Quantum Dynamical Semigroups on the Trace Class

A conservative quantum dynamical semigroup is a continuous evolution on normal states whose maps compose according to elapsed time and preserve complete positivity and trace. The canonical Quantum Dynamical Semigroups article develops its physical interpretation. This extension fixes the Banach space and continuity convention and explains how a master equation remains meaningful when its generator is unbounded.

Required background. Quantum Channels in Infinite Dimensions defines the state maps. Helpful background. Strong Continuity distinguishes pointwise state continuity from operator-norm continuity.

Conservative semigroups on the trace class

On a separable Hilbert space $\mathcal H$, let $\Phi_t:\mathcal S_1(\mathcal H)\to\mathcal S_1(\mathcal H)$, $t\ge0$, satisfy:

• $\Phi_0=\mathrm{id}$ and $\Phi_{t+s}=\Phi_t\Phi_s$;
• each $\Phi_t$ is CP and trace preserving;
• $\|\Phi_t(X)-X\|_1\to0$ as $t\downarrow0$ for every trace-class $X$.

These are the conventions used here. Some literature allows trace-decreasing maps under the name “quantum dynamical semigroup” and adds conservative or Markov for the trace-preserving case.

Each map is a contraction in trace norm. The adjoints $T_t=\Phi_t^\dagger$ form a semigroup of normal unital CP maps on $B(\mathcal H)$, continuous pointwise in the ultraweak topology. The duality is

$$\operatorname{Tr}[\Phi_t(X)A]=\operatorname{Tr}[XT_t(A)].$$

Normality is continuity with respect to monotone operator limits at a fixed time. Time continuity is a separate condition on the family. Neither should be silently substituted for the other.

A one-parameter semigroup is time homogeneous. A two-time propagator $\Phi(t,s)$ can instead obey $\Phi(t,r)\Phi(r,s)=\Phi(t,s)$ while depending on both endpoints. The mathematical language of Markovianity separates these structures.

Differentiable trajectories start in the generator domain

Define

$$D(\mathcal L)=\left\{X: \lim_{h\downarrow0}\frac{\Phi_h(X)-X}{h} \text{ exists in }\mathcal S_1\right\}, \qquad \mathcal L X=\lim_{h\downarrow0}\frac{\Phi_h(X)-X}{h}.$$

This is a dense linear domain, but it need not contain every density operator. If $X\in D(\mathcal L)$, the semigroup law gives

$$\frac{\Phi_h(\Phi_tX)-\Phi_tX}{h} =\Phi_t\frac{\Phi_hX-X}{h}.$$

Boundedness of $\Phi_t$ proves that $\Phi_tX\in D(\mathcal L)$ and $\mathcal L\Phi_tX=\Phi_t\mathcal LX$. The trajectory is differentiable and solves

$$\frac{d}{dt}\Phi_tX=\mathcal L\Phi_tX,\qquad \Phi_0X=X.$$

For $X\notin D(\mathcal L)$ the trajectory still exists and is trace-norm continuous. It is a mild solution, represented by the integrated relation

$$\int_0^t\Phi_sX\,ds\in D(\mathcal L),\qquad \Phi_tX-X=\mathcal L\int_0^t\Phi_sX\,ds.$$

The integral is a trace-class Bochner integral. The domain and resolvent proof is given in Generators of CP Semigroups. Writing $\dot\rho=\mathcal L\rho$ for every initial state without this qualification is stronger than the semigroup theorem warrants.

Worked example: resetting to a normal state

Fix any density operator $\tau$ on $\mathcal H$ and $\gamma>0$. The reset channel $R_\tau(X)=\operatorname{Tr}(X)\tau$ satisfies $R_\tau^2=R_\tau$. Hence

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

is a channel for each $t$. Decompose $X=R_\tau X+(I-R_\tau)X$. The first component is fixed and the second is multiplied by $e^{-\gamma t}$, proving the semigroup law directly.

The generator is the bounded operator

$$\mathcal L=\gamma(R_\tau-\mathrm{id}),\qquad \|\mathcal L\|_{1\to1}\le2\gamma.$$

For every state,

$$\|\Phi_t(\rho)-\tau\|_1 =e^{-\gamma t}\|\rho-\tau\|_1.$$

Thus $\tau$ is the unique stationary density operator and attraction is uniform over states. This construction works in infinite dimension because $\tau$ is a specified trace-class state. Replacing it by a purported normalized identity would be invalid when $\operatorname{Tr}I=\infty$.

An explicit countable Kraus representation for $R_\tau$ follows if $\tau=\sum_mq_m|m\rangle\langle m|$: use $K_{mn}=\sqrt{q_m}|m\rangle\langle n|$. The strong sum of $K_{mn}^\dagger K_{mn}$ is $I$. This exhibits a perfectly bounded generator even though its natural list of jump operators is infinite.

Semigroup structure does not establish microscopic validity

The abstract axioms prove positivity, normalization, composition, and continuity within a reduced model. They do not prove that a given system–environment Hamiltonian produces that model over a chosen time window.

In particular, tracing out a fixed environment after joint unitary evolution generally does not give the semigroup law. Reusing the environment retains correlations, whereas composing a reduced channel often corresponds to preparing a fresh environment for the next interval. The mathematical open-system overview works through this distinction explicitly.

Also, a semigroup of one-time reduced maps does not by itself specify every probability for experiments with interventions at intermediate times. That stronger statement requires a multitime process model.

Exercises

Fixed point and generator kernel. Show that a state is stationary for all $t$ if and only if it belongs to $D(\mathcal L)$ and $\mathcal L\rho=0$.

Solution

A stationary orbit has zero difference quotient, so it is in the domain with zero generator. Conversely, domain invariance gives $d\Phi_t(\rho)/dt=\Phi_t(\mathcal L\rho)=0$. The initial condition then fixes the orbit to $\rho$.

A reset map at a random time. Give a probabilistic construction of the reset semigroup and explain why the reset state must be the same at every event.

Solution

Let reset events occur at Poisson rate $\gamma$. With probability $e^{-\gamma t}$ no event occurs, so the input survives. With complementary probability at least one occurs, and every later reset leaves the same state $\tau$. Averaging gives $\Phi_t$. If the reset target changes with absolute time, the resulting propagator generally depends on both time endpoints.

References

• Engel, Klaus-Jochen, and Rainer Nagel. One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194, Springer, 2000. doi:10.1007/b97696. Generator domains, classical solutions, and integrated trajectories.
• Holevo, Alexander S. Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed. De Gruyter, 2019. doi:10.1515/9783110642490. Normal channels 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. Norm-continuous quantum semigroups.