Completely Positive Maps and Causal Quantum Channels

A quantum channel is compatible with relativity only when its action is both completely positive and localized. Complete positivity controls extensions by ancillas; localization controls which observable algebras can change. Neither property implies the other, and a Lindblad representation is not required for a finite spacetime intervention.

Required background. Field master equations and Markov generators provide the continuous-time comparison. Local measurement instruments supplies CP operations and their effects.

Helpful background. Positivity, monotonicity, and data processing explains the information inequalities preserved by channels.

Channel pictures and support

A Schrödinger channel $\mathcal E$ is completely positive and trace preserving. Its Heisenberg dual $\mathcal E^*$ is completely positive and unital:

$$\operatorname{tr}[\mathcal E(\rho)A] =\operatorname{tr}[\rho\mathcal E^*(A)], \qquad \mathcal E^*(\mathbf1)=\mathbf1.$$

Let $\mathcal A(O)$ denote the algebra of a region $O$. A nonselective operation localized in a compact coupling region $K$ must act trivially on observables in regions causally disjoint from $K$,

$$\mathcal E_K^*(B)=B, \qquad B\in\mathcal A(O),\quad O\subset K^\perp.$$

This is an action criterion. Kraus representations are nonunique, so the spatial appearance of one decomposition is not a localization invariant.

Complete positivity is checked at the instrument stage; causal support is checked by the channel’s action on complementary algebras. Both are needed for a local intervention. The diagram is schematic.

Spacelike and timelike composition

For two localized Gaussian channels, it is convenient to track first and second moments. Let $[R_j,R_k]=i\Omega_{jk}$ and define $V_{jk}=\langle\{R_j-m_j,R_k-m_k\}\rangle/2$. A Gaussian channel acts as

$$m\mapsto Xm+d, \qquad V\mapsto XVX^T+Y,$$

with complete positivity condition

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

Suppose $\mathcal E_A$ and $\mathcal E_B$ arise from disjoint spacelike couplings. Their physical dilations should causally factor, giving

$$\mathcal E_A^*\circ\mathcal E_B^* =\mathcal E_B^*\circ\mathcal E_A^*.$$

For timelike-related supports, an ordered composition is expected and the two orders need not agree. In a regulated Gaussian model, verify the claim both at the matrix level and directly on a set of local observables. Matrix commutation within an arbitrarily chosen mode decomposition is not enough if the modes themselves have nonlocal tails.

Ancillas sharpen the distinction. Complete positivity guarantees that $\mathcal E\otimes\operatorname{id}_R$ remains positive for an arbitrary spectator $R$. Relativistic locality requires, separately, that a spectator algebra in the causal complement is unchanged by the local physical operation.

Nonlocal Kraus descriptions

The same CP map may admit Kraus lists related by an isometry. A decomposition with delocalized operators does not prove that the map is delocalized. Conversely, a visually local Kraus list does not prove identity action on every complementary observable. Test $\mathcal E^*$ on the net, or exhibit a supported dilation whose scattering map has the required factorization.

The system–probe construction of Fewster and Verch 2020, §§ 3–5 supplies such a dilation and clarifies the causal ordering of multiple couplings. Abstract local CP operations and their approximation properties require additional assumptions on the net; Kitajima 2017, §§ 3–4 gives a focused algebraic analysis.

A channel can pass the CP test and still fail causal support; a chosen Kraus representation can look nonlocal while the channel action remains local. The operative check is representation independent. The map is schematic.

Channel versus generator

A finite intervention $\mathcal E_K$ need not lie on a time-homogeneous semigroup $e^{t\mathcal L}$. Inferring a Lindblad generator requires divisibility, time parametrization, domain control, and usually an approximation such as weak coupling or Markovianity. Use generator language only when those conditions are part of the model.

References

• Fewster, C. J., and Verch, R. (2020). “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378, 851–889. DOI. Open PDF.
• Kitajima, Y. (2017). “Local Operations and Completely Positive Maps in Algebraic Quantum Field Theory.” arXiv:1704.01229.