Relativistic Causality and No-Signaling

Relativistic causality constrains physical influence between spacetime regions. Operational no-signaling says that a local choice cannot change a spacelike-separated observer’s locally accessible statistics. Bell correlations challenge a stronger factorization principle; they do not by themselves supply a faster-than-light channel. This page connects the finite-system No-Signaling result with local field observables.

Helpful background. Causality and Light Cones supplies the spacetime distinction between timelike and spacelike separation.

Locality of observables

In a relativistic field theory, physical observables associated with spacelike-separated regions obey microcausality,

$$[A(\mathcal O_A),B(\mathcal O_B)]=0.$$

For fermion fields, graded commutation relations occur at the field level; the physical local observable algebra has the appropriate spacelike commutativity. The statement concerns operator localization, not the vanishing of correlations. A state can have $\langle AB\rangle\ne\langle A\rangle\langle B\rangle$ while $A$ and $B$ commute.

A finite tensor-product model encodes commutativity by assigning operators $A\otimes I$ and $I\otimes B$. Local algebras in quantum field theory need not factorize as two ordinary finite subsystem Hilbert spaces, so the algebraic statement is the more general one.

A local operation leaves distant probabilities unchanged

Suppose Alice’s physically localized nonselective instrument has Kraus operators $K_a$ that commute with Bob’s effect $B$, and $\sum_aK_a^\dagger K_a=I$. Its Heisenberg action on Bob’s effect is

$$\sum_aK_a^\dagger B K_a =B\sum_aK_a^\dagger K_a=B.$$

Therefore Bob’s expectation is unchanged for every initial state. If Alice retains one outcome, the conditioned expectation can differ; Bob cannot identify that subensemble without her outcome label.

The localization and implementability assumptions matter. An abstract instantaneous projection onto a spatially extended entangled observable is not automatically an allowed local measurement. One must model how the apparatus communicates across its support.

Correlations across spacelike regions

For a singlet pair, equal-axis spin measurements are perfectly anticorrelated. Each observer still sees a fair local result. Alice choosing a different axis changes the joint correlation pattern but leaves Bob’s marginal fixed. Comparing that pattern requires bringing the records together by causal communication.

Different Lorentz frames can disagree on the time ordering of spacelike measurement events. Commuting local instruments give the same joint probabilities under either mathematical ordering. A chosen state-update hypersurface is therefore not directly an observable signal front.

Peres and Terno emphasize that quantum information must be localized and detector operations physically realizable. Relativistic one-particle wavefunctions alone are not a complete account of local creation, detection, and interactions; see Locality and Causality Warnings.

Exercise

Let two projectors $P$ and $Q$ commute. Show that the sequential joint probability for their yes outcomes is independent of their order. Why is this insufficient for arbitrary timelike measurements?

Solution

Lüders measurement gives $\operatorname{tr}(QP\rho PQ)=\operatorname{tr}(\rho PQ)$, since $P^2=P$, $Q^2=Q$, and $PQ=QP$. Reversing the order gives the same result. Timelike-separated instruments can involve noncommuting observables and real intervening dynamics, so this algebra need not apply. Commutation also does not by itself certify that an abstract instrument can be implemented in a claimed spacetime region.

References

• Peres, A., and Terno, D. R., “Quantum information and relativity theory,” Reviews of Modern Physics 76, 93–123 (2004), doi:10.1103/RevModPhys.76.93 — local operations, relativistic information, and detector constraints.