Does entanglement through a common source allow signalling?

A common source can correlate without communicating

The Single Monad proposal invites a distinction between the origin of correlations and the operational transmission of a message. Two distant records may share generative structure even when a reader cannot control one of them to send information to the other. Faster-than-light signalling requires a controllable choice at one location that changes accessible statistics at another before an ordinary causal signal can arrive.

That definition includes more than the final outcome label. Arrival times, missing events, completion flags and correlations with previously stored local records may all carry information. A source account of entanglement therefore needs to specify which local variables an observer can access and how those variables depend on interventions at the remote apparatus. Stating that generation occurs at an inner level does not settle this operational question.

The composition and entanglement guide supplies the conventional joint-state setting. Within that setting, no-signalling follows from a concrete algebraic property of complete local operations. It provides a useful benchmark for a DTT completion interface.

Why a complete local operation leaves the remote state unchanged

Let ρ_AB be a joint density matrix. Alice applies a local instrument with operators Aₖ satisfying ΣAₖ†Aₖ=I. Bob tests an effect E_B. If Alice's outcome is not supplied to Bob, his event probability after the operation is the sum over every outcome k.

The equality uses the completeness relation and ordinary trace algebra. It holds for entangled states as well as product states, and for a different complete instrument chosen by Alice. Thus a local trace-preserving operation cannot change Bob's reduced-state statistics. The argument assumes the stated tensor-product setting and Born probabilities; a source reconstruction must obtain or consistently realize those ingredients.

An accepted completion branch alone usually does not preserve trace. Applying the same calculation to only one Aₖ therefore removes the crucial sum. A changed conditional remote state is then possible, but Bob needs to know which of his trials belongs to that branch before he can use the conditional distribution.

The Bell-pair example makes the distinction visible

Prepare |Φ⟩=(|00⟩+|11⟩)/√2. If Alice measures Z and reports outcome zero, Bob's conditional state is |0⟩. If she reports outcome one, it is |1⟩. Each subensemble is different from his original state I/2. Without Alice's label, the two equally likely subensembles combine back into I/2.

If Alice instead measures X, Bob is conditionally in |+⟩ or |−⟩. Their equal mixture is again I/2. Bob's isolated measurements cannot determine whether Alice chose X or Z. The preparation contains strong correlations that become evident when their records are compared; it does not provide Alice with control over her randomly obtained label.

A source mechanism that deterministically selects Alice's preferred branch while retaining the standard conditional state at Bob would change this conclusion. It would no longer be the complete local quantum instrument used above. Such a model must specify its replacement dynamics and confront the resulting signalling statistics. A normalized postselected state cannot silently stand in for that replacement.

No-signalling is also weaker than a local common-cause model

Consider two possible binary measurements at each site, with outcomes ±1. A deterministic local common-cause model assigns values a₀, a₁, b₀ and b₁ for each hidden preparation. For that preparation, the combination a₀(b₀+b₁)+a₁(b₀−b₁) is always ±2: exactly one of the two bracketed expressions vanishes. Averaging over a setting-independent preparation distribution therefore bounds the absolute CHSH combination by two.

For the Bell state above, choose Alice's observables Z and X, and Bob's (Z+X)/√2 and (Z−X)/√2. Direct evaluation gives the CHSH value 2√2 while each site's single-observable statistics remain balanced. The quantum calculation violates the local common-cause bound without violating the complete-operation identity. The four-setting comparison follows the Clauser–Horne–Shimony–Holt formulation of the distinction established by Bell's original local hidden-variable analysis.

A DTT proposal should consequently identify which premise of the local common-cause comparator it modifies. Possible discussions concern the separability of the source description, the causal meaning of its hidden variables, or its admissible interventions. Merely calling the source common explains neither the quantitative Bell value nor the protection of the remote marginal.

Completion times belong in the no-signalling test

Suppose accepted outcomes look harmless but Alice's setting changes the chance that Bob's detector completes a record in an early time window. Bob could estimate that rate locally over many trials. The absence of a change in the normalized distribution of accepted labels would not prevent the timing channel from carrying information.

The appropriate local output is therefore a joint variable: outcome, completion status and local arrival time, together with any retained local history. For each admissible Bob procedure and each setting b, its full distribution should be independent of Alice's choice a over the spacelike comparison region. In a discrete notation, summing the joint distribution over Alice's complete outcomes must give one Bob marginal for every a.

Rejected events cannot simply be removed from this accounting. If an analysis uses a common time window or a coincidence criterion, state how that rule is selected and whether it can depend on both parties' data. Selection after comparing remote records describes a conditioned data set. It does not establish what either observer could infer before receiving the other's information.

A constructive target for the generative model

Build one finite bipartite source with two local settings per observer, explicit completion flags and a declared clock. Let the six nested inner roles specify dependencies in generating the joint state; let the seventh expose the permitted records. Calculate the full outcome-and-time table for all setting pairs. No observer should acquire an extra control merely because the model's author can inspect hidden variables.

The first useful theorem is simultaneous: recover the specified nonclassical correlation and prove setting independence of every accessible remote marginal, including failures and timing. An approximate construction should give a quantitative bound on any residual marginal dependence and show how that bound scales with finite resources. A counterexample is equally valuable if it pinpoints the exact completion or memory rule that creates an unintended communication channel.

This task connects the ontology of a common source to a recognizable physical constraint. It asks how strong shared correlations can survive while available records obey causal restrictions, and it turns a broad claim about nonlocality into a reproducible finite calculation.

Continue reading

• How phase-compatible systems compose and become entangled →
• What spatial readout forgets →
• Compare generation before and after registration →
• When a readable record is enough to predict the future →

Read this discussion as part of a path

• From completion to quantum predictions →