Quantum Dynamical Semigroups

A quantum dynamical semigroup is a one-parameter family of quantum channels $\{\Phi_t:t\ge0\}$ satisfying

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

The parameter $t$ is elapsed time. The composition law says that evolving for time $s$ and then for time $t$ is the same as evolving once for time $t+s$.

When the maps are completely positive, trace preserving, and continuous at $t=0$, the finite-dimensional generator is characterized by the Lindblad Theorem. This page explains the semigroup property itself: what it means, what it rules out, and how to recognize it in examples.

Helpful background. Quantum channels describe one interval; the Lindblad theorem characterizes a compatible continuous family in finite dimension.

Channel composition and continuity

Let $\mathcal H$ be finite dimensional unless stated otherwise. A quantum dynamical semigroup on states is a family of linear maps

$$\Phi_t:\mathcal T(\mathcal H)\to\mathcal T(\mathcal H), \qquad t\ge0,$$

such that:

• $\Phi_0=\mathrm{id}$;
• $\Phi_{t+s}=\Phi_t\Phi_s$ for $t,s\ge0$;
• each $\Phi_t$ is trace preserving;
• each $\Phi_t$ is completely positive;
• $t\mapsto\Phi_t$ is continuous at $t=0$ in the appropriate topology.

In finite dimensions, the continuity assumption gives a bounded generator

$$\mathcal L = \lim_{t\downarrow0} \frac{\Phi_t-\mathrm{id}}{t},$$

and

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

The corresponding master equation is

$$\frac{d\rho(t)}{dt} = \mathcal L(\rho(t)), \qquad \rho(t)=\Phi_t(\rho(0)).$$

For the finite-dimensional numerical version of this generator picture, including $e^{t\mathcal L}$, steady states, and Liouvillian spectra, see Solving Lindblad Equations.

Physical Meaning

The semigroup property combines two ideas.

First, the evolution is memoryless at the reduced level. If the system state at time $s$ is $\rho(s)$, then the future state after an additional time $t$ is

$$\rho(s+t) = \Phi_t(\rho(s)).$$

This is sufficient to predict the unperturbed one-time reduced states within the model. It does not establish that the same propagator describes arbitrary measurements, resets, and interventions in an actual correlated environment; those require a multi-time process description.

Second, the evolution is time homogeneous. The map for an interval of length $t$ is the same no matter when the interval begins. A driven or aging environment may still be Markovian in a broader sense, but it is not described by a time-homogeneous one-parameter semigroup.

Semigroup Versus Group

Closed-system unitary dynamics forms a group:

$$\mathcal U_t(\rho) = U(t)\rho U^\dagger(t), \qquad U(t)=e^{-iHt/\hbar},$$

with

$$\mathcal U_{t+s}=\mathcal U_t\mathcal U_s$$

and inverses $\mathcal U_{-t}$.

In finite dimension $e^{t\mathcal L}$ is always invertible as a linear map for finite $t$, with inverse $e^{-t\mathcal L}$. That inverse is generally not positive on all states and therefore is not a physical channel. For dephasing, the inverse multiplies transverse Bloch components by $e^{\Gamma t}>1$; applying it to a pure equatorial state leaves the Bloch ball.

The inverse is positive on states known to lie in the forward image, but it cannot act as a channel on arbitrary inputs. Exact information erasure by a singular map, such as complete dephasing, is reached only as an infinite-time limit of this bounded-generator semigroup.

Generator and Relaxation Modes

If

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

then $\mathcal L$ is the infinitesimal generator. Eigenoperators of $\mathcal L$ evolve simply. If

$$\mathcal L(X_\alpha)=\lambda_\alpha X_\alpha,$$

then

$$\Phi_t(X_\alpha)=e^{\lambda_\alpha t}X_\alpha.$$

Stationary states satisfy

$$\mathcal L(\rho_{\mathrm{ss}})=0,$$

so they are fixed by the semigroup:

$$\Phi_t(\rho_{\mathrm{ss}})=\rho_{\mathrm{ss}}.$$

Every eigenvalue obeys $\operatorname{Re}\lambda_\alpha\le0$ in a finite-dimensional CPTP semigroup. Bounded evolution forbids growing modes. Eigenvalues on the imaginary axis have no nontrivial Jordan blocks; decaying eigenvalues may have such blocks, producing polynomial factors multiplying exponentials. Their real parts give decay rates, and their imaginary parts give oscillation frequencies.

The general fixed-point and relaxation-mode language is collected in Steady States and Relaxation. In Reservoir Engineering, the zero eigenmodes and the smallest nonzero decay rates become design data: they determine whether a target state is actually stationary, unique, and attracting on a useful timescale.

Lindblad Form

The finite-dimensional CPTP semigroup generator has the form

$$\mathcal L(\rho) = - \frac{i}{\hbar}[H,\rho] + \sum_j \left( L_j\rho L_j^\dagger - \frac12\{L_j^\dagger L_j,\rho\} \right).$$

The Hamiltonian term generates reversible coherent motion. The dissipators generate irreversible or noisy reduced dynamics while preserving trace and complete positivity.

The Lindblad–GKSL Equation is the working form of this generator. The Lindblad Theorem states the finite-dimensional characterization.

Classical Analogy

A classical continuous-time Markov chain has

$$p(t)=e^{tQ}p(0),$$

where $Q$ is a rate matrix. The transition matrices satisfy

$$P(t+s)=P(t)P(s), \qquad P(t)=e^{tQ}.$$

Quantum dynamical semigroups are the density-operator analogue. The generator $\mathcal L$ plays the role of $Q$, but it must preserve quantum positivity in the stronger complete-positivity sense.

Example: Pure Dephasing

A qubit dephasing semigroup in the $\sigma_z$ basis has

$$\rho_{01}(t) = e^{-\Gamma_\phi t}\rho_{01}(0), \qquad \rho_{00}(t)=\rho_{00}(0).$$

The semigroup law is just multiplication of coherence factors:

$$e^{-\Gamma_\phi(t+s)} = e^{-\Gamma_\phi t} e^{-\Gamma_\phi s}.$$

For this scalar, population-preserving model, continuous composition enforces an exponential coherence factor (possibly with a constant phase frequency). More general scalar dephasing functions need not form a semigroup. This does not mean every observable in a general semigroup must be a single exponential.

See Dephasing Channel for finite-time channel constraints.

Example: Amplitude Damping

For zero-temperature qubit amplitude damping,

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

The decay probability by time $t$ is

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

If damping for time $s$ is followed by damping for time $t$, the total decay probability is

$$p(t+s) = p(s)+p(t)-p(s)p(t).$$

This agrees with

$$1-e^{-\Gamma(t+s)} = 1-e^{-\Gamma s}e^{-\Gamma t}.$$

Thus the exponential survival probability is the semigroup-compatible parameter. The finite-time Amplitude-Damping Channel can be written for any $0\le p\le1$, but the Markovian semigroup path has $p(t)=1-e^{-\Gamma t}$.

Example: Depolarizing Semigroup

A $d$-dimensional depolarizing semigroup has

$$\Phi_t(\rho) = e^{-\Gamma t}\rho + \left( 1-e^{-\Gamma t} \right) \frac{I}{d}.$$

The shrink factor is

$$\lambda(t)=e^{-\Gamma t}.$$

Composition multiplies shrink factors:

$$\lambda(t+s)=\lambda(t)\lambda(s).$$

The full finite-time depolarizing channel allows a wider range of $\lambda$, including some negative values. Those negative values can be completely positive channels, but they are not reached by this simple continuous-time relaxation semigroup with $\Gamma\ge0$. See Depolarizing Channel.

Time-Dependent Markovian Evolution

If the generator depends on time,

$$\frac{d\rho}{dt} = \mathcal L_t(\rho),$$

then the solution is generally a two-time propagator

$$\Phi(t,s), \qquad \rho(t)=\Phi(t,s)\rho(s).$$

It satisfies the propagator composition rule

$$\Phi(t,r)\Phi(r,s)=\Phi(t,s),$$

but it need not satisfy the one-parameter semigroup law $\Phi_{t+s}=\Phi_t\Phi_s$. The map depends on both start and end times, not only elapsed time.

Time-dependent Lindblad form with nonnegative rates is often called time-dependent Markovian or CP-divisible dynamics. It is not the same as a time-homogeneous quantum dynamical semigroup.

Markovian and Non-Markovian Noise compares a fitted finite channel, repeated fixed steps, semigroups, CP-divisible evolution, and multi-time memory.

Relation to Memory

Memory-kernel equations have explicit dependence on earlier reduced states:

$$\frac{d\rho(t)}{dt} = \int_0^t ds\, \mathcal K(t,s)\rho(s).$$

They do not generally define a semigroup because the future cannot be computed from $\rho(t)$ alone using a fixed elapsed-time map.

Time-convolutionless equations are subtler. They are time local:

$$\frac{d\rho(t)}{dt} = \mathcal K_{\mathrm{TCL}}(t)\rho(t),$$

but a local generator may describe processes with environmental memory. Time dependence alone does not prove memory: controlled time-dependent GKSL generators also exist. Time local does not automatically mean semigroup.

For the contrast, see Memory Kernels, Time-Convolutionless Master Equations, CP Divisibility, and the terminology guide What Non-Markovian Means.

A nonexponential population inside a semigroup

Take a three-level cascade with jumps $\sqrt\Gamma|1\rangle\langle2|$ and $\sqrt\Gamma|0\rangle\langle1|$, no Hamiltonian, and initial state $|2\rangle$. The constant generator gives

$$\dot p_2=-\Gamma p_2,\qquad \dot p_1=\Gamma p_2-\Gamma p_1.$$

Multiplying the second equation by $e^{\Gamma t}$ and integrating yields

$$p_2=e^{-\Gamma t},\qquad p_1=\Gamma t\,e^{-\Gamma t},\qquad p_0=1-(1+\Gamma t)e^{-\Gamma t}.$$

The middle-level population first rises, peaks at $t=1/\Gamma$ with value $1/e$, then falls. This is an exact time-homogeneous semigroup: the nonexponential observable results from feeding and draining a level, not from environmental memory. Test the composition of full maps, rather than inferring a process from one decay curve.

Exercises

Exponential maps form a semigroup

Show that $\Phi_t=e^{t\mathcal L}$ satisfies

$$\Phi_{t+s}=\Phi_t\Phi_s$$

for a time-independent generator $\mathcal L$.

Solution

Because $\mathcal L$ commutes with itself at all times,

$$e^{(t+s)\mathcal L} = e^{t\mathcal L}e^{s\mathcal L}.$$

Thus

$$\Phi_{t+s} = e^{(t+s)\mathcal L} = e^{t\mathcal L}e^{s\mathcal L} = \Phi_t\Phi_s.$$

Dephasing composition

A dephasing channel multiplies $\rho_{01}$ by $\eta(t)$. What condition on $\eta$ is required by the semigroup law?

Solution

Composition multiplies coherence factors, so the semigroup law requires

$$\eta(t+s)=\eta(t)\eta(s), \qquad \eta(0)=1.$$

For continuous positive real decay, the solution is

$$\eta(t)=e^{-\Gamma t}$$

with $\Gamma\ge0$.

Amplitude-damping probabilities

If $p(t)=1-e^{-\Gamma t}$, verify that two damping intervals combine as

$$p(t+s)=p(t)+p(s)-p(t)p(s).$$

Solution

Substitute:

$$p(t)+p(s)-p(t)p(s) = (1-e^{-\Gamma t}) + (1-e^{-\Gamma s}) - (1-e^{-\Gamma t})(1-e^{-\Gamma s}).$$

Expanding and canceling gives

$$1-e^{-\Gamma t}e^{-\Gamma s} = 1-e^{-\Gamma(t+s)} = p(t+s).$$

Time-dependent rate

For

$$\dot\rho = \gamma(t)\mathcal D[L]\rho,$$

why is the evolution generally not a time-homogeneous semigroup?

Solution

The accumulated decay depends on the integral

$$\int_s^t du\,\gamma(u),$$

which generally depends on the start time $s$, not only the elapsed time $t-s$. Therefore the propagator is a two-time map $\Phi(t,s)$ rather than a one-parameter family $\Phi_{t-s}$ with $\Phi_{t+s}=\Phi_t\Phi_s$.

References

• R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Springer (1987).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002). Publisher.
• E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976).
• 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). DOI.
• G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics 48, 119–130 (1976). DOI.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012). Author manuscript.