, so the result is automatically CPTP.

Go deeper · Derivation B · 1 minBloch equations and the T2 ≤ 2T1 relation for a damped qubit

Derivation BBloch equations and the T2 ≤ 2T1 relation for a damped qubit

The goal is to turn the abstract generator into the two decay times an experimentalist reports, and to prove the inequality that constrains every qubit ever built. We project the qubit Lindblad equation onto its matrix elements, letting an amplitude-damping and a pure-dephasing jump act together, and read off the population and coherence rates separately — the first defines $T_1$ , the second $T_2$ , and their relation exposes why $T_2$ can never exceed $2T_1$ . Take $H=\tfrac{\hbar\omega_0}{2}\sigma_z$ , $L_1=\sqrt{\gamma}\,\sigma_-$ , and $L_2=\sqrt{\gamma_\phi/2}\,\sigma_z$ .

Step 1. From $L_1$ , evaluate the dissipator on the excited population. Since $\sigma_+\sigma_-=|1\rangle\langle1|$ acts only on $\rho_{11}$ , the sandwich feeds population down and the anticommutator removes it: $\dot\rho_{11}=-\gamma\rho_{11}$ , so $\rho_{11}(t)=e^{-\gamma t}\rho_{11}(0)$ and $T_1=1/\gamma$ . Why: energy relaxation is a one-way flow $|1\rangle\to|0\rangle$ , appearing as pure exponential decay of the excited occupation.

Step 2. For the coherence $\rho_{01}$ under $L_1$ , the sandwich does not regenerate the off-diagonal, so only the anticommutator acts: $\dot\rho_{01}\big|_{L_1}=-\tfrac{\gamma}{2}\rho_{01}$ . Why: the no-jump back-action damps coherence at exactly half the population rate — the geometric origin of the $1/(2T_1)$ term in $T_2$ .

Step 3. For $L_2$ , $L_2^\dagger L_2=\tfrac{\gamma_\phi}{2}\mathbb{1}$ , so the anticommutator gives $-\tfrac{\gamma_\phi}{2}\rho_{01}$ while $\tfrac{\gamma_\phi}{2}\sigma_z\rho\sigma_z$ flips off-diagonal signs, adding another $-\tfrac{\gamma_\phi}{2}\rho_{01}$ ; together $\dot\rho_{01}\big|_{L_2}=-\gamma_\phi\rho_{01}$ , populations untouched. Why: pure dephasing scrambles phase without moving energy — the decoherence that survives even an infinitely long-lived qubit.

Step 4. Add both plus Hamiltonian precession: $\dot\rho_{01}=\big(i\omega_0-\tfrac{\gamma}{2}-\gamma_\phi\big)\rho_{01}$ , so $1/T_2=\gamma/2+\gamma_\phi=1/(2T_1)+1/T_\phi$ . Since $\gamma_\phi\ge0$ , $T_2\le2T_1$ , with equality only when pure dephasing vanishes. Why: coherence decays at least as fast as half the energy relaxation, so no engineering pushes $T_2$ past twice $T_1$ — the ceiling is set by the anticommutator, not the noise.

Limiting cases

Specializing the jump operators of a single qubit reproduces the canonical noise channels — the entire zoo is one Lindblad equation with different $L_k$ .

Case	Result	Meaning
Amplitude damping ( $T_1$ )	$L=\sqrt{\gamma}\,\sigma_-$ , $\ \dot\rho=\gamma(\sigma_-\rho\sigma_+-\tfrac12\{\sigma_+\sigma_-,\rho\})$	Excited population decays as $e^{-\gamma t}$ , so $T_1=1/\gamma$ ; integrating gives the amplitude-damping channel with $p=1-e^{-\gamma t}$ .
Pure dephasing ( $T_2$ )	$L=\sqrt{\gamma_\phi/2}\,\sigma_z$ , $\ \dot\rho_{01}=-\gamma_\phi\rho_{01}$	Off-diagonal coherence decays at rate $\gamma_\phi$ with populations frozen; this is the phase-damping channel.
Combined relaxation	$\dfrac{1}{T_2}=\dfrac{1}{2T_1}+\dfrac{1}{T_\phi}$	Adding $\sigma_-$ and $\sigma_z$ jumps gives the universal relation; $\gamma_\phi\ge0$ forces $T_2\le2T_1$ .
Depolarizing	$\dot\rho=\dfrac{\gamma}{4}\sum_{j=x,y,z}(\sigma_j\rho\sigma_j-\rho)$	Isotropic Pauli jumps shrink the Bloch vector uniformly, $\vec r(t)=e^{-\gamma t}\vec r(0)$ ; the fixed point is the maximally mixed state $\mathbb{1}/2$ .
Finite temperature	$L_\downarrow=\sqrt{\gamma(1+\bar n)}\,\sigma_-$ , $\ L_\uparrow=\sqrt{\gamma\bar n}\,\sigma_+$	Detailed balance gives the thermal (Gibbs) steady state, $\rho_{11}^{ss}=\bar n/(1+2\bar n)=1/(e^{\beta\hbar\omega_0}+1)$ .
Zero coupling	all $L_k\to0$	$\mathcal{L}=-\tfrac{i}{\hbar}[H,\cdot]$ ; the semigroup becomes a unitary group — closed-system dynamics.

Common trapReading a negative "rate" off the dissipator, or treating the H and L split as physical

Two traps live in the same equation. First, the split of $\mathcal{L}$ into a Hamiltonian $H$ and jumps $\{L_k\}$ is not unique: you may shift $L_k\to L_k+c_k\mathbb{1}$ and absorb a Hermitian term into $H$ , or mix the jumps by any unitary $L_k\to\sum_j u_{kj}L_j$ , without changing the physics. Only the full generator $\mathcal{L}$ is physical; the "number of jumps" and the "Hamiltonian" are conventions. Second, a rate read off a non-diagonal dissipator can be momentarily negative without any problem — common in exact time-local (time-convolutionless) master equations. Complete positivity is positivity of the whole matrix $a_{mn}$ , not of each term. Conversely, forcing every term to look dissipative — the secular step — is what can distort short-time dynamics.

Worked examples

1 · Extracting the pure-dephasing time from measured T1 and T2 (fully worked)

A transmon is characterized by two pulse sequences: inversion recovery gives $T_1=100\ \mu s$ , and Ramsey gives $T_2=120\ \mu s$ . What pure-dephasing time $T_\phi$ does the environment impose, and is the result physically consistent?

Step 1. The Lindblad model with $\sigma_-$ and $\sigma_z$ jumps gives $1/T_2=1/(2T_1)+1/T_\phi$ (Derivation 2). Solve for the pure-dephasing rate: $1/T_\phi=1/T_2-1/(2T_1)$ .

Step 2. Insert the numbers: $1/T_\phi=\dfrac{1}{120}-\dfrac{1}{2\cdot100}=\dfrac{1}{120}-\dfrac{1}{200}=\dfrac{200-120}{24000}=\dfrac{80}{24000}=\dfrac{1}{300}\ \mu s^{-1}$ , so $T_\phi=\mathbf{300\ \mu s}$ .

Step 3. Check consistency. The relation forces $T_2\le2T_1=200\ \mu s$ ; the measured $120\ \mu s$ sits below the ceiling, so the data are physical — a fit returning $T_2>200\ \mu s$ would signal non-Markovian noise or miscalibration, not a real $T_\phi$ . The budget splits as relaxation $1/(2T_1)=1/200$ and dephasing $1/300$ : comparable, neither alone limiting. $T_\phi=300\ \mu s$ , and $T_2=120\ \mu s<2T_1=200\ \mu s$ , so the characterization is self-consistent.

2 · The thermal steady state of a qubit (fully worked)

A qubit at $\omega_0/2\pi=5\ \mathrm{GHz}$ in a dilution refrigerator at $T=20\ \mathrm{mK}$ is coupled to the thermal photon bath by $L_\downarrow=\sqrt{\gamma(1+\bar n)}\,\sigma_-$ and $L_\uparrow=\sqrt{\gamma\bar n}\,\sigma_+$ . What excited-state population survives in equilibrium?

Step 1. In steady state the up-rate into $|1\rangle$ balances the down-rate: $\gamma(1+\bar n)\rho_{11}=\gamma\bar n(1-\rho_{11})$ , so $\rho_{11}^{ss}=\dfrac{\bar n}{1+2\bar n}=1/(e^{\beta\hbar\omega_0}+1)$ — the Lindblad fixed point reproduces the Gibbs population and detailed balance automatically.

Step 2. Compute the mean thermal occupation $\bar n=1/(e^{\hbar\omega_0/k_BT}-1)$ . The exponent is $\dfrac{\hbar\omega_0}{k_BT}=\dfrac{hf}{k_BT}=\dfrac{(6.626\times10^{-34})(5\times10^{9})}{(1.381\times10^{-23})(0.020)}=\mathbf{12.0}$ . So $\bar n=1/(e^{12}-1)\approx e^{-12}=\mathbf{6.1\times10^{-6}}$ .

Step 3. Since $\bar n\ll1$ , $\rho_{11}^{ss}\approx\bar n=\mathbf{6.2\times10^{-6}}$ : the qubit is frozen into its ground state to a few parts per million. This is why millikelvin operation matters — at $\hbar\omega_0/k_BT=12$ any measured excited population much above $10^{-5}$ points to non-thermal sources (stray infrared, quasiparticles), not the equilibrium bath. $\bar n=6.1\times10^{-6}$ , $\rho_{11}^{ss}\approx6.2\times10^{-6}$ — a near-perfect ground state.

3 · From the generator to the amplitude-damping channel

Integrating equation (6) over a time $t$ yields the two-Kraus amplitude-damping channel. The solutions $\rho_{11}(t)=e^{-\gamma t}\rho_{11}(0)$ and $\rho_{01}(t)=e^{-\gamma t/2}\rho_{01}(0)$ (Derivation 2 with $\gamma_\phi=0$ ) are reproduced exactly by $K_0=\left(\begin{smallmatrix}1&0\\0&\sqrt{1-p}\end{smallmatrix}\right)$ , $K_1=\left(\begin{smallmatrix}0&\sqrt{p}\\0&0\end{smallmatrix}\right)$ with $p=1-e^{-\gamma t}$ , since $K_0^\dagger K_0+K_1^\dagger K_1=\mathbb{1}$ . So $\Lambda_t=e^{\mathcal Lt}$ is the amplitude-damping channel with damping probability $p(t)=1-e^{-\gamma t}$ — generator and channel are one object at two resolutions.

Physics of the portrait

The Lindblad equation is one of those rare results that arrived twice in the same year, from two directions, and turned out to be the same theorem — the meeting point of rigorous operator mathematics and the physicist's need for a master equation that never lies about probabilities.

Origins · 1 minGöran Lindblad and how this was discovered

Göran Lindblad

1940–2022 · open quantum systems and quantum dynamical semigroups

Göran Lindblad was a Swedish theoretical physicist at the KTH Royal Institute of Technology in Stockholm whose name is now attached to the most-used equation in the theory of open quantum systems. In 1976 he published "On the generators of quantum dynamical semigroups" in Communications in Mathematical Physics, proving that any norm-continuous semigroup of completely positive, trace-preserving maps has a generator of exactly the canonical dissipative form, for bounded generators in arbitrary dimension. In the same year, and independently, Vittorio Gorini, Andrzej Kossakowski, and E. C. George Sudarshan proved the finite-dimensional version from the Kossakowski matrix and its positive-semidefiniteness — the four names together give the GKSL label. Lindblad's route was the more abstract, grounded in the completely-positive-map theory Stinespring had built in 1955 and Kraus was formalizing in parallel. He went on to foundational work on quantum entropy, including the monotonicity of relative entropy under CPTP maps, which quietly underpins modern quantum information theory. His single 1976 paper is cited tens of thousands of times by physicists modelling everything from spontaneous emission to superconducting qubits. He died in 2022, having given decoherence its equation of motion.

History

1928

— Pauli writes a master equation for transition probabilities among energy levels

1960

— Zwanzig casts open-system dynamics as an exact projection-operator generalized master equation

1974

— Davies builds a rigorous theory of Markovian quantum dynamical semigroups

1976

— Lindblad, and independently Gorini, Kossakowski and Sudarshan, fix the canonical GKSL generator

1992

— Dalibard, Castin and Mølmer unravel the master equation into stochastic quantum trajectories

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, §8.2–8.4 · Preskill, Lecture Notes on Quantum Computation, Ch. 3 · Watrous, The Theory of Quantum Information, Ch. 2 & 4 · Wilde, Quantum Information Theory, Ch. 4 · Breuer & Petruccione, The Theory of Open Quantum Systems, Ch. 3 (the definitive microscopic treatment) · Lindblad, Commun. Math. Phys. 48, 119 (1976); Gorini, Kossakowski, Sudarshan, J. Math. Phys. 17, 821 (1976). All equations are standard GKSL and quantum-optics results. Key consistency checks: the theorem $a\succeq0\Leftrightarrow$ complete positivity is established in forms (1) and (2); trace preservation (4) and the adjoint/unital duality (5) follow from the Hilbert–Schmidt pairing; the qubit relations $T_1=1/\gamma$ and $1/T_2=1/(2T_1)+1/T_\phi$ , hence $T_2\le2T_1$ , are derived in the Bloch-equation derivation; the worked values $T_\phi=300\ \mu s$ (from $T_1=100\ \mu s$ , $T_2=120\ \mu s$ ) and $\bar n=6.1\times10^{-6}$ , $\rho_{11}^{ss}\approx6.2\times10^{-6}$ at $5$ GHz, $20$ mK ( $\hbar\omega_0/k_BT=12.0$ ) are computed in the worked examples; and the infinitesimal-Kraus expansion recovers $\mathcal{L}$ to first order in $\mathrm{d}t$ . No non-standard constants are used.

Prerequisites. Open Quantum Systems and Reduced Dynamics · Quantum Operations and CPTP Maps · Kraus Representation of Quantum Channels · Markovian vs Non-Markovian Dynamics. Leads to. Relaxation and Dephasing: T1 and T2 Times · Amplitude Damping Channel · Phase Damping Channel · Depolarizing Channel · Master Equation Derivations: Born–Markov and Secular Approximations · Process Tomography and the Choi–Jamiołkowski Isomorphism.

Ask the tutor: Ask me to derive the amplitude-damping Kraus operators from the generator, to work through why the Kossakowski matrix must be positive-semidefinite, or to set up the Lindbladian for your own qubit's measured $T_1$ and $T_2$ .

What is the difference between the Lindblad and the Redfield master equations?

Both begin from the same Born–Markov treatment of a system weakly coupled to a memoryless bath. The Lindblad (GKSL) equation adds one step: the secular (rotating-wave) approximation, which discards cross terms oscillating at differences of Bohr frequencies. That step forces the rate matrix to be positive-semidefinite, so the evolution is completely positive and $\rho$ never develops a negative eigenvalue. The Redfield equation keeps those cross terms; it is often more accurate at short and intermediate times, but pays for that accuracy by occasionally producing negative populations. Rule of thumb: use Lindblad when you need a guaranteed-valid density matrix, and Redfield (or a time-convolutionless scheme) when you need short-time accuracy and can tolerate small positivity violations.

Are the Hamiltonian H and the jump operators Lk unique for a given dynamics?

No — only the full generator $\mathcal{L}$ , and its action on states, is physical. Two gauge freedoms leave $\mathcal{L}$ untouched. First, adding a constant to a jump, $L_k\to L_k+c_k\mathbb{1}$ , and absorbing the leftover Hermitian piece into $H$ — this can even give a "purely dissipative" model an effective Hamiltonian. Second, mixing the jumps by any unitary, $L_k\to\sum_j u_{kj}L_j$ , which changes their number and appearance but not the dissipator. So "this system has three jump operators" or "the Hamiltonian is such-and-such" are conventions unless you fix a gauge (say, traceless jumps). What is invariant is the spectrum of the Kossakowski matrix — the physical rates.

Why can the coherence time T2 never exceed twice the relaxation time T1?

Because energy relaxation by itself already destroys coherence at half its own rate. When the excited state decays via $L=\sqrt{\gamma}\,\sigma_-$ , the no-jump back-action $-\tfrac12\{L^\dagger L,\rho\}$ damps the off-diagonal at rate $\gamma/2$ , even with no pure dephasing at all. So $1/T_2=1/(2T_1)+1/T_\phi$ , and since $1/T_\phi\ge0$ , we get $T_2\le2T_1$ , with equality only in the ideal $T_1$ -limited case. This is why $T_2\approx2T_1$ is a badge of honor: all coherence loss comes from unavoidable relaxation, none from stray phase noise. A measured $T_2>2T_1$ is a red flag for non-Markovian effects or calibration error.

What does the semigroup property Λt Λs = Λt+s actually encode?

It is "no memory" made precise. If evolving for $s$ then for $t$ equals evolving once for $t+s$ , the map forward depends only on the current state, never on how the system got there — the environment keeps no record to feed back. That is the Markov assumption, and it is why the dynamics compresses into a single time-independent generator $\mathcal{L}$ . Note it is a semigroup, not a group: unlike unitary evolution there is generally no $\Lambda_{-t}$ , because you cannot un-mix information that has leaked into the environment. That missing inverse is the arrow of time — entropy grows, purity falls, and the system forgets its initial condition as it relaxes.

How does the smooth Lindblad equation relate to random quantum jumps?

They are the same physics at two levels. The Lindblad equation gives the ensemble average — the density matrix from averaging infinitely many runs. But you can "unravel" it into stochastic trajectories: a pure state evolving under the non-Hermitian effective Hamiltonian $H-\tfrac{i}{2}\sum_k L_k^\dagger L_k$ between sudden jumps $\psi\to L_k\psi/\|L_k\psi\|$ , which fire at rates $\langle L_k^\dagger L_k\rangle$ . Average many trajectories and you recover $\rho(t)$ exactly. This Monte Carlo wavefunction method (Dalibard–Castin–Mølmer, 1992) is both a computational tool — it propagates a vector of size $N$ rather than a matrix of size $N^2$ — and a faithful model of what a single continuously monitored experiment, like a photon-counted atom, actually records.

Can dissipation ever be useful rather than just harmful?

Yes — one of the liveliest ideas in the field. Because the Lindblad equation has a fixed point $\mathcal{L}\rho_{ss}=0$ , you can engineer the jump operators so that steady state is something you want, then let the environment prepare and hold it for free, with no active feedback. Choose $L_k$ whose unique dark state is a Bell state and the system relaxes there from any start — this "reservoir engineering" underlies autonomous quantum error correction and the dissipatively stabilized cat qubit. Far from being only the enemy of coherence, a shaped dissipator can be the very mechanism that creates and defends it.