1. The existence of system configurations: A physical system’s configurations form a set,1 and the system is in one of its possible configurations at any given instant. The configuration set of a composite system is the Cartesian product of the corresponding subsystem configuration sets.

2. A stochastic description of time-directed dynamics: The laws governing a physical system’s evolution are described via time-directed conditional probabilities relating probabilities of configurations at a given conditioning time to probabilities of configurations at target times. The dynamics are generically non-Markovian. We will often take the conditioning time to be an initial time preceding a later target time, but there need not be any fundamental breaking of time-reversal symmetry or fundamental arrow of time. Subsystems are governed by marginal distributions derived from the laws of their parent systems.

3. Causal locality: Physical influences as described by a system’s stochastic dynamics cannot propagate faster than the speed of light. That is, the stochastic dynamics of a localized system during a specific time interval cannot be conditionally dependent on any other systems beyond the range of light-like signals during that time interval.

In [13,14,15], Barandes showed that the first two postulates are sufficient to describe all known physical systems, including quantum systems. Here, we illustrate how stochastic systems that respect the third postulate can violate Bell inequalities up to Tsirelson’s bound. Note the implication that any non-signaling scenario involving Tsirelson-violating correlations, such as PR boxes, cannot be modeled using the indivisible stochastic systems of [13].

Section 2 reviews the CHSH game and the no-signaling (NS) box framework that demonstrates the existence of probabilistic strategies that win the game all the way up to the theoretical maximum, assuming suitable non-signaling resources shared by Alice and Bob. Section 3 outlines the stochastic-quantum correspondence and some of the key ideas for the definition of causal locality that we use in this work. In Section 4, we recast an arbitrary round of the CHSH game in stochastic terms. We then prove that adopting causal locality yields the Tsirelson bound. In Section 5, we consider some questions about the nature of causality and locality in quantum theory versus the stochastic picture, as well as suggesting some directions for future research.

2 Review of the CHSH Game

In a given round of the CHSH game, two players, Alice and Bob, are spacelike separated. Alice randomizes a bit x ∈ {0, 1}, which she sends to Charlie, along with another bit q ∈ {0, 1} of her choosing. Alice is aware of both bit values. Bob follows the same procedure, randomizing y ∈ {0, 1} and choosing r ∈ {0, 1}, sending both to Charlie.

Charlie tests whether the condition q ⊕ r = xy holds, where the operation ⊕ denotes addition modulo 2. If Alice and Bob win, they are awarded a point; if they lose, they are docked a point.

Alice and Bob share a bipartite resource, prepared by Charlie in advance according to instructions that Alice and Bob agreed to before the start of the game. Charlie sends Alice one part of the resource and Bob the other. Only one bit of information is extractable from each part of the resource.

Let P (q, r, x, y) denote the probability that Alice and Bob return the bit values q, r, x, and y. Their expected score in any given round of the game is ⟨score⟩ =

x,y,q,r P (q, r, x, y)δ(q ⊕ r = xy) -P (q, r, x, y)δ(q ⊕ r ̸ = xy) = x,y,q,r (-1) xy-q⊕r P (q, r|x, y)P (x, y)

x,y (-1) xy P (0, 0|x, y) -P (0, 1|x, y) -P (1, 0|x, y) + P (1, 1|x, y) P (x, y), (1) where P (x, y) is the marginal joint probability of the bit value pair (x, y) and P (q, r|x, y) ≡ P (q, r, x, y)/P (x, y). The function δ(condition) = 1 if the condition in its argument is true, and is 0 otherwise. We can re-express the score by substituting P (1, 1|x, y) = 1 -P (0, 0|x, y) -P (0, 1|x, y) -P (1, 0|x, y) into the last expression, yielding

x,y (-1) xy P (0, 1|x, y) + P (1, 0|x, y) P (x, y)

x,y (-1) xy P (0, 1|x, y) + P (1, 0|x, y) P (x, y), (2) where the last equality in (2) follows from

x,y (-1) xy P (x, y) = P (0, 0) + P (0, 1) + P (1, 0) -P (1, 1) = 1 -2P (1, 1).

(

If Alice and Bob generate the values x and y according to a uniform distribution, then P (x, y) = 1/4. Thus,

x,y (-1) xy P (0, 1|x, y) + P (1, 0|x, y) .

Equivalently, from (1), we have ⟨score⟩ = 1 4 x,y (-1) xy P (0, 0|x, y) -P (0, 1|x, y) -P (1, 0|x, y) + P (1, 1|x, y) .

(5)

The conditional probability distribution P (q, r|x, y) describes all possible correlations between Alice and Bob’s response bits, given the pair of input bits (x, y) they randomly generate and their use of the shared resource. Note that at this stage, P (q, r|x, y) is merely descriptive, and does not commit us to any particular underlying dynamical picture. The no-signaling principle is a physical criterion that is meant to constrain P (q, r|x, y), and thus indirectly encode some physical information in the correlations the conditional probabilities describe. These no-signaling correlations satisfy P (q|x, y) = P (q|x), P (r|x, y) = P (r|y), (6) where the marginal probabilities P (q|x, y) and P (r|x, y) are defined by P (q|x, y) ≡ r P (q, r|x, y), P (r|x, y) ≡ q P (q, r|x, y).

The no-signaling constraints describe the intuition that outcomes seen only locally by either Alice or Bob should only be affected by local influences. In the CHSH context, Alice’s outcomes for q should only be correlated with the bit value x, and, similarly, Bob’s outcomes for r should only be correlated with y.

P N S (q, r|x, y) = 1 4 1 + (-1) xy-q⊕r E .

Notice that this distribution is locally unbiased in that Alice and Bob each have marginal probabilities of 1/2 for any of the outcomes q = 0, 1 and r = 0, 1, respectively. In particular, it satisfies the no-signaling conditions (6). Substituting (8) into (5) shows that the parameter E is equivalent to ⟨score⟩. The case E = 0 corresponds to Alice and Bob ignoring their random bit values, x and y, respectively, and simply selecting q and r randomly with the equivalent of fair coin tosses. When E = 1/2, Alice and Bob have chosen a strategy that allows them to win 3/4ths of the time, saturating Bell’s inequality. The quantum upper limit corresponds to E = 1/ √ 2, while the ‘PR box’ construction of [9,10] corresponds to E = 1.

The no-signaling principle places no restrictions on the score parameter E in ( 8), and thus lacks the explanatory resources to single out the Tsirelson bound.

3 The Stochastic-Quantum Correspondence and Causal Locality

Let C be the set of configurations of a system Q. For simplicity, we assume that C is finite, and let q t = 1, 2, . . . , N label the possible configuration of Q at time t. When we refer to a specific time t n that carries an index of its own, we employ the notation q n ≡ q tn . We assume at least one conditioning time t 0 at which Q has no correlations with any other systems.

Conditional probabilities p(q t |q 0 ) encode stochastic dynamics. Given an initial standalone probability distribution over configurations p(q 0 ) ≡ p(q 0 ; t 0 ), we may define a joint probability distribution, p(q t , q 0 ) = p(q t |q 0 )p(q 0 ), (9) and marginalize over the configuration at the initial time to arrive at the standalone probability for configuration q at time t, p(q t ) = q0 p(q t |q 0 )p(q 0 ).

The formalism works for either t < t 0 or t > t 0 , but, for specificity, we will assume that t 0 is the earliest time under consideration at which our system is uncorrelated with anything else and that all other times in the process occur later.

Generically, systems evolve indivisibly, except for specific instants, including t 0 , called division events or division times. Indivisible evolution means that the system’s configurations cannot be tracked, even in principle, which is reflected by the lack of any assignment of probabilities to system trajectories. By contrast, a divisible stochastic process during the time interval t 0 → t has division times at each instant within the given time interval, up to the given level of precision.

We define a stochastic matrix Γ(t ← t 0 ) with components Γ qtq0 (t ← t 0 ) ≡ p(q t |q 0 ). An indivisible stochastic process is called divisible at a time t 1 if the stochastic matrix factorizes at that time,

where Γ(t ← t 1 ) is a well-defined stochastic matrix of its own. In that case, we axiomatically take the matrix elements of Γ(t ← t 1 ) to be conditional probabilities so that, given the standalone distribution at time t 1 > t 0 , p(q 1 ) = q0 p(q 1 |q 0 )p(q 0 ), (12) we may consistently write p(q t ) = q1 p(q t |q 1 )p(q 1 ), (13) where t > t 1 . As with t 0 , the time t 1 is called a division event or division time. Correlations between the system Q and other systems are effectively washed out during a division event. Division events are often only approximate or effective, such as when they arise due to interactions between a system and a larger environment. We adopt a pragmatic perspective that all dynamical probabilistic statements are defined up to some given level of precision, which demarcates the validity of approximating a process as a division event.

As stipulated in the first postulate in Section 1, a configuration of a composite system QR can be expressed as an ordered pair (q t , r t ) of configurations q t and r t of the individual systems Q and R, respectively. Thus, the configurations of QR form a Cartesian product,

where C Q and C R are the configuration sets of the systems Q and R, respectively. Given an initial division event at t 0 , the joint stochastic dynamics for QR are defined by conditional probabilities of the form p(q t , r t |q 0 , r 0 ) that describe the evolution of the initial probability distribution to later times

p(q t , r t |q 0 , r 0 )p(q 0 , r 0 ). ( 14)

Marginalizing yields mixed stochastic dynamics for Q and R,

If the conditional probabilities factorize,

then we say that the subsystems Q and R do not interact during the interval t 0 → t. If Q and R interact at a later time, then their composite stochastic dynamics will cease to factorize until a subsequent division event. Note that interactions are an example of the broader notion of causal influence defined in [16], and described in more detail in Section 3.3.

Trace Formula for Conditional Probabilities A system’s stochastic dynamics can be (non-uniquely) represented in terms of complex-valued elements of a matrix

where θ qtq0 (t) is an arbitrary phase. For the moment, let us consider the case where Θ(t ← t 0 ) = U (t ← t 0 ) is a unitary matrix. In that case, the corresponding matrix Γ(t ← t 0 ) of conditional probabilities is called unistochastic [18]. The matrix U (t ← t 0 ) acts linearly on a Hilbert space H C of dimension N . The configurations of the system define a preferred basis, {|q⟩} q , for H C . However, U (t ← t 0 ) generically carries configuration-basis vectors to other vectors that are superpositions of configuration-basis vectors. These are not regarded as physical configurations of the system, but rather as a way to characterize the behavior of the system when it undergoes indivisible evolution.

Returning to the case of generic Θ(t ← t 0 ), we let P q = |q⟩⟨q| be a projection matrix onto the configuration-basis vector |q⟩. The stochastic dynamic probabilities can now be expressed via the trace

The relation (18) is the dictionary of the stochastic-quantum correspondence.

Any indivisible stochastic system’s quantum description can be dilated so that the enlarged system evolves according to a unitary time-evolution matrix U (t ← t 0 ) with matrix elements U qtq0 , as demonstrated in [14]. Processes that can be represented unitarily are called unistochastic processes. Their conditional probabilities satisfy

Unitarity permits us to define U (t ← t ′ ) ≡ U (t ← t 0 )U † (t ′ ← t 0 ) for any t ′ < t, which implies that U (t ← t 0 ) = U (t ← t ′ )U (t ′ ← t 0 ), and hence

Here the index q ′ corresponds to the intermediate configuration label at time t ′ , and the matrix qCor(t ← t ′ ← t 0 ) captures the discrepancy between actual indivisible unistochastic evolution and fictitious unistochastic evolution with a division event at t ′ . Such discrepancies are typically referred to as quantum interference effects. The stochasticquantum correspondence thus situates quantum interference within the broader context of discrepancies between exact non-Markovian processes and any Markovian approximation.

Stochastic evolution is generically indivisible. Given a system Q, let T div denote the set of division times where

and t 0 < t 1 < . . . < t n < t. The stochastic-quantum correspondence associates effective division events with environmental decoherence [13]. 2 Such decoherence can be modeled in terms of dephasing channels D i that remove density-matrix coherence terms at the times t i ∈ T div as expressed in the configuration basis. We refer to these as division channels.

Consider two systems Q and R. Following [16], we say that R does not causally influence Q during the time interval starting with the initial division time t 0 until the later target time t if p(q t |q 0 , r 0 ) = p(q t |q 0 ), (22) where p(q t |q 0 , r 0 ) is the marginal conditional probability distribution derived from the joint conditional probability distribution p(q t , r t |q 0 , r 0 ). Otherwise, R does causally influence Q. Note that no external observers or measurements are invoked in this approach to causality. The laws formulated as conditional probabilities themselves provide the causal description in a non-interventionist variant of causal Bayesian network theory [19].

If Q and R are causally independent, then their marginal conditional probability distributions satisfy p(q t |q 0 , r 0 ) = p(q t |q 0 ), (23) p(r t |q 0 , r 0 ) = p(r t |r 0 ). ( 24) The conditions (23) are weaker than the factorization condition ( 16) that characterizes non-interaction. Thus, interaction is a special case of causal influence.

Suppose that Q and R are spatially separated. They form a causally local system if they are causally independent when c • (t -t 0 ) is shorter than their separation distance, with c being the speed of light. Notice how the stochastic picture, in postulating a level of physical reality beneath the Hilbert spaces of standard quantum theory, makes it possible to formulate this new criterion of causal locality, which will turn out to be stronger than the no-signaling principle and will make it possible to derive the Tsirelson bound. Let t 0 denote a division event for all of the systems initiating one round of the game. At this time, Charlie has particles Q and R in configurations q 0 and r 0 , respectively. During the interval t 0 → t 1 , Charlie implements the instructions given to him by Alice and Bob before the round began. At t 1 , Charlie’s configuration will include information about the preparation of the composite system QR, which we denote c 1 (Ψ, . . .). The preparation may involve joining Q and R in an indivisible process, in which case the label Ψ in Charlie’s memory is correlated with a process rather than with any particular configuration (q 1 , r 1 ) of the joint system QR.

Charlie sends Q to Alice and R to Bob at time t 1 , with the particles arriving at time t 2 . On receiving their particles, Alice and Bob generate the random bits x and y, respectively. The bit values are reflected in Alice and Bob’s configurations as a 2 (x, . . .) and b 2 (y, . . .). Alice and Bob perform local operations on Q and R, respectively, during the interval t 2 → t 3 . Note that Alice and Bob’s choices of local operations may be influenced by the values of x and y, respectively, as part of their prearranged strategy. Causal locality plays an important role here, ensuring that Alice’s choice of operation depends solely on x and Bob’s solely on y.

At time t 3 , Alice and Bob measure the configurations of Q and R to be q and r, respectively. Alice and Bob will choose their discretionary bits to be their measurement-outcome bits q and r, respectively. They communicate all of their bit values x, y, q, and r to Charlie, who receives the information at time t 4 and whose configuration is now of the form c 4 (q, r, x, y, Ψ, . . .). Charlie is thus able to check the win condition for the round and keep a tally of the score in memory.

The overall process described above results in a hybrid stochastic distribution for q and r at times t ≥ t 3 conditioned on Alice’s knowledge a 2 (x) of x, Bob’s knowledge b 2 (y) of y, and Charlie’s preparation c 1 (Ψ),

where a 2 (x), b 2 (y), and c 1 (Ψ) lie along a single Cauchy hypersurface. The left-hand side of (25) introduces a notational shorthand, while the right-hand side makes explicit the conditional dependencies inherent in the stochastic process we are describing. We use lower-case ‘p’ to distinguish the stochastic distribution (25) above from the generic distribution P (q, r|x, y) introduced in Section 2.1.

In the CHSH context, causal locality implies further that p xy q|Ψ ≡ p x q|Ψ = p q; t|a 2 (x), c 1 (Ψ) (26)

p xy r|Ψ ≡ p y r|Ψ = p r; t|b 2 (y), c 1 (Ψ) .

Causal locality thus defines what it means for Alice and Bob’s operations to be local: Configuration r is outside of the future light cone of random bit x, and thus cannot be causally influenced by x. Similarly, configuration q is outside of the future light cone of random bit y, and thus cannot be causally influenced by y. However, the marginal probabilities for outcomes q and r both generically depend on the result of the preparation Ψ at time t 1 , due to that indivisible process being in both of their causal pasts.

The no-signaling principle by itself does not constrain the nature of the processes underlying the descriptive joint conditional probability distribution P (q, r|x, y) of Section 2. By contrast, causal locality is more stringent, requiring a stochastic picture with time-directed conditional probabilities that describe the underlying laws governing the system.

Tsirelson’s bound: The stochastic-quantum correspondence implies that the CHSH game score associated with a causally local unistochastic process given by p xy (q, r|Ψ) can be no greater than 1/ √ 2. Equivalently, the total win probability cannot exceed (1 + √ 2)/2 √ 2.

Figure 2: At time t 3 , the configuration q of Q may depend on the random bit x as it lies in the blue-shaded future light cone in the diagram. Similarly, the configuration r of R at time t 3 lies in the red-shaded future light cone of random bit y, and may therefore depend on y. Both configurations q and r will generically depend on the preparation Ψ as it lies in the overlap of their past light cones under the intersection of the dotted lines in the diagram. However at t 3 , causal locality implies that r cannot depend on x and q cannot depend on y as each lies outside of the future light cones of x and y, respectively.

Suppose that QR evolves unistochastically up until Alice and Bob’s measurements at time t 3 . According to the stochastic-quantum dictionary established in [13], we have

where P q = |q⟩⟨q|, P r = |r⟩⟨r|, (AB) xy is a unitary representation of Alice and Bob’s joint operations on QR, and C Ψ is a unitary representation of Charlie’s preparation of QR.

Using properties of the trace, we may re-express (28) as, p xy q, r|Ψ = ⟨Ψ (AB) † xy P q ⊗ P r (AB) xy Ψ⟩ .

Causal locality requires Alice and Bob’s joint operation to factorize: (AB) xy = A x ⊗ B y . Therefore,

We define the Hermitian operators

We calculate the CHSH score for this setup by substituting P (q, r|x, y) = p xy (q, r|Ψ) into the formula (5),

where

The quantity,

is the usual CHSH operator, satisfying the Tsirelson bound,

and thus ⟨score⟩ ≤ 1/ √ 2.

Recalling that the CHSH game awards +1 point for winning and -1 point for losing, the expected score per round is related to the probability of winning the round via

where P win is the probability of winning the round and P loss = 1 -P win is the probability of losing the round. Substituting Tsirelson’s bound for the score and solving (37) for the win probability yields

Our work demonstrates that treating the CHSH game as a causally local unistochastic process reproduces the Tsirelson bound for the CHSH game. In particular, our intuitively natural condition of causal locality threads a fine needle between violating the CHSH inequality up to the Tsirelson bound while preserving the no-signaling principle.

Does causal locality imply that quantum systems whose dynamics satisfy the no-signaling condition are themselves local, despite some famous arguments to the contrary? The trouble with drawing this conclusion is that textbook quantum theory does not supply a way to define local causal influence at a microphysical level, as has been argued

in [16]. The standard Copenhagen interpretation refrains from assigning any picture whatsoever to the microphysical goings-on of our systems. As such, it precludes any possibility of a causally local explanation underlying the behavior of the system from one measurement to another.

Causal locality relies on a different conceptual framework, one that, in contrast to standard quantum theory and the Copenhagen interpretation, does paint a picture of underlying laws governing the behavior of systems as they evolve from one configuration to another. All of the game’s actions can be formulated in terms of generically indivisible stochastic processes that occur entirely within a light cone that encompasses the entangling of qubit systems Q and R, their transmission to Alice and Bob, Alice and Bob’s subsequent local manipulations and measurements of Q and R, and, ultimately, the tallying of the score by Charlie.

Our work implies that a violation of the Tsirelson bound requires either a violation of the principle of causal locality, and thus manifestly non-local dynamical laws, or requires laws that cannot be formulated in terms of unistochastic processes. Are there frameworks for dynamical laws that go beyond the indivisible stochastic systems formulated in [13]? We leave that question to future exploration.

Other approaches specialize from the very beginning to quantum systems that are explicitly formulated in terms of unitary time-evolution operators on Hilbert spaces [20]. According to these alternative approaches, one needs to impose a technical condition to obtain violations of Tsirelson’s bound. This technical condition is that for a composite system consisting of localized subsystems at spacelike separation, the overall time-evolution operator should fail to tensor-factorize into unitary factors that individually encode the time evolution of each localized subsystem. One can show that such arrangements would lead to faster-than-light signaling. Our work, by contrast, does not assume that

We do not assume a richer mathematical structure, such as a Hilbert space. The case of systems with finite sets of configurations is sufficient for our purposes.

Exact division events can also occur. See the example given in[13].