Generators of CP Semigroups

An unbounded quantum dynamical generator is a closed operator on a dense subspace of the trace class. Its domain is part of the generator, not supplementary notation. Averaged trajectories and Laplace transforms provide a direct route to this structure, while preserving the complete positivity needed in quantum mechanics. They also produce bounded channel generators that approximate an already established semigroup.

Required background. Quantum Dynamical Semigroups defines the trace-class $C_0$ setting and the derivative domain. Strong Continuity explains why the generator need not be bounded.

Time averages lie in the generator domain

Let $\Phi_t$ be a conservative CP $C_0$ semigroup on $\mathcal S_1(\mathcal H)$, with generator $\mathcal L$. For $X\in\mathcal S_1$ and $h>0$, form the Bochner average

$$X_h=\frac1h\int_0^h\Phi_sX\,ds.$$

Strong continuity implies $X_h\to X$ in trace norm. The semigroup law also gives

$$\frac{\Phi_kX_h-X_h}{k} =\frac1{hk}\left( \int_h^{h+k}\Phi_sX\,ds-\int_0^k\Phi_sX\,ds \right).$$

As $k\downarrow0$, continuity of the integrands yields

$$X_h\in D(\mathcal L),\qquad \mathcal LX_h=\frac{\Phi_hX-X}{h}.$$

Thus the domain is dense. Multiplying this identity by $h$ also proves the integrated master equation for every initial trace-class operator.

For a density operator $\rho$, $\rho_h$ is again a density operator: it is a convex time average of states. Every state can therefore be approximated in trace norm by states whose trajectories are differentiable at zero. This approximation does not imply that the original state has a derivative.

Closedness controls limits of differentiated states

Suppose $X_n\in D(\mathcal L)$, $X_n\to X$, and $\mathcal LX_n\to Y$, both in trace norm. For domain elements,

$$\Phi_tX_n-X_n=\int_0^t\Phi_s\mathcal LX_n\,ds.$$

Contraction permits passage to the limit: $\Phi_tX-X=\int_0^t\Phi_sY\,ds$. Divide by $t$ and use strong continuity at zero. The right-hand side tends to $Y$, proving $X\in D(\mathcal L)$ and $\mathcal LX=Y$. This is closedness.

Approximating an operator without approximating its generator is insufficient. A core is a subspace dense in $D(\mathcal L)$ for the graph norm $\|X\|_1+\|\mathcal LX\|_1$. Trace-norm density alone does not make a set a core. This distinction is central to unbounded-generator conservation problems.

The Laplace resolvent is completely positive

For real $\lambda>0$, define

$$R_\lambda X=\int_0^\infty e^{-\lambda t}\Phi_tX\,dt.$$

The integral converges in trace norm and $\|R_\lambda\|_{1\to1}\le1/\lambda$. Changing variables in $\Phi_hR_\lambda X$ gives

$$\Phi_hR_\lambda X =e^{\lambda h}\left(R_\lambda X- \int_0^h e^{-\lambda s}\Phi_sX\,ds\right).$$

Its derivative at zero proves that $R_\lambda X\in D(\mathcal L)$ and $(\lambda I-\mathcal L)R_\lambda X=X$. Integration by parts for $X\in D(\mathcal L)$ proves the reverse identity $R_\lambda(\lambda I-\mathcal L)X=X$. Thus $R_\lambda=(\lambda I-\mathcal L)^{-1}$ is the resolvent.

At every finite ancilla level, the integral is a positive weighted integral of positive outputs. Hence $R_\lambda$ is CP. Trace preservation gives

$$\operatorname{Tr}R_\lambda X=\lambda^{-1}\operatorname{Tr}X, \qquad Q_\lambda=\lambda R_\lambda\ \text{is a channel}.$$

Operationally, $Q_\lambda$ is the original evolution stopped after an independent exponentially distributed time of mean $1/\lambda$.

Bounded Yosida generators retain channel structure

The Yosida approximation is

$$\mathcal L_\lambda =\lambda\mathcal L R_\lambda =\lambda^2R_\lambda-\lambda I =\lambda(Q_\lambda-I).$$

It is bounded and generates

$$e^{t\mathcal L_\lambda} =e^{-\lambda t} \sum_{n=0}^\infty\frac{(\lambda t)^n}{n!}\,Q_\lambda^n.$$

This is a Poisson mixture of channel compositions, so each approximating evolution is a channel. The standard Yosida approximation theorem gives $e^{t\mathcal L_\lambda}X\to\Phi_tX$ in trace norm, uniformly for $t$ in compact intervals, for every fixed $X$ as $\lambda\to\infty$.

The theorem does not assert convergence uniformly over all input states. It also starts from an established $C_0$ semigroup or equivalent generator hypotheses. Merely writing a formal differential expression and naming its resolvent does not establish those hypotheses.

Worked example: the resolvent slows a reset process

For the reset semigroup with $R_\tau(X)=\operatorname{Tr}(X)\tau$ and $\mathcal L=\gamma(R_\tau-I)$, its fixed and decaying subspaces give

$$R_\lambda =\frac1\lambda R_\tau +\frac1{\lambda+\gamma}(I-R_\tau).$$

Consequently,

$$Q_\lambda =\frac{\lambda}{\lambda+\gamma}I +\frac{\gamma}{\lambda+\gamma}R_\tau,\qquad \mathcal L_\lambda =\frac{\gamma\lambda}{\lambda+\gamma}(R_\tau-I).$$

The bounded approximation is itself a reset generator with rate $\gamma_\lambda=\gamma\lambda/(\lambda+\gamma)<\gamma$. Its rate approaches the exact one from below. In this bounded example convergence also holds in superoperator norm; the general unbounded theorem only promises the stated orbitwise convergence.

Exercises

Conservation on the full domain. Suppose a CP trace-nonincreasing $C_0$ semigroup has $\operatorname{Tr}\mathcal LX=0$ for every $X\in D(\mathcal L)$. Prove it is trace preserving.

Solution

Domain invariance gives $d\,\operatorname{Tr}\Phi_tX/dt=\operatorname{Tr}\mathcal L\Phi_tX=0$ on the domain. Density and boundedness extend trace preservation to all trace-class $X$. The conclusion requires the identity on the actual generator domain, or on a proved graph-norm core from which it extends.

Resolvent contraction. Show that $Q_\lambda\rho$ tends to $\rho$ in trace norm for every state as $\lambda\to\infty$.

Solution

Write $Q_\lambda\rho-\rho=\int_0^\infty\lambda e^{-\lambda t}(\Phi_t\rho-\rho)\,dt$. On a short interval the integrand norm is uniformly small by continuity; the remaining exponential tail contributes at most $2e^{-\lambda\delta}$. Choose the interval first and then $\lambda$.

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. Closed generators, Laplace resolvents, and Yosida approximation.
• Holevo, Alexander S. Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed. De Gruyter, 2019. doi:10.1515/9783110642490. Quantum semigroups and normal-state channel structure.