In 2012, with Howard Wiseman, Cavalcanti wrote...

The 1964 theorem of John Bell shows that no model that reproduces the predictions of quantum mechanics can simultaneously satisfy the assumptions of locality and determinism. On the other hand, the assumptions of signal locality plus predictability are also sufficient to derive Bell inequalities. This simple theorem, previously noted but published only relatively recently by Masanes, Acin and Gisin, has fundamental implications not entirely appreciated. Firstly, nothing can be concluded about the ontological assumptions of locality or determinism independently of each other—it is possible to reproduce quantum mechanics with deterministic models that violate locality as well as indeterministic models that satisfy locality. On the other hand, the operational assumption of signal locality is an empirically testable (and well-tested) consequence of relativity. Thus Bell inequality violations imply that we can trust that some events are fundamentally unpredictable, even if we cannot trust that they are indeterministic.

"Bell Nonlocality, Signal Locality and Unpredictability (or What Bohr Could Have Told Einstein at Solvay Had He Known About Bell Experiments)" Foundations of Physics vol.42, pp.1329–1338

Signal locality is the same as the no-signaling theorem. The Bell inequalities show that widely separated events have non-local correlations that cannot have been caused by signals between the events or actions of one event on the other ("spooky" action at a distance).

Seven years later, with Raymond Lai, Cavalcanti wrote...

Bell's 1964 theorem causes a severe problem for the notion that correlations require explanation, encapsulated in Reichenbach's Principle of Common Cause. Despite being a hallmark of scientific thought, dropping the principle has been widely regarded as a much less bitter medicine than the perceived alternative---dropping relativistic causality. Recently, however, some authors have proposed that modified forms of Reichenbach's principle could be maintained even with relativistic causality. Here we break down Reichenbach's principle into two independent assumptions---the Principle of Common Cause proper, and Factorisation of Probabilities. We show how Bell's theorem can be derived from these two assumptions plus Relativistic Causality and the Law of Total Probability for actual events, and we review proposals to drop each of these assumptions in light of the theorem...

To better explore the connection between the Principle of Common Cause, Bell’s notion of Local Causality (LC), and the multiple responses to the dilemma imposed by Bell’s theorem, we will use a weaker definition of the R- PCC, that does not imply separability of probabilities, as follows.

Definition 1 (Principle of Common Cause (PCC)). If two events A and B are correlated, i.e., if P (A, B) > P (A)P (B), then either: (i) A and B are directly causally connected, i.e. either A causes B or B causes A, or (ii) A and B share a common cause that explains the correlation.

Definition 2 (Factorisation of probabilities (FP)). Two events have a common cause if and only if there exists a sufficient specification of variables λ corresponding to events in the common causal past of A and B such that conditioned on those variables the joint probability of A and B factorises:

P (A, B|λ) = P (A|λ)P (B|λ). (1)

The above is also referred to as a screening-off condition. The conjunction of Definition 1 and Definition 2 is Reichenbach’s Principle of Common Cause (R-PCC). What counts as the ‘common causal past’ of the two events in question of course depends on the causal structure assumed in a theory.

"On modifications of Reichenbach's principle of common cause in light of Bell's theorem" ArXiv 1311.6852v2, 3 Sep 2019

Relativistic causality is the physical principle that a cause must always happen before its effect for every observer. In space-time diagrams, the cause of an event must be in the past light cone of the event. Recently Jacob Barandes and Travis Norsen have drawn illustrative diagrams...

Barandes "especially thanks" Norsen, who mentions Bell's formulation of local causality and illustrates Barandes' point about overlapping past light cones.

Fig. 8.4 Space-time regions relevant to Bell’s formulation of local causality. Bell writes: “Full specification of what happens in 3 makes events in 2 irrelevant for predictions about 1 in a locally causal theory.” This is what Cavalcanti refers to as the screening-off condition."

Ref. "La Nouvelle Cuisine," in Speakable and Unspeakable in Quantum Mechanics, Cambridge, p.240

But Norsen then explicitly shows how a common cause from the initial entanglement is still in the past light cone of the "separated" measurements at A and B.

Fig. 8.5 Space-time diagram for the Bell experiment. The particle pair is emitted at the “flash’' at the bottom of the diagram; world-lines for the two individual particles flying apart in opposite directions are represented by the gray dashed lines. The (large!) region 3 encompasses both particles at some intermediate time and "screens off" the two measurement regions, 1 and 2. from their overlapping past light cones in the way that is required in Bell's formulation of locality.

Barandes provides a similar diagram of the diverging paths of particles he calls Q and R leaving an initial entanglement at time t' and traveling to measurement devices at A and B at time t.

He describes the time evolution of the particles.

Suppose that the two subsystems Q and R are not kept at spacelike separation during the physical process in question, but locally interact at some intermediate time t′ between 0 and t. Then, again following standard textbook arguments, the overall system’s unitary time-evolution operator U_QR(t) will fail to tensor-factorize at t′:

U_QR(t′) ≠ U_Q(t′) ⊗ U_R(t′). (59)

Because the corresponding transition matrix Γ_QR(t) encodes cumulative statistical effects starting at the initial time 0, the transition matrix will continue to fail to tensor-factorize for all times t ≥ t′ (at least until the next division event):

Γ_QR(t) ≠ Γ_Q(t) ⊗ Γ_R(t) [for t ≥ t′]. (60)

The breakdown in tensor-factorization for t ≥ t′ is precisely entanglement, as manifested at the level of the underlying indivisible stochastic process... so one can conclude that the two subsystems Q and R exert causal influences on each other, stemming from their local interaction at the time t′.

Notice that this local interaction, despite being the ‘common cause’ of the correlations between Q and R, is not the sort of ‘variable’ that can be plugged into the unistochastic theory’s microphysical conditional probabilities. Reichenbach’s principle of common causes therefore does not hold.

"New Prospects," p.12

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