Complete Positivity of Dynamics

Complete positivity is a requirement on the finite-time evolution and its extension to an untouched ancilla. A generator is a derivative at the identity and satisfies a different positivity condition. Confusing these statements leads both to incorrect theorem hypotheses and to numerical steps that leave the state space. The canonical Completely Positive Maps article introduces the channel condition; the mathematical definition handles normality and matrix amplification.

Required background. Quantum Channels in Infinite Dimensions specifies trace-class maps and the convergence used below.

Completely positive evolution survives composition and limits

If $\Phi$ and $\Psi$ are CP maps on compatible trace-class spaces, then

$$(\Psi\Phi)\otimes\mathrm{id}_m =(\Psi\otimes\mathrm{id}_m)(\Phi\otimes\mathrm{id}_m)$$

is positive for every finite ancilla dimension $m$. Positive linear combinations are CP as well. Composition and convex mixing therefore generate new channels without testing their matrices afresh.

There is also a useful closure statement. Suppose channels $\Phi_n$ converge in trace norm on every trace-class input to a linear map $\Phi$. Channel contractivity implies $\|\Phi\|_{1\to1}\le1$, and continuity of trace gives trace preservation. For a positive block operator $X=[X_{ij}]$ with finitely many blocks,

$$\|[(\Phi_n-\Phi)(X_{ij})]\|_1 \le\sum_{i,j}\|(\Phi_n-\Phi)(X_{ij})\|_1\longrightarrow0.$$

The positive cone of the trace class is norm closed. Each amplified limit is therefore positive, proving that $\Phi$ is CP. This is a theorem about pointwise trace-norm limits on the full trace class. A weak limit that allows probability to escape to high levels need not preserve trace.

These facts justify many constructions by splitting and repeated interactions, provided convergence is independently proved. If bounded generators $\mathcal L_1,\mathcal L_2$ each generate channels, the bounded-operator product formula yields

$$e^{t(\mathcal L_1+\mathcal L_2)} =\lim_{n\to\infty} \left(e^{t\mathcal L_1/n}e^{t\mathcal L_2/n}\right)^n.$$

Every approximant is a channel, and the limit is a channel. With unbounded generators, the existence, domain, and convergence of this formula require a separate theorem.

A conservative generator cannot itself be a nonzero positive map

Let $\Phi_t=e^{t\mathcal L}$ be a norm-continuous trace-preserving semigroup. Differentiation gives

$$\operatorname{Tr}\mathcal L(X)=0$$

for every trace-class $X$. If $\mathcal L$ were positive, then for each density operator $\rho$, $\mathcal L(\rho)$ would be positive with trace zero, hence zero. Density operators linearly span the trace class, so $\mathcal L=0$.

Thus a nontrivial conservative generator is not even positive, although every $\Phi_t$ is CP. Infinitesimal positivity constrains how the derivative points away from the identity map; it does not ask the derivative to be a physical operation.

Conditional positivity appears at the boundary

Write $T_t=\Phi_t^\dagger$ and let $\mathcal G$ be its bounded generator on $B(\mathcal H)$. For arbitrary finite families $A_i\in B(\mathcal H)$ and $u_i\in\mathcal H$, complete positivity gives

$$f(t)=\sum_{i,j}\langle u_i,T_t(A_i^\dagger A_j)u_j\rangle\ge0.$$

If $\sum_iA_i u_i=0$, then $f(0)=0$. The right derivative must therefore be nonnegative:

$$\sum_{i,j}\langle u_i,\mathcal G(A_i^\dagger A_j)u_j\rangle\ge0 \quad\text{whenever }\sum_iA_i u_i=0.$$

This conditional complete positivity is one form of the infinitesimal condition behind the bounded GKSL–Lindblad Theorem. The constraint is indispensable.

A related consequence follows by differentiating the unital CP Schwarz inequality at $t=0$:

$$\Gamma(A)=\mathcal G(A^\dagger A) -\mathcal G(A^\dagger)A-A^\dagger\mathcal G(A)\ge0.$$

The Hamiltonian part cancels from this expression. For a bounded Lindblad representation, $\Gamma(A)=\sum_j[L_j,A]^\dagger[L_j,A]$, with the appropriate strong convergence. This measures dissipative departure from a derivation.

Worked example: Euler stepping breaks a unitary channel

Take a qubit, set $\hbar=1$, and use $H=(\omega/2)\sigma_z$. The exact evolution $e^{-itH}\rho e^{itH}$ is unitary, but its forward Euler approximation is $E_h(\rho)=\rho-ih[H,\rho]$. For $\rho=|+\rangle\langle+|$,

$$E_h(\rho)=\frac12 \begin{pmatrix} 1&1-i\omega h\\ 1+i\omega h&1 \end{pmatrix}.$$

Its eigenvalues are

$$\lambda_\pm=\frac{1\pm\sqrt{1+\omega^2h^2}}2.$$

For every $h\ne0$ and $\omega\ne0$, $\lambda_-<0$. Trace and Hermiticity are preserved exactly, yet positivity fails. Reducing the step makes the negative eigenvalue small, of order $h^2$, without making the step a channel.

An exponential unitary step or a product of exact channel steps preserves the state space. Positivity is a structural property of the update, not a consequence of small truncation error alone.

Exercises

Loss of trace in a weaker limit. Let $\Phi_n(X)=\operatorname{Tr}(X)|n\rangle\langle n|$ on $\ell^2$. Show that all compact-observable expectations tend to zero while the output trace stays one for every state.

Solution

For compact $K$, $\langle n|K|n\rangle\to0$, whereas $\operatorname{Tr}\Phi_n(\rho)=1$. The identity observable is not compact. Convergence tested only against compact operators can therefore miss all probability at high levels; it is not the trace-norm convergence assumed in the closure theorem.

Poissonized channel steps. For any channel $C$ and $\gamma\ge0$, prove that $\mathcal L=\gamma(C-\mathrm{id})$ generates a channel semigroup.

Solution

The bounded exponential is $e^{t\mathcal L}=e^{-\gamma t}\sum_{n=0}^\infty(\gamma t)^n C^n/n!$. The coefficients form a probability distribution and the series converges in operator norm. Each $C^n$ is a channel, so composition, convexity, and norm closure prove the assertion.

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. Bounded generators and product formulas.
• Lindblad, Göran. “On the generators of quantum dynamical semigroups.” Communications in Mathematical Physics 48, 119–130 (1976). doi:10.1007/BF01608499. Infinitesimal complete positivity and bounded generator structure.
• Watrous, John. The Theory of Quantum Information. Cambridge University Press, 2018. Author’s text. Completely positive operations and their finite-dimensional amplifications.