Signaling, No-Signaling, and Causal Composition

Operational signaling means that a sender’s choice changes a receiver’s unconditional statistics. It is therefore a statement about interventions, not merely about a correlated state, a nonfactorizing joint probability, or a Bell inequality. This distinction is the starting point for composing relativistic operations: spacelike-supported nonselective operations cannot transmit a choice, while operations in causal succession must be composed in their physical order.

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

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

Signaling is an intervention contrast

Fix the initial preparation, receiver operation, and receiver measurement. Let $a$ label one of the sender’s trace-preserving operations $\mathcal E_a^A$, and let $\{E_y^B\}_y$ be the receiver POVM. The observable classical channel is

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

There is signaling from $A$ to $B$ exactly when this distribution depends on $a$. For two choices, the total-variation contrast is

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

If the receiver has access to its whole local algebra, let $\omega_B^{(a)}$ be the restriction of the output state to that algebra. The representation-independent distinguishability is half the norm distance of normal functionals,

$$D_B(a,a') =\frac12\left\lVert \omega_B^{(a)}-\omega_B^{(a')} \right\rVert, \qquad \lVert\varphi\rVert =\sup_{\lVert B\rVert\leq1}|\varphi(B)|.$$

Every fixed receiver POVM satisfies $S_B\leq D_B$, and equality is attained by an optimal binary discrimination measurement. For a type-I detector or cutoff with density operators, this reduces to $D_B=\lVert\rho_B^{(a)}-\rho_B^{(a')}\rVert_1/2$. A type-III local algebra need not have an intrinsic density matrix or trace, so the normal-state norm is the correct continuum statement. In either setting, $D_B=0$ says that no receiver measurement can reveal the sender’s choice. It is stronger and cleaner than testing one convenient observable.

Correlation asks a different question. A joint law can obey $p(x,y)\ne p(x)p(y)$ even when every receiver marginal $p(y\mid a)$ is independent of the sender setting. The chapter task map locates signaling, harvesting, entanglement distribution, and Bell tests on separate branches; the canonical protocol table lists their distinct outputs, and the failure-control map shows where postselection and support tails enter.

A defensible intervention comparison holds more fixed than the state symbol $\rho$ suggests. The receiver’s support, detector setting, readout time, and accessible outcome set must be identical in both branches. The sender setting must be an externally controllable choice rather than a label inferred afterward from a correlated record. Any shared randomness or common cause belongs to the preparation and must have the same distribution in the two branches. Finally, the comparison is made before any sender outcome is communicated. These conditions turn $p(y\mid a)$ into a counterfactual test of influence: what would the same receiver have observed had the sender chosen $a'$ instead?

In an experiment or simulation, one estimates $S_B$ with uncertainty. A nonzero point estimate does not by itself defeat a causal theorem: finite sampling, cutoff error, profile tails, and perturbative truncation all create residuals. The appropriate null test gives a confidence interval or deterministic error bound and checks whether it contains zero. Exact algebraic no-signaling and numerical consistency with zero are different levels of evidence, even though they use the same operational statistic.

Why a localized channel cannot signal across a spacelike gap

The shortest proof is in the Heisenberg picture. Let $\mathcal E_a^{A*}$ be the dual of a nonselective sender channel. Trace preservation makes the dual unital. Localization in a compact interaction region $K_A$ requires

$$\mathcal E_a^{A*}(B)=B, \qquad B\in\mathcal A(K_A^{\perp}),$$

where $K_A^{\perp}$ is the causal complement. If the receiver effect is localized in a spacelike region $K_B\subset K_A^{\perp}$, then

$$\begin{aligned} p(y\mid a) &=\rho\!\left(\mathcal E_a^{A*}(E_y^B)\right)\\ &=\rho(E_y^B), \end{aligned}$$

independently of $a$. This argument allows an entangled or thermally correlated initial state: no product-state assumption was used.

Microcausality, $[A,B]=0$ for spacelike local observables, is essential but is not by itself a complete protocol proof. One must also show that the operation is localized and that unreported outcomes have been summed. A selective instrument component $\mathcal I_x^{A*}$ is generally not unital, because $\mathcal I_x^{A*}(\mathbf1)$ is the effect associated with outcome $x$. Its normalized conditional state can therefore change remote conditional probabilities without enabling a message.

This operational hierarchy mirrors the finite-dimensional distinction between causal and physically localizable bipartite maps developed by Beckman, Gottesman, Nielsen, and Preskill 2001, §§ II–III. In QFT, compact system–probe couplings and their induced instruments provide the spacetime-supported construction; Fewster and Verch 2020, §§ 3–5 establish localization and causal composition under an explicit causal-factorization hypothesis.

First QFT benchmark: two detector interventions

Consider two finite probes that have interacted with a scalar field in compact regions $K_A$ and $K_B$. The reduced probe state may be correlated because the field state has nonzero spacelike correlations. The following exact two-qubit calculation isolates that fact from influence.

Take the post-interaction probe state

$$|\Phi^+\rangle_{AB} =\frac{|00\rangle+|11\rangle}{\sqrt2},$$

and let the sender encode $a=0,1$ by $U_0=\mathbf1_A$ or $U_1=X_A$. The receiver state is

$$\rho_B^{(0)}=\rho_B^{(1)}=\frac{\mathbf1_B}{2},$$

so $D_B(0,1)=0$. Nevertheless, a later comparison of the two detector records gives

$$\langle Z_AZ_B\rangle_{a=0}=+1, \qquad \langle Z_AZ_B\rangle_{a=1}=-1.$$

The joint correlation perfectly records the local flip, while the receiver alone learns nothing. This is the correct pattern for spacelike detector interventions: shared noise or harvested correlations may be nonzero, but the unconditional receiver contrast must vanish.

The QFT support check is the smeared commutator

$$\Delta_{AB} =\int d^4x\,d^4y\, f_A(x)f_B(y)\,\Delta(x,y),$$

where $f_A,f_B$ include the compact switching and spatial smearing and $[\phi(x),\phi(y)]=i\Delta(x,y)\mathbf1$. If $K_A$ and $K_B$ are spacelike separated, microcausality gives $\Delta_{AB}=0$ exactly. When $K_B\subset J^+(K_A)$, the same integral need not vanish; in perturbative detector models it supplies the causal-response part of the receiver change. Symmetrized two-point functions instead contribute field-state-dependent noise and correlations. The separation of these terms is order- and model-dependent, but the intervention definition of $D_B$ is exact. Explicit detector channels and their spacelike null result are calculated by Cliche and Kempf 2010, §§ II–V.

For an ideal timelike control, prepare $A$ in $|a\rangle$, prepare $B$ in $|0\rangle$, and let causal propagation implement a controlled flip from $A$ to $B$. Then

$$\rho_B^{(0)}=|0\rangle\langle0|, \qquad \rho_B^{(1)}=|1\rangle\langle1|, \qquad D_B(0,1)=1.$$

This controlled gate is an exactly solvable endpoint, not a claim that a weak detector–field coupling realizes a perfect gate. A concrete field model must compute the induced channel and will generally give $0\lt D_B\lt1$ inside the causal future. The benchmark’s purpose is to verify the analysis pipeline: the spacelike configuration returns zero influence, while a causally connected configuration is allowed to return a nonzero value.

Causal composition and common causes

Suppose two compact measurement schemes have scattering maps that satisfy causal factorization. If $K_A$ is entirely earlier than $K_B$, the composite instrument is ordered as

$$\mathcal E_{B\leftarrow A}=\mathcal E_B\circ\mathcal E_A.$$

Reversing the algebraic order would describe a different protocol. If the regions are causally disjoint, the two admissible orders coincide,

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

The second equality is a conclusion of the supported scattering construction, not a license to assume that arbitrary CP maps commute. It also concerns the nonselective operations. Conditional state-update rules can look order-dependent because they refer to different classical records, even though the unconditioned spacelike experiment is order independent.

It is often useful to test both directions separately. Holding the preparation fixed, vary the operation in $A$ and compare receiver states in $B$; then vary $B$ and compare states in $A$. Causally disjoint localized operations give zero in both directions. If $K_A$ precedes $K_B$, an $A\to B$ contrast may be nonzero while the reverse contrast must remain zero for the same schedule. This directional pair prevents an order-dependent numerical routine from turning ordinary back-action in the later apparatus into apparent retrocausality.

The resulting causal arrow is protocol relative. A different receiver observable, switching function, or decoding time may reveal a response that the first receiver missed. Therefore $D_B=0$ for one restricted detector proves no signaling only to that detector’s accessible algebra. Algebraic no-signaling is stronger because it fixes every effect in the full causal-complement algebra. A careful report states which of these two levels has actually been established.

A variable in $J^-(K_A)\cap J^-(K_B)$ can correlate the apparatuses or their settings. Such a common cause belongs in the preparation model. It can explain $p(x,y)\ne p(x)p(y)$, but it is not an $A\to B$ influence. The resolution of the Fermi two-atom puzzle likewise distinguishes local excitation probabilities from a controllable causal effect; see Buchholz and Yngvason 1994, Eqs. (1)–(6), pp. 613–615.

Adversarial control: a rare postselected outcome

Let

$$|\psi_\varepsilon\rangle =\sqrt{1-\varepsilon}\,|00\rangle +\sqrt{\varepsilon}\,|11\rangle, \qquad 0\lt\varepsilon\lt1.$$

The sender measures $Z_A$. Conditional on the rare outcome $x=1$, which occurs with probability $\varepsilon$, the receiver state is $|1\rangle\langle1|$. Before the outcome is communicated, however,

$$\rho_B =(1-\varepsilon)|0\rangle\langle0| +\varepsilon|1\rangle\langle1|,$$

whether the sender performs the measurement and discards its result or does nothing. At $\varepsilon=0.01$, the conditional state for $x=1$ lies at trace distance $0.99$ from $\rho_B$, yet the signaling contrast between the two nonselective sender choices is exactly zero. A calculation that reports the $0.99$ conditional change as a spacelike signal has silently supplied the receiver with the postselection label.

Common pitfalls

Testing one observable. Agreement of one receiver expectation value does not establish no-signaling. Test equality of receiver states or equality on the whole receiver algebra.

Using Gaussian support as exact support. Gaussian switching and smearing have nonzero tails. A numerical value “close to zero” then certifies only a declared tail tolerance; exact spacelike no-signaling requires compact support or a separate bound on the tails.

Equating time order with causal order. Coordinate-time ordering can change between frames for spacelike regions. Only causal order is invariant, and causally disjoint localized operations must give the same composite in either order.

Exercises

1. Optimize the receiver test

Show that the largest total-variation contrast over all binary receiver POVMs equals $D_B(a,a')$.

Solution

For a type-I receiver, write $\delta=\rho_B^{(a)}-\rho_B^{(a')}$. For a binary effect $0\le E\le\mathbf1$, the contrast is $|\operatorname{tr}(E\delta)|$. Decompose $\delta=\delta_+-\delta_-$ into positive and negative parts. Because $\operatorname{tr}\delta=0$, $\operatorname{tr}\delta_+=\operatorname{tr}\delta_-=\lVert\delta\rVert_1/2$. Choosing $E$ as the support projector of $\delta_+$ attains $\lVert\delta\rVert_1/2=D_B$; no other effect can exceed it. For a general von Neumann algebra, the Jordan decomposition of the normal functional $\omega_B^{(a)}-\omega_B^{(a')}$ gives the same conclusion with the normal-state norm.

2. Prove spacelike invariance without a product state

Let $\rho$ be any normal state, possibly entangled across the two regions. Prove that a localized nonselective map cannot change the probability of a receiver effect $E_y^B$.

Solution

Localization gives $\mathcal E_a^{A*}(E_y^B)=E_y^B$. Therefore

$$\operatorname{tr}[E_y^B\mathcal E_a^A(\rho)] =\rho(\mathcal E_a^{A*}(E_y^B)) =\rho(E_y^B).$$

No step factorizes $\rho$, so pre-existing entanglement and common-cause correlations are allowed.

3. Reproduce the postselection numbers

For $\varepsilon=0.01$, calculate the probability of $x=1$, the conditional receiver state, and its trace distance from the unconditional receiver state.

Solution

The Born probability is $p(x=1)=\varepsilon=0.01$, and the conditional state is $|1\rangle\langle1|$. Both states are diagonal in the same basis, so their trace distance is the classical total-variation distance:

$$\frac12\left(|0-(1-\varepsilon)|+|1-\varepsilon|\right) =1-\varepsilon=0.99.$$

The nonselective measurement leaves the receiver state unchanged, hence its signaling contrast with “do nothing” is zero.

References

• Beckman, D., Gottesman, D., Nielsen, M. A., and Preskill, J. (2001). “Causal and Localizable Quantum Operations.” Physical Review A 64, 052309. DOI. Open PDF.
• 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.