Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) Theorem

The GKSL characterization extends beyond matrices when the semigroup is normal and continuous in superoperator norm. Its generator is then bounded, even though the Hilbert space and the list of noise operators may be infinite. The canonical Lindblad Theorem gives the finite-dimensional statement and physical interpretation; this page states the precise bounded infinite-dimensional version and proves that its operator data generate channels.

Required background. Quantum Dynamical Semigroups fixes state and observable pictures. Kraus Representations explains strongly convergent completeness sums.

Normal norm-continuous semigroups have bounded Lindblad data

Let $\mathcal H$ be separable. A semigroup $T_t$ of normal unital CP maps on $B(\mathcal H)$ satisfying $\|T_t-\mathrm{id}\|_{\infty\to\infty}\to0$ has a bounded generator of the form

$$\begin{aligned} \mathcal G(A) &=i[H,A]+\Psi(A)-\frac12(RA+AR),\\ \Psi(A)&=\sum_j L_j^\dagger A L_j,\qquad R=\sum_jL_j^\dagger L_j. \end{aligned}$$

Here $H=H^\dagger$ is bounded, each $L_j$ is bounded, and the finite or countable positive sum defining $R$ converges strongly to a bounded operator. The CP map $\Psi$ is normal. Conversely, any such data generate a normal unital CP norm-continuous semigroup. Units are chosen so that $\hbar=1$; $H$ therefore appears as a frequency operator.

The Schrödinger generator is the bounded predual

$$\mathcal L(\rho) =-i[H,\rho]+\sum_jL_j\rho L_j^\dagger -\frac12(R\rho+\rho R).$$

For positive $\rho$, the jump sum converges in trace norm, and linear extension defines it on all trace-class operators. In particular,

$$\|\mathcal L\|_{1\to1}\le2\|H\|+2\|R\|.$$

The hypothesis is boundedness of the strong sum $R$, not $\sum_j\|L_j\|^2<\infty$. The latter is sufficient but unnecessarily restrictive.

What the characterization theorem contributes

Necessity is the structural part of Lindblad’s theorem. Differentiating complete positivity first yields conditional complete positivity, rather than positivity of $\mathcal G$ itself. The representation theorem separates that conditional positive form into a normal CP part and a derivation plus its normalization terms. Bounded derivations of $B(\mathcal H)$ are implemented by bounded commutators, producing $H$.

These structural steps are invoked here as the theorem, not inferred from trace preservation alone. The converse direction admits a direct construction that also explains why the finite-time map remains positive.

A convergent jump expansion proves sufficiency

Write

$$G=-iH-\frac12R,\qquad \mathcal A(\rho)=G\rho+\rho G^\dagger,\qquad \mathcal J(\rho)=\sum_jL_j\rho L_j^\dagger.$$

The no-jump evolution $S_t(\rho)=e^{tG}\rho e^{tG^\dagger}$ is CP and trace nonincreasing, since $G+G^\dagger=-R\le0$. All operators here are bounded, and $\|\mathcal J\|_{1\to1}=\|R\|$.

Define $V_0(t)=S_t$ and recursively

$$V_{n+1}(t)=\int_0^t S_{t-s}\mathcal J V_n(s)\,ds.$$

Composition and positive integration make every $V_n(t)$ CP. Contractivity of $S_t$ gives $\|V_n(t)\|\le t^n\|\mathcal J\|^n/n!$. Therefore

$$\Phi_t=\sum_{n=0}^\infty V_n(t)$$

converges in superoperator norm and is CP. The bounded integral equation identifies it uniquely with $e^{t(\mathcal A+\mathcal J)}$.

Finally,

$$\operatorname{Tr}\mathcal L(\rho) =\operatorname{Tr}[(G+G^\dagger+R)\rho]=0.$$

Boundedness allows differentiation of the trace along every trajectory, so $\operatorname{Tr}\Phi_t(\rho)=\operatorname{Tr}\rho$. This supplies trace preservation, the semigroup law, and norm continuity. The construction fails to justify an unbounded formula merely by substitution: domains, convergence, and possible probability escape must then be treated separately.

Worked example: infinitely many bounded jumps with a bounded generator

Let $P_n=|n\rangle\langle n|$ be rank-one basis projections and put $L_n=\sqrt{\gamma}P_n$, with $\gamma>0$. Then

$$\sum_nL_n^\dagger L_n=\gamma I \quad\text{strongly},\qquad \sum_n\|L_n\|^2=\infty.$$

If $\Delta(\rho)=\sum_nP_n\rho P_n$ is the normal dephasing channel, the generator is $\mathcal L=\gamma(\Delta-\mathrm{id})$. Since $\Delta^2=\Delta$,

$$\Phi_t=e^{-\gamma t}\mathrm{id} +(1-e^{-\gamma t})\Delta.$$

Every off-diagonal matrix element decays at the same rate, while the diagonal distribution is fixed. The channel is uniformly continuous with a bounded generator despite the failure of norm-square summability of the individual jumps. This illustrates the exact summability condition in the theorem.

Exercises

A scalar noise operator If $L=cI$ for a scalar $c$, compute its dissipator.

Solution

$L\rho L^\dagger=|c|^2\rho$ and $\frac12\{L^\dagger L,\rho\}=|c|^2\rho$, so the dissipator vanishes. A representation can contain a nonzero noise operator that contributes nothing. Lindblad operators are not uniquely observable parameters; a choice of representation matters.

Dissipation of products. Verify that the Heisenberg generator satisfies $\mathcal G(A^\dagger A)-\mathcal G(A^\dagger)A-A^\dagger\mathcal G(A)=\sum_j[L_j,A]^\dagger[L_j,A]$.

Solution

The commutator obeys the Leibniz rule and cancels. For each $j$, expanding the three dissipative terms leaves $L_j^\dagger A^\dagger A L_j-L_j^\dagger A^\dagger L_jA-A^\dagger L_j^\dagger A L_j+A^\dagger L_j^\dagger L_jA$, which is the claimed square. Finite partial sums converge strongly; the bounds from $R$ control the limit.

References

• Gorini, Vittorio, Andrzej Kossakowski, and E. C. George Sudarshan. “Completely positive dynamical semigroups of N-level systems.” Journal of Mathematical Physics 17, 821–825 (1976). doi:10.1063/1.522979. Finite-dimensional characterization.
• Holevo, Alexander S. Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed. De Gruyter, 2019. doi:10.1515/9783110642490. Normal CP maps and quantum dynamical semigroups.
• Lindblad, Göran. “On the generators of quantum dynamical semigroups.” Communications in Mathematical Physics 48, 119–130 (1976). doi:10.1007/BF01608499. Bounded generator theorem on $B(\mathcal H)$.