QBism recasts quantum theory as a normative framework for an agent's probability assignments, with the Born rule taking the form of a consistency condition known as the Urgleichung. Motivated by this perspective, qplex theories provide a broader class of probabilistic models in which the sets of valid states and measurements are constrained by QBist-inspired geometric conditions. While qplexes have been extensively studied for single systems, their implications for bipartite correlations remain largely unexplored. In this work, we investigate bipartite correlations in qplex theories by expressing joint expectation values as inner products between suitably defined $C$-vectors. This geometric formulation allows Bell-type inequalities to be studied as optimization problems over qplex-compatible probability assignments. We first analyze the CHSH scenario and show that the shared inner-product structure of the $C$-vectors restricts the maximal value to the Tsirelson bound $2\sqrt{2}$. We then turn to the three-outcome CGLMP inequality $I_{2233}$ and find that the same qplex-derived norm and inner-product constraints allow the algebraic maximum of 4, thereby exhibiting superquantum correlations. These results show that qplex geometry captures enough structure to reproduce an important quantum bound in the two-outcome case, but not enough to recover the full set of quantum correlation constraints. The analysis therefore suggests that additional principles are needed to complete the QBist reconstruction of quantum theory.
QBism recasts quantum theory as a normative framework for an agent's probability assignments, with the Born rule taking the form of a consistency condition known as the Urgleichung. Motivated by this perspective, qplex theories provide a broader class of probabilistic models in which the sets of valid states and measurements are constrained by QBist-inspired geometric conditions. While qplexes have been extensively studied for single systems, their implications for bipartite correlations remain largely unexplored. In this work, we investigate bipartite correlations in qplex theories by expressing joint expectation values as inner products between suitably defined C-vectors. This geometric formulation allows Bell-type inequalities to be studied as optimization problems over qplex-compatible probability assignments. We first analyze the CHSH scenario and show that the shared inner-product structure of the C-vectors restricts the maximal value to the Tsirelson bound 22. We then turn to the three-outcome CGLMP inequality I2233 and find that the same qplex-derived norm and inner-product constraints allow the algebraic maximum of 4, thereby exhibiting superquantum correlations. These results show that qplex geometry captures enough structure to reproduce an important quantum bound in the two-outcome case, but not enough to recover the full set of quantum correlation constraints. The analysis therefore suggests that additional principles are needed to complete the QBist reconstruction of quantum theory.
When two particles are created together in a single quantum event, they can become entangled: their properties remain correlated no matter how far apart they travel. Measure one, and the outcome of measuring the other is no longer independent. This correlation is the central puzzle of quantum mechanics. Classical physics sets a hard limit on how strong such correlations can be. The CHSH inequality (Clauser, Horne, Shimony, and Holt, 1969) formalises that limit: for any theory in which each particle carries its own independent set of predetermined answers, the correlation measure S cannot exceed 2. Bell-type experiments consistently violate this bound, with the maximum ideal value predicted by quantum mechanics being S = 2√2. Something does not fit. Several explanations have been proposed. Some invoke extra dimensions. Others appeal to wormhole geometries connecting the particles. The mainstream position is that the outcomes "collapse and become correlated," which states the observation without explaining the mechanism. All of these either add dramatic ontological furniture or repackage the mystery as an answer. This paper proposes a simpler reading. The value 2√2 is not a mystery to be explained by exotic physics. It is the geometric signature of a jointly structured bipartite state whose correlation was fixed at the source, at the moment of entanglement, not at measurement time. The two particles are not two independent objects that later communicate. They are two readout surfaces of a single joint preparation. The cosine correlation law, the Tsirelson bound, and the full Bell violation follow directly from the geometry of the two-qubit state space and the relative orientation of the measurement settings. No signal, no extra dimension, no new physics is required. Three companion papers in this series explored the opposite hypothesis: The Speed of Light and Its Limits [1] examined whether entanglement could circumvent relativistic constraints; Quantum Entanglement as a Communication Channel [2] proposed a synchronized-clock experiment to detect timing signatures; and Parametrizing Correlation Timing [3] developed a Lorentz-structured ansatz for concurrence-dependent delays. If those experimental proposals return null, the present reading is reinforced: not by proof, but by elimination of the alternatives that require more. A final section develops geometric and regime-based frameworks consistent with the source-structural reading, including the measurement torus as the natural geometry of Bell experiments, a jointness index based on Concurrence, and connections to emergent spacetime programmes. These are offered as directions for future development.
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres’s dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. The central variety of mathematical entity in our reconstruction is the qplex, a very particular type of subset of a probability simplex. Along the way, by closely studying the symmetry properties of qplexes, we derive a condition for the existence of a d-dimensional SIC.
Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models. Quantum mechanics exceeds this boundary through non-commutativity, reaching the Tsirelson bound 22 for CHSH. We show that this polytope structure is not specific to quantum foundations: it appears identically in the causal inference literature, where the instrumental inequality, the Balke--Pearl linear programming bounds, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope. Fine's theorem -- that CHSH inequalities hold if and only if a joint distribution exists -- is precisely the pivot: the instrumental variable model in causal inference is structurally equivalent to the Bell local hidden-variable model, with the instrument playing the role of the measurement setting and the latent confounder playing the role of the hidden variable λ. We develop this correspondence in detail, extending it to algorithmic (Kolmogorov complexity) and entropic formulations of Bell inequalities, the NPA semidefinite programming hierarchy, and the MIP∗=RE undecidability result. We further show that the Born-rule / Bayes-rule duality underlying quantum Bayesian computation exploits the same non-commutativity that enables Bell violation, providing polynomial speedups for posterior inference. The framework yields a concrete dictionary between quantum information theory, causal econometrics, and Bayesian computation, and suggests new directions including NPA-based quantum causal inference algorithms and quantum architectures for function approximation.
This dissertation investigates statistical constraints on measurement outcomes in quantum mechanics, addressing two fundamental features that distinguish quantum theory from classical physics: non-reality and uncertainty.We examine these features through two perspectives: observable-dependent statistical constraints on quantum correlations and estimationbased measurement uncertainty in infinite-dimensional quantum systems.In the first part, we study realizability problems in both local realism and quantum theory, focusing on observable-dependent constraints.While previous studies have mainly characterized quantum correlations independently of observables, we derive a necessary and sufficient condition for quantum realizability of correlations in the Clauser-Horne-Shimony-Holt (CHSH) scenario when observables are fixed in advance.Our main result provides an explicit inequality that imposes stronger constraints than previously known observable-independent bounds.We demonstrate that the maximum CHSH inequality violation depends on observable choice and apply our results to Bell's original scenario, revealing that whether Bell's original inequality is violated depends sensitively on observable selection.These findings demonstrate that non-reality does not manifest equally in all measurement configurations.In the second part, we extend the estimation-based formulation of measurement uncertainty by Watanabe, Sagawa, and Ueda to infinite-dimensional quantum systems.The original formulation provided clear physical interpretation but excluded fundamental observables such as position and momentum.Using pseudo-inverse forms of positive-semidefinite sesquilinear forms, we develop classical and quantum estimation theory for statistical models whose parameter space is the set of full-rank quantum states on infinite-dimensional Hilbert spaces.We derive classical and quantum Cramér-Rao inequalities and define estimationbased measurement errors for both bounded and unbounded operators.For each error type, we establish error-error uncertainty relation inequalities stronger than the original Watanabe-Sagawa-Ueda bound due to an improved derivation utilizing the full Cauchy-Schwarz inequality.Our work provides a universal definition of measurement errors with a clear physical interpretation while maintaining mathematical rigor.Our results contribute to quantum foundations by deepening the understanding of how measurement outcome constraints arise from quantum structure and observable properties.The methodological tools introduced-including characteristic functions, quasi-joint probability distributions, symmetry-based reduction techniques, and pseudo-inverse forms-may prove useful for future investigations in quantum foundations and quantum information theory.