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.

Definition

Let $\mathcal H$ be a system Hilbert space. 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

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

No extra environment variables, preparation history, or hidden correlations are needed in the reduced 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 $\mathcal U_{-t}$.

Dissipative quantum dynamical semigroups usually only exist for $t\ge0$. The inverse map may not be positive or even well defined on the range. Information can be lost from the reduced system into the ignored environment, so backward time evolution need not be a physical channel.

This is why the word is semigroup, not group.

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:

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

Nonzero modes typically have $\operatorname{Re}\lambda_\alpha\le0$ in stable finite-dimensional CPTP semigroups. 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:

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

This is why exponential decay is the natural time-homogeneous Markovian form. More general dephasing functions can still define valid channels, but they need not form a semigroup.

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 owns the QI-facing tests that distinguish a fitted finite channel, fixed-step powers, a time-homogeneous semigroup, time-dependent CP-divisible evolution, and multitime memory; this page retains semigroup and generator theory.

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 the generator can carry memory through its time dependence or singularities. 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.

Diagnostic Questions

Before calling a model a quantum dynamical semigroup, check:

• Is the map family indexed by elapsed time $t\ge0$ only?
• Does $\Phi_0$ equal the identity?
• Does $\Phi_{t+s}=\Phi_t\Phi_s$ hold for all $t,s\ge0$?
• Is each $\Phi_t$ trace preserving and completely positive?
• Is the generator time independent?
• Does the same generator apply throughout the interval of interest?
• Are initial correlations, driving, feedback, or bath aging being hidden?
• Does the fitted finite-time channel embed into a continuous semigroup?

Failure of any one item does not make the model useless. It means the word “semigroup” should not be used for that structure.

Common Mistakes

Confusing a channel with a semigroup

A single CPTP map is not a semigroup. A semigroup is a whole compatible family $\{\Phi_t\}$ with a composition law.

Confusing time local with semigroup

A time-local equation with $\mathcal L_t$ can be useful without being time homogeneous. Semigroup dynamics requires a fixed generator.

Requiring a physical inverse

Dissipative semigroups usually cannot be extended to CPTP maps at negative times. Irreversible reduced dynamics is compatible with a semigroup for $t\ge0$.

Treating exponential decay as merely a fit

In semigroup models, exponential decay follows from time-homogeneous composition. Nonexponential decay is often a signal of time dependence, memory, finite reservoirs, or a changing effective rate.

Inferring microscopic assumptions from the semigroup alone

A semigroup can be a phenomenological model. Microscopic validity still requires assumptions about coupling strength, bath correlations, secularization, temperature, and preparation.

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

• 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).
• 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).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).