Joint Measurements and Spacelike Composition

Two spacelike measurements admit an order-independent joint instrument when their complete localized operations causally factor. Commutation of their POVM effects is necessary for compatible probabilities but is not, by itself, enough: the update maps and apparatus supports must also be compatible.

Required background. Causal quantum channels supplies the localization and composition criteria.

Helpful background. Operational locality supplies the supported spacetime protocol and record graph.

Joint probabilities from operations

Let $\{\mathcal I_x^A\}$ and $\{\mathcal J_y^B\}$ be instruments supported in spacelike regions $K_A$ and $K_B$. If their branch maps commute,

$$\mathcal I_x^A\circ\mathcal J_y^B =\mathcal J_y^B\circ\mathcal I_x^A,$$

then the joint probability

$$p(x,y)=\operatorname{tr}\!\left[ (\mathcal J_y^B\circ\mathcal I_x^A)(\rho) \right]$$

is independent of which mathematical ordering is used. Marginalization gives the local outcome laws, and initial field correlations can make $p(x,y)\ne p_A(x)p_B(y)$ without creating a signal.

For instruments induced by compactly supported system–probe couplings, causal factorization of the scattering maps provides the physical reason for this commutation. It also identifies the relevant support as the coupling region, not the later location where two classical data files are compared.

Each spacelike branch has its own supported coupling and readout. A joint record is formed only in their common causal future, although the branch maps can be composed in either mathematical order. The diagram is schematic.

Effects can commute while updates do not

Consider commuting effects $E_x^A$ and $F_y^B$ but outcome-dependent unitaries in the updates,

$$\mathcal I_x^A(\rho)=U_x^A\sqrt{E_x^A}\rho\sqrt{E_x^A}U_x^{A\dagger},$$ $$\mathcal J_y^B(\rho)=V_y^B\sqrt{F_y^B}\rho\sqrt{F_y^B}V_y^{B\dagger}.$$

If $U_x^A$ or $V_y^B$ acts outside its advertised local algebra, the branches can fail to commute even though $[E_x^A,F_y^B]=0$. Immediate joint-effect statistics then do not certify a relativistically implementable joint measurement. Test the complete superoperators on a spanning set of regulated observables or derive both from supported dilations.

The nonselective maps require a separate check:

$$\mathcal E_A=\sum_x\mathcal I_x^A, \qquad \mathcal E_B=\sum_y\mathcal J_y^B.$$

No signaling concerns the remote marginal under $\mathcal E_A$ or $\mathcal E_B$; joint conditional distributions concern individual branches. Conflating the two is a common source of apparent contradictions.

A two-order validation

For a pair of detector instruments, compute $p_{A\prec B}(x,y)$ and $p_{B\prec A}(x,y)$ using the same initial state, cutoffs, and perturbative order. Report

$$\delta_{xy}=p_{A\prec B}(x,y)-p_{B\prec A}(x,y)$$

with a numerical and truncation uncertainty. For exactly spacelike compact supports, each $\delta_{xy}$ should be consistent with zero. Repeat the test after introducing a controlled support overlap; the resulting order dependence verifies that the test is capable of detecting the failure.

Fewster and Verch 2020, § 5 establish the causal composition of probe-induced measurements. Bostelmann, Fewster, and Ruep 2021, §§ II–IV explain why an apparently local ideal measurement may fail if its required apparatus is not localizable.

Spacelike composition fails when apparatus supports overlap or update maps extend outside their declared regions. Commuting effects alone do not close either loophole. The map is schematic.

Exercises

Assume every branch map commutes across $A$ and $B$. Show that the nonselective maps commute and that both orderings give the same marginals.

Solution

Sum $\mathcal I_x^A\mathcal J_y^B=\mathcal J_y^B\mathcal I_x^A$ over $x$ and $y$ to obtain $\mathcal E_A\mathcal E_B=\mathcal E_B\mathcal E_A$. Summing the common joint distribution over $y$ uses trace preservation of $\mathcal E_B$ and yields $\operatorname{tr}\mathcal I_x^A(\rho)$; similarly for the other marginal.

References

• Bostelmann, H., Fewster, C. J., and Ruep, M. H. (2021). “Impossible Measurements Require Impossible Apparatus.” Physical Review D 103, 025017. DOI. Open PDF.
• Fewster, C. J., and Verch, R. (2020). “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378, 851–889. DOI. Open PDF.