Algebraic Quantum Channels and Localized Operations

A quantum operation is localized by how it acts on the observable net, not by a chosen Kraus decomposition. For a nonselective operation in region $K$, observables in the causal complement must remain fixed; a supported dilation or causal factorization then supplies the physical implementation. Normality and the representation of the local algebra are part of the statement.

Required background. Signaling and causal composition supplies the operational criterion that localization must protect.

Helpful background. Local regions and algebras supplies the net $O\mapsto\mathcal A(O)$.

The action criterion

Let $\mathcal E^*$ be a normal unital CP map on the quasilocal algebra. A basic localization condition for $K$ is

$$\mathcal E^*(B)=B, \qquad B\in\mathcal A(K^\perp).$$

A stronger formulation controls the map jointly on local and complementary observables and its composition with other supported operations. The exact equivalences depend on additivity, split or funnel properties, and the representation. These hypotheses must be stated rather than hidden behind finite-dimensional tensor notation.

The localized operation sits at the sender or receiver coupling. Its support is certified by action on the net or by a supported dilation, not by the visual shape of an operator list. The diagram is schematic.

A regulated Gaussian example

On a finite interval regulator, select canonical variables $R_K$ associated with wavepackets supported in $K$ and complementary variables $R_{K^\perp}$. Use $[R_j,R_k]=i\Omega_{jk}$ and $V_{jk}=\langle\{R_j-m_j,R_k-m_k\}\rangle/2$. A local Gaussian noise channel may act as

$$R_K\mapsto XR_K+\xi, \qquad V_K\mapsto XV_KX^T+Y,$$

while leaving $R_{K^\perp}$ unchanged. Complete positivity requires

$$Y+\frac{i}{2}\left(\Omega_K-X\Omega_KX^T\right)\ge0.$$

Check separately that all cross-observable expectations transform consistently and that the regulated wavepackets’ tails satisfy the claimed localization tolerance. A mode partition is a regulator-dependent realization, not the definition of the continuum local algebra.

Kraus and dilation independence

If $\mathcal E(\rho)=\sum_jK_j\rho K_j^\dagger$, another set $L_\alpha=\sum_j u_{\alpha j}K_j$ with an isometry $u$ defines the same channel. Individual $L_\alpha$ can appear more or less delocalized than the $K_j$. Therefore localization by Kraus support is representation dependent.

By contrast, identity action on $\mathcal A(K^\perp)$ is a property of the map. A local system–probe dilation is stronger evidence: coupling in $K$ and tracing the probe produces a map whose causal action follows from the scattering construction. Not every abstract channel is guaranteed such a compact realization; Okamura and Ozawa 2015, §§ 3–5 and Kitajima 2017, §§ 3–4 make the relevant extension qualifications explicit.

A nonlocal-looking Kraus list is not itself a failure; nonidentity action on a causal-complement observable is. Conversely, compact notation cannot hide a dilation with long support tails. The map is schematic.

Check your understanding

Why is a map $\mathcal E^*(A)=U^\dagger AU$ with a nonlocal unitary $U$ not localized in $K$ merely because it leaves one chosen observable in $K^\perp$ fixed?

Solution

Localization must hold for the whole complementary algebra, not one test observable. A nonlocal $U$ may commute accidentally with that observable while changing another $B\in\mathcal A(K^\perp)$. One must verify the algebraic action or exhibit a supported physical dilation.

References

• Kitajima, Y. (2017). “Local Operations and Completely Positive Maps in Algebraic Quantum Field Theory.” arXiv:1704.01229.
• Okamura, K., and Ozawa, M. (2015). “Measurement Theory in Local Quantum Physics.” Journal of Mathematical Physics 56, 015209. DOI. Open PDF.