Quantum Channels in Infinite Dimensions

An infinite-dimensional channel acts continuously on trace-class states, but a sequence of accurate finite cutoffs need not approximate it uniformly over all states. The distinction matters whenever a numerical model discards high-energy input levels. The basic operational definition belongs to Quantum Operations; this page gives a precise cutoff construction, its convergence theorem, and its limitation.

Required background. Completely Positive Maps supplies normality and the predual relation. Trace Class Operators supplies finite-rank approximation.

Normal channels act on every trace-class input

Let $\mathcal H_A,\mathcal H_B$ be separable Hilbert spaces. A channel here is a bounded CP trace-preserving linear map

$$\Phi:\mathcal S_1(\mathcal H_A)\longrightarrow\mathcal S_1(\mathcal H_B).$$

Its adjoint $T:B(\mathcal H_B)\to B(\mathcal H_A)$ is normal, unital, and CP, with $\operatorname{Tr}[\Phi(X)A]=\operatorname{Tr}[XT(A)]$. Conversely, a normal unital CP map has such a predual. Normality is essential to this converse; positivity and unitality alone do not promise trace-class output states.

The Schwarz inequality for unital CP maps gives $\|T(A)\|\le\|A\|$. Trace duality then yields

$$\|\Phi(X)\|_1\le\|X\|_1$$

for every trace-class $X$, including non-self-adjoint inputs. This bounded linear extension is useful even though only positive trace-one operators are physical states.

We say $\Phi_n$ converges pointwise in trace norm to $\Phi$ if $\|\Phi_n(X)-\Phi(X)\|_1\to0$ for every $X\in\mathcal S_1$. It is enough to test density operators, because their complex linear span is the whole trace class. Uniform contraction bounds also make convergence on a trace-norm dense linear subspace sufficient.

A cutoff can preserve the channel axioms exactly

Choose finite-rank orthogonal projections $P_n\uparrow I$ strongly and a fixed output density operator $\tau$. Define

$$\Phi_n(X)=\Phi(P_nXP_n) +\operatorname{Tr}[(I-P_n)X]\tau.$$

The second term assigns a normalized output to the input weight outside the cutoff. Compression, $\Phi$, and this replacement operation are CP; their sum is CP. Moreover,

$$\operatorname{Tr}\Phi_n(X) =\operatorname{Tr}(P_nX)+\operatorname{Tr}[(I-P_n)X] =\operatorname{Tr}X.$$

Thus every approximant is a channel on the original input space. Simply deleting the omitted branch would give a trace-nonincreasing operation instead.

For fixed $X$, trace-class approximation implies $\|X-P_nXP_n\|_1\to0$. One proof first checks finite-rank $X$, where strong convergence controls finitely many vectors, and then uses the uniformly bounded compressions to pass to general $X$. Consequently,

$$\|\Phi_n(X)-\Phi(X)\|_1 \le\|P_nXP_n-X\|_1 +|\operatorname{Tr}[(I-P_n)X]| \longrightarrow0.$$

The input dependence of the bound is consequential. No rate valid for all trace-one inputs follows merely from $P_n\uparrow I$.

Worked example: perfect pointwise convergence with maximal worst-case error

Take $\Phi$ to be the identity on $\ell^2(\mathbb N_0)$, let $P_n$ project onto $|0\rangle,\ldots,|n\rangle$, and choose $\tau=|0\rangle\langle0|$. The cutoff channels converge on every fixed state. Yet for the moving input $\rho_n=|n+1\rangle\langle n+1|$,

$$\Phi_n(\rho_n)=\tau,\qquad \|\Phi_n(\rho_n)-\rho_n\|_1=2.$$

Therefore $\|\Phi_n-\mathrm{id}\|_{1\to1}=2$ for every $n$. The unrestricted diamond distance is also $2$: an unentangled input already attains the maximal channel discrimination distance. Increasing the cutoff moves the bad input to a higher level without eliminating it.

This is compatible with pointwise convergence. Its quantifiers fix the state before sending $n$ to infinity; a worst-case norm chooses a new state after seeing the cutoff.

An energy constraint controls the missing tail

Suppose an input Hamiltonian $H\ge0$ has discrete spectrum with finite-dimensional spectral subspaces below each finite energy, and $P_n$ contains all levels below a threshold $E_n\to\infty$. Then $H\ge E_n(I-P_n)$, so a state with $\operatorname{Tr}(\rho H)\le E$ obeys

$$\varepsilon_n=\operatorname{Tr}[\rho(I-P_n)]\le E/E_n.$$

The projection estimate $\|\rho-P_n\rho P_n\|_1\le2\sqrt{\varepsilon_n}$ gives

$$\|\Phi_n(\rho)-\Phi(\rho)\|_1 \le2\sqrt{\varepsilon_n}+\varepsilon_n \le2\sqrt{E/E_n}+E/E_n.$$

To see the projection estimate, purify $\rho$ and split the purification into its orthogonal retained and omitted components. Their squared norms are $1-\varepsilon_n$ and $\varepsilon_n$. The trace norm of the difference between the original pure projector and the unnormalized retained projector is $\sqrt{4\varepsilon_n-3\varepsilon_n^2}\le2\sqrt{\varepsilon_n}$. Partial trace cannot increase this norm.

The same proof permits an arbitrary reference system: apply $P_n$ only to the input system, and constrain its marginal energy. The replacement branch then outputs $\tau$ tensored with the reference’s discarded marginal. Thus the estimate is uniform even for entangled probes under this energy constraint. The assumption on the energy subspaces is what makes a finite cutoff useful; an infinitely degenerate low-energy sector would invalidate that inference.

Exercises

Finite-rank outputs. Modify the construction to approximate the output as well as the input while keeping every approximant trace preserving.

Solution

Choose output projections $Q_m\uparrow I$ and a density operator $\sigma$ supported in $Q_1$. Compose $\Phi_n$ with $C_m(Y)=Q_mYQ_m+\operatorname{Tr}[(I-Q_m)Y]\sigma$. This is a channel with output supported in $Q_m$. Pointwise convergence follows by contraction and the triangle inequality, first for the input cutoff and then for the output cutoff of $\Phi(X)$. A uniform output-energy estimate requires additional information about the channel.

Dense testing. Why does convergence on matrix units suffice when all maps are channels?

Solution

Finite linear combinations of matrix units are trace-norm dense. For an approximation $F$ to $X$, the two channel contractions give an error at most $2\|X-F\|_1$ plus the already convergent error on $F$. Choose $F$ first, then the channel index.

References

• Holevo, Alexander S. Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed. De Gruyter, 2019. doi:10.1515/9783110642490. Normal channels and infinite-dimensional state spaces.
• Watrous, John. The Theory of Quantum Information. Cambridge University Press, 2018. Author’s text. Trace-norm channel estimates and discrimination norms; the cutoff argument here supplies the infinite-dimensional extension.