A Deterministic Interface Model for Quantum Statistics: Born Rule, Interference, and Measurement-Dependence Budgets for Loophole-Free Bell Correlations

In the paper's words

Device-independent quantum key distribution and certified random-number generation derive their security guarantees from the magnitude of Bell violations, and those guarantees are only as strong as the assumption of measurement independence between setting choices and hidden variables. No published computable alternative model tells the certification engineer how much violation a bounded amount of measurement dependence can fake. We build such a model: a continuous deterministic state space with a measure-preserving, mixing flow, observed through a finite measurement partition.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. From these premises we derive the Born rule as the invariant measure of a measurement basin, reproduce single- and sequential-measurement interference as $p_A(\varphi) = (1+\cos\varphi)/2$ (giving $p_A = 0.75$ at $\varphi = \pi/3$), and rederive the CHSH bounds algebraically: the local bound $S \le 2$ and the quantum value $S = 2\sqrt{2} \approx 2.82842712$, an excess of $41.42\%$ over the local bound.

   from Abstract

2. Within a binary conspiratorial split, exact reproduction of the quantum correlations requires full measurement dependence ($\delta = 1$), while matching each correlator to precision $\varepsilon$ requires a budget $\delta \ge \varepsilon/2$; a necessary budget for the Tsirelson value is $\delta \ge 0.10355339$.

   from Abstract

3. An algebraic derivation of the CHSH local bound $S \le 2$ and the quantum value $S = 2\sqrt{2} \approx 2.82842712$, an excess of $41.42\%$ over the local bound (Section 4.3).

   from 1. Introduction

4. A measurement-dependence budget analysis: exact quantum correlations require $\delta = 1$ in the binary-split construction, matching correlators to precision $\varepsilon$ requires $\delta \ge \varepsilon/2$, and the Tsirelson value requires $\delta \ge 0.10355339$ as a necessary condition (Section 4.4).

   from 1. Introduction

5. This establishes a link between deterministic ontologies and the quantitative ceiling of quantum correlations, which our Section 4.3 rederives algebraically inside the interface model. [2] formulates quantum mechanics without an a priori notion of time, defining time as the ratio $|x|/|v|$ of position magnitude to velocity and showing that the uncertainty principle is preserved under this definition.

   from 2. Background and Related Work

6. Our measurement-dependence budget is the quantitative form of that confrontation: Section 4.4 states exactly how much setting–system correlation a deterministic model must spend to reproduce observed violations, turning a philosophical dispute into a bounded design parameter. [7] treats the implementation of generalized measurements, defining POVM elements $\{E_i\}$ with $\sum_i E_i = \openone$ and outcome probabilities $p(i) = \operatorname{Tr}(\rho E_i)$, and demonstrates an implementation on a qudit via quantum walks.

   from 2. Background and Related Work

How it is done

1. From these premises we derive the Born rule as the invariant measure of a measurement basin, reproduce single- and sequential-measurement interference as $p_A(\varphi) = (1+\cos\varphi)/2$ (giving $p_A = 0.75$ at $\varphi = \pi/3$), and rederive the CHSH bounds algebraically: the local bound $S \le 2$ and the quantum value $S = 2\sqrt{2} \approx 2.82842712$, an excess of $41.42\%$ over the local bound.

   from Abstract

2. The practitioners facing this problem are certification engineers and protocol designers, and the cost of the gap is concrete: a certification proof that assumes measurement independence is only as strong as that assumption, and no published alternative model tells the cryptographer how much violation a bounded amount of measurement dependence can fake.

   from 1. Introduction

3. The programme of deriving quantum mechanics from deterministic premises has a long history, and the present model sits inside it while differing from each of its neighbours in a specific, checkable way. [1] derives the Tsirelson bound $2\sqrt{2}$ for the CHSH quantity from global determinism, showing that counterfactual parameter independence inside a globally deterministic universe leads directly to the quantum limit on non-locality.

   from 2. Background and Related Work

4. The ontology is a probability space $(\Omega, \Sigma, \mu)$ of continuous deterministic states $\omega \in \Omega$, together with a flow $\Phi_t : \Omega \to \Omega$ that is measure-preserving, $\mu(\Phi_t^{-1}(A)) = \mu(A)$ for all $A \in \Sigma$, and mixing, meaning that for all $A, B \in \Sigma$, $$ \lim_{t \to \infty} \mu\bigl(\Phi_t^{-1}(A) \cap B\bigr) = \mu(A)\,\mu(B). $$ A measurement act with outcome set $\{1, \dots, m\}$ is a finite partition $\Pi = \{B_1, \dots, B_m\}$ of $\Omega$ into measurable basins.

   from 3.1 Ontology and measurement interface

What would falsify it

1. A falsification criterion for the model, expressed as a bound on certifiable CHSH values given an independently bounded $\delta$ (Sections 5 and 6).

   from 1. Introduction

2. Together with [4], this shows that apparently exotic dynamics are unitary quantum mechanics in disguise, so a deterministic interface model must be judged on its observable statistics, not on its formal dress — the standard we apply in Sections 4 and 5. [6] argues that a deterministic universe can accommodate the appearance of free will, while acknowledging that such a view must confront Bell-type experimental evidence.

   from 2. Background and Related Work

3. The two parametrizations are not interconverted (Section 3.3); the comparison holds because both are upper bounds on the same observable quantity $S$, so a model respecting the tighter information benchmark cannot reach the value that the looser $\delta$-budget permits in principle.

   from 4.5 Mutual-information benchmark comparison

Where the claim stops

1. What the model does and does not claim.

   from 6. Discussion

2. It does not claim that the ontology is true, that the measurement partition is unique, or that the $\delta$-budget construction is optimal — the budgets in R4 are necessary conditions within one construction, not proven minima over all local deterministic models.

   from 6. Discussion

3. The result rests on the following assumptions, each with a stated sensitivity: Mixing and measure preservation of the flow.

   from 6. Discussion

Where to read

1. Abstract257 words · 1 min
2. 1. Introduction513 words · 2 min
3. 2. Background and Related Work540 words · 2 min
4. 3. Methods445 words · 2 min
5. 4. Analysis1,276 words · 6 min
6. 5. Results352 words · 2 min
7. 6. Discussion772 words · 3 min
8. 7. Practical Implications and Engineering Pathways668 words · 3 min
9. 8. Conclusion221 words · 1 min
10. Provenance and AI Involvement225 words · 1 min