Local Measurement Instruments in QFT

A local measurement instrument is a measure-valued family of completely positive operations. It contains more information than its POVM: the effects determine outcome probabilities, while the operations determine disturbance and conditional states. Relativistic locality adds a further demand—identity action on causally complementary observables for the nonselective local intervention, together with causal composability.

Required background. Influence functionals supplies the system–environment reduction viewpoint. System–probe scattering derives an instrument from a localized coupling.

Helpful background. Symmetry-constrained operations explains when charge or reference-frame restrictions narrow the admissible instrument set.

Effects, operations, and states

In the Schrödinger picture, an instrument $X\mapsto\mathcal I_X$ maps trace-class states to positive trace-class operators and is countably additive over outcomes. It obeys

$$p_\rho(X)=\operatorname{tr}\mathcal I_X(\rho), \qquad \operatorname{tr}\mathcal I_\Omega(\rho)=1.$$

The normalized posterior state, when $p_\rho(X)>0$, is

$$\rho_X=\frac{\mathcal I_X(\rho)}{p_\rho(X)}.$$

In the Heisenberg picture, the associated effect is $E(X)=\mathcal I_X^*(\mathbf1)$, so $p_\rho(X)=\operatorname{tr}[\rho E(X)]$. Two instruments can have the same $E(X)$ and different $\mathcal I_X^*(A)$ for $A\ne\mathbf1$. Consequently, the POVM fixes probabilities for the current readout but not later statistics, backreaction, or signaling tests.

The induced effect and the update map share a physical origin but answer different questions. Calibration of probabilities cannot replace validation of disturbance and causal support. The diagram is schematic.

Gaussian unsharp measurement

Let $Q=\Phi(f)$ be a self-adjoint smeared field quadrature in a regulated representation. A continuous Gaussian measurement with resolution $\Delta$ can be represented by

$$M_x=(2\pi\Delta^2)^{-1/4} \exp\!\left[-\frac{(x-Q)^2}{4\Delta^2}\right], \qquad \mathcal I_x(\rho)=M_x\rho M_x^\dagger.$$

Functional calculus gives

$$\int_{-\infty}^{\infty}dx\,M_x^\dagger M_x=\mathbf1,$$

so the instrument is normalized, and each branch is completely positive. The probability density is the sharp spectral distribution of $Q$ convolved with a Gaussian of variance $\Delta^2$. This algebraic form is useful, but its localization is not automatic: a physical implementation must realize $Q$ and the update through a supported probe coupling, and the sharp limit $\Delta\to0$ may have unbounded energetic or domain cost.

As a decisive comparison, keep $E_x=M_x^\dagger M_x$ fixed and append an outcome-dependent unitary $U_x$ after the measurement:

$$\widetilde{\mathcal I}_x(\rho) =U_xM_x\rho M_x^\dagger U_x^\dagger.$$

The immediate probability density is unchanged, but future observables, energy, and causal support can change. If $U_x$ is nonlocal or is applied using an inaccessible record, the modified instrument cannot inherit the localization of the effect.

Locality is an action criterion

For a nonselective operation localized in $K$, the Heisenberg map should satisfy

$$\mathcal I_\Omega^*(B)=B$$

for observables $B$ localized in a causally disjoint region, subject to the precise net and representation hypotheses. A Kraus decomposition is not unique, so localization cannot be defined by demanding that one chosen list of Kraus operators look local. The action of the map on the observable net is the representation-independent datum.

Okamura and Ozawa 2015, §§ 3–5 formulate local measurement theory through completely positive instruments and explain the extension assumptions needed in local quantum physics. Fewster and Verch 2020, § 5 derive selective and nonselective updates from probe coupling without imposing a global projection postulate.

Complete positivity, normalization, and localization are independent checks. In particular, a valid CP update can still be physically nonlocal. The map is schematic.

Exercises

Show that $\mathcal I_x$ and $\widetilde{\mathcal I}_x$ above have the same effect density but generally different nonselective channels.

Solution

For either instrument, the effect is obtained by applying the dual branch to $\mathbf1$. Since $U_x^\dagger U_x=\mathbf1$, one gets $M_x^\dagger M_x$ in both cases. The nonselective maps are $\int dx\,M_x\rho M_x^\dagger$ and $\int dx\,U_xM_x\rho M_x^\dagger U_x^\dagger$, which differ unless the outcome-dependent unitaries act trivially on every branch or cancel under special symmetries.

References

• Fewster, C. J., and Verch, R. (2020). “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378, 851–889. DOI. Open PDF.
• Okamura, K., and Ozawa, M. (2015). “Measurement Theory in Local Quantum Physics.” Journal of Mathematical Physics 56, 015209. DOI. Open PDF.