Signaling, No-Signaling, and Causal Composition

Operational signaling is a change in a receiver’s unconditional statistics caused by a sender’s intervention. It is not the same as correlation, conditional dependence, or Bell nonlocality. Relativistic causal composition requires localized operations to commute at spacelike separation and to factor in causal order when their supports are timelike related.

Required background. Relativistic causality and spacelike compatibility supply the field-algebra constraint. Causal quantum channels supplies localized CP maps, and data processing supplies the operational distinguishability test.

Helpful background. Spacelike joint measurements separates commuting effects from commuting instruments.

Intervention contrast

Let $a$ label a sender operation $\mathcal E_a^A$ and $E_y^B$ a receiver effect. The receiver law is

$$p(y\mid a)=\operatorname{tr}\!\left[ E_y^B\,\mathcal E_a^A(\rho) \right].$$

The protocol signals from $A$ to $B$ if this distribution depends on $a$ with every shared preparation and receiver setting held fixed. A convenient one-shot strength is the total-variation contrast

$$S(a,a')=\frac12\sum_y \left|p(y\mid a)-p(y\mid a')\right|.$$

For spacelike supported trace-preserving operations, causal locality gives $S=0$. Initial correlations can still make a joint law $p(x,y)$ nonfactorizing.

Signaling occupies the causal-propagation branch: a sender choice changes a receiver marginal only when the full supported channel permits influence. Correlation branches do not establish that change. The diagram is schematic.

Spacelike and timelike detector interventions

For two compact detector couplings $K_A,K_B$, compute both a correlation statistic and $S(a,a')$. In the spacelike arrangement, the probes can inherit correlated noise from the field state while $S$ vanishes. Move $K_B$ into $J^+(K_A)$ while leaving the apparatus otherwise fixed; the retarded commutator can then transmit the sender choice and $S$ may become nonzero.

At leading detector orders, symmetrized field correlations control much of the shared-noise term, whereas the commutator controls causal response. This decomposition is model-dependent beyond the stated order, but the intervention definition of signaling is exact.

Postselection is the decisive adversarial test. If $x$ is a rare sender outcome, $p(y\mid x,a)$ may change across $x$ even at spacelike separation. Before $x$ is communicated, however, the receiver observes

$$p(y\mid a)=\sum_xp(x,y\mid a),$$

which remains independent of $a$ for a local spacelike instrument. Sorting by an unavailable $x$ tests correlation, not signaling.

Causal composition

Let $\mathcal E_A$ and $\mathcal E_B$ be localized nonselective channels. For spacelike supports,

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

For $K_A$ entirely earlier than $K_B$, the physical composite is ordered. Common causes in $J^-(K_A)\cap J^-(K_B)$ may correlate settings or apparatus noise; they must be part of the preparation, not misdrawn as an $A\to B$ arrow.

The distinction between excitation probability and causal influence in the two-atom problem is worked out by Buchholz and Yngvason 1994, Eqs. (1)–(6), pp. 613–615. Explicit detector-mediated signaling channels appear in Cliche and Kempf 2010, §§ III–V, while the general causal factorization of localized measurement schemes is proved in Fewster and Verch 2020, §§ 3–5.

Postselected conditional change and shared vacuum correlation belong to the correlation column. Only an intervention contrast in the unconditional receiver law licenses a signaling claim. The map is schematic.

References

• Buchholz, D., and Yngvason, J. (1994). “There Are No Causality Problems for Fermi’s Two-Atom System.” Physical Review Letters 73, 613–616. DOI.
• Cliche, M., and Kempf, A. (2010). “The Relativistic Quantum Channel of Communication through Field Quanta.” Physical Review A 81, 012330. 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.