Kraus Representations

A normal quantum operation between separable Hilbert spaces admits a countable Kraus representation. Its Schrödinger sum converges in trace norm, while the corresponding completeness relation generally converges only strongly. The canonical Kraus Representation article covers the operator-sum interpretation and finite-dimensional algorithms; this page establishes the convergence needed for infinitely many Kraus operators.

Helpful background. Completely positive maps supplies the normal map/predual pair; trace-class operators supplies ideal estimates.

The normal operator-sum theorem

Let $\mathcal H_{\mathrm{in}}$ and $\mathcal H_{\mathrm{out}}$ be separable Hilbert spaces. Let $\Phi$ be a CP trace-nonincreasing map on trace-class operators, with normal dual $\Psi$. There are bounded operators $K_j:\mathcal H_{\mathrm{in}}\to\mathcal H_{\mathrm{out}}$, in a finite or countable family, such that

$$\Phi(T)=\sum_jK_jTK_j^*, \qquad \Psi(A)=\sum_jK_j^*AK_j,$$

and

$$\sum_jK_j^*K_j=\Psi(I)\leq I.$$

The first series converges in trace norm. The second and the completeness relation converge strongly; in particular they converge ultraweakly. Trace preservation is equivalent to equality $\sum_jK_j^*K_j=I$.

Normality is part of the theorem. An arbitrary CP map on bounded operators may fail normality and need not have this representation by bounded Kraus operators with the stated convergence. Separability supports the countable index set; more general spaces require a correspondingly general formulation.

Reading Kraus operators from a dilation

The normal Stinespring representation of $\Psi$ can be written

$$\Psi(A)=V^*(A\otimes I_{\mathcal K})V,$$

where $V:\mathcal H_{\mathrm{in}}\to \mathcal H_{\mathrm{out}}\otimes\mathcal K$ is a contraction, and it is an isometry for a channel. For separable input and output one can choose a separable minimal environment $\mathcal K$.

Choose an orthonormal basis $(b_j)$ of $\mathcal K$ and set $K_j=(I\otimes\langle b_j|)V$. Then

$$\sum_j\|K_j\psi\|^2=\|V\psi\|^2,$$

which proves the strong completeness relation. Inserting environment basis projections into the dilation gives the Heisenberg series. Tracing out the environment gives the Schrödinger series.

The basis labels are representation data. They become recorded measurement outcomes only after specifying an instrument or measuring an environment observable. A channel alone does not identify a unique collection of physical events.

Why trace-norm convergence follows

For positive $T\in\mathcal S_1$, every term $K_jTK_j^*$ is positive trace class, and

$$\sum_j\operatorname{Tr}(K_jTK_j^*) =\operatorname{Tr}(T\Psi(I)) \leq\operatorname{Tr}T.$$

The norm of a positive tail equals its trace, so partial sums are Cauchy in trace norm. For a general trace-class $T$, decompose its real and imaginary parts into positive and negative parts. Applying the same argument to those four positive operators proves convergence for all $T$.

Conversely, any countable bounded-operator family satisfying the strong inequality $\sum_jK_j^*K_j\leq I$ defines a CP trace-nonincreasing map by this construction. Complete positivity holds for finite partial sums and survives trace-norm limits at every finite matrix level. This provides a direct way to verify a proposed infinite operator-sum model.

Complete dephasing needs infinitely many terms

On $\ell^2(\mathbb N)$ let $K_j=P_j=|j\rangle\langle j|$. Then

$$\Phi(\rho)=\sum_j\langle j,\rho j\rangle P_j$$

removes all off-diagonal matrix entries. The completeness relation $\sum_jP_j=I$ holds strongly, but its finite remainders have operator norm one. Therefore demanding operator-norm convergence of completeness would incorrectly exclude this valid channel.

For a fixed state, the trace-norm error from omitting outcomes above $N$ is the tail probability $\sum_{j>N}\langle j,\rho j\rangle$. This tends to zero state by state. It does not tend to zero uniformly over all states: a state supported entirely above $N$ has omitted weight one.

This example separates pointwise convergence of a Kraus approximation from uniform approximation of a channel over its whole infinite-dimensional input state space.

Representation freedom

Changing the environment basis mixes Kraus operators by a unitary on the index space and leaves the map unchanged. Starting from a minimal dilation, embedding its environment isometrically in a larger space generates nonminimal representations.

Thus neither the number of displayed operators nor their particular matrix entries is generally an invariant. In finite dimensions the minimal number is the Choi rank. In infinite dimensions there is no normalizable maximally entangled state furnishing the same finite-matrix Choi construction; one must use the appropriate operator-algebraic formulation.

Exercises

1. For the dephasing channel above, evaluate the truncated output on $\rho=|N+1\rangle\langle N+1|$.

Solution

The first $N$ Kraus terms give zero, while the full channel returns $\rho$. The trace-norm error is one. This shows why state-specific tail control is needed in channel simulations.

2. Let $K_j=c_jU$, where $U$ is unitary and $\sum_j|c_j|^2=1$. What channel results?

Solution

The sum is $\Phi(T)=UTU^*$. Infinitely many displayed Kraus operators represent a channel with a single-operator representation. The displayed family is not minimal.

References

• Busch, Paul, Pekka Lahti, Juha-Pekka Pellonpää, and Kari Ylinen, Quantum Measurement, Springer, 2016 — normal operations, dilations, and Kraus representations. Publisher.
• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd revised and expanded ed., De Gruyter, 2019 — channels on trace-class operators and infinite-dimensional representations. Publisher.