Semi-Quantum Games Unify Bell Inequalities and Quantum Communication — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 11, 2026• 0 citations

Abstract

FINDING: Buscemi's semi-quantum games generalize Bell inequalities by replacing local measurements with quantum channels, unifying entanglement and classical communication in a single statistical comparison framework. | MATH: Bell inequality violation condition: $ \sum_{a,b,x,y} P(a,b|x,y) \, \mu(a,b,x,y) \leq 1 $ (classical bound); Buscemi's extension replaces $P(a,b|x,y)$ with quantum channel correlations $ \mathrm{Tr}[(E_a^x \otimes F_b^y) \rho] $ where $E_a^x, F_b^y$ are POVMs, and the "free" operations become arbitrary quantum instruments, yielding a hierarchy of semi-device-independent bounds. The nondegeneracy property (arXiv:1903.05303) defines when a Bell inequality is violated only by a state $\rho$ and not by its orthogonal complement $\rho^\perp$, requiring $\mathrm{Tr}[\rho \rho^\perp]=0$ in the fixed measurement basis. | CONNECTION: The statistical comparison framework maps to convex cones — the set of classical correlations forms a polytope whose facets are Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23292697

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Semi-Quantum Games Unify Bell Inequalities and Quantum Communication — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 11, 2026• 0 citations

Publisher

Abstract

FINDING: Buscemi's semi-quantum games generalize Bell inequalities by replacing local measurements with quantum channels, unifying entanglement and classical communication in a single statistical comparison framework. | MATH: Bell inequality violation condition: ( \sum_{a,b,x,y} P(a,b|x,y) , \mu(a,b,x,y) \leq 1 ) (classical bound); Buscemi's extension replaces (P(a,b|x,y)) with quantum channel correlations ( \mathrm{Tr}[(E_a^x \otimes F_b^y) \rho] ) where (E_a^x, F_b^y) are POVMs, and the "free" operations become arbitrary quantum instruments, yielding a hierarchy of semi-device-independent bounds. The nondegeneracy property (arXiv:1903.05303) defines when a Bell inequality is violated only by a state (\rho) and not by its orthogonal complement (\rho^\perp), requiring (\mathrm{Tr}[\rho \rho^\perp]=0) in the fixed measurement basis. | CONNECTION: The statistical comparison framework maps to convex cones — the set of classical correlations forms a polytope whose facets are Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23292697

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B

and fixed local measurements

\{M_A^a, M_B^b\}

, if a state

\rho

of dimension

d

violates

B

(i.e.,

\langle B\rangle_\rho > \beta_{\text{local}}

), then its orthogonal complement

\rho^\perp

(with

\text{Tr}(\rho\rho^\perp)=0

) cannot also violate

B

under the same measurements. This yields a dimension-witness bound:

\text{Ent}(\rho) \geq f(\langle B\rangle_\rho, d)

where

f

is monotonic in the Bell violation. | CONNECTION: The orthogonal-state separation mirrors root-system orthogonality (e.g.,

A_2

root lattice angles at 60° and 120°), and the dimension constraint

d

acts like a lattice rank — violation strength maps to a radial coordinate in a convex set of quantum corre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables) - Quantum maximum:

S_{\text{QM}} = 2\sqrt{2} \approx 2.828

(Tsirelson bound) - Ratio:

2\sqrt{2}/2 = \sqrt{2} \approx 1.414

— not a golden ratio, but note

\sqrt{2}

is the diagonal of unit square (crystallographic root system

A_1 \times A_1

) - Buscemi's extension: replaces measurement choices with quantum inputs, yielding *nonlocal* inequalities with no classical analogue — the bound becomes state-dependent, often expressed via semidefinite programming (SDP) hierarchies (NPA hierarchy). CONNECTION: - The Tsirelson bound

2\sqrt{2}

is the maximal algebraic value of a Clifford algebra generator pair — linked to the octonionic/split-octonionic s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables); quantum violation

S_{\text{max}} = 2\sqrt{2} \approx 2.828

. Bell's original:

P(\vec{a},\vec{b}) - P(\vec{a},\vec{b}') \leq 1 + P(\vec{b},\vec{b}')

. Buscemi's unification: replaces measurement settings with quantum inputs, yielding semi-device-independent bounds via teleportation-replication constraints. | CONNECTION: The CHSH maximum

2\sqrt{2}

is the ratio of the diagonal of a unit square to its side — a direct geometric expression of the square's symmetry (dihedral group

D_4

). The violation threshold

2

2\sqrt{2}

maps to the ratio

1:\sqrt{2}

, which is the aspect ratio of A-series paper (ISO 216) and t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Bell inequality tests confirm quantum non-locality; new generalized rules (Buscemi) unify entanglement tests via device-independent frameworks. MATH: CHSH inequality:

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables); quantum violation max

S = 2\sqrt{2} \approx 2.828

. Bell's original inequality:

P(\vec{a},\vec{b}) - P(\vec{a},\vec{c}) \leq 1 + P(\vec{b},\vec{c})

. Buscemi's unification: replaces measurement settings with quantum inputs, yielding semi-device-independent bounds—generalizes to

S_q \leq 2\sqrt{2}

for qubits, but for higher-dimensional systems, Tsirelson bounds scale as

2\sqrt{d}

for maximally entangled states. CONNECTION: The maximal quantum violation

2\sqrt{2}

relates to the golden ratio via

2\sqrt{2} = 2 \times 1.414

, and

1.414^2 = 2

—a Pythagorean doubling. More directly, the CHSH bound

2\sqrt{2}

appears in the geometry of the square (diagonal/side ratio), linking to 45° rotations in 2D lattice symme Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\mathcal{P}_{\text{loc}} = \text{conv}\{p(a,b|x,y) = p(a|x)p(b|y)\}

. Buscemi's unification introduces "nonlocal games" with quantum strategies yielding success probabilities

\omega_q(G) > \omega_c(G)

, where the gap is bounded by the Grothendieck constant

K_G

(1.676 ≤

K_G

≤ 1.782) for correlation-type inequalities. The facet structure is characterized by the dimension of the probability simplex and the rank of the correlation matrix

C_{ab} = \langle A_x B_y \rangle

, with Tsirelson bounds

\langle B \rangle_q \leq 2\sqrt{2}

for CHSH. | CONNECTION: The polytope facets correspond to root systems of type

A_n

(for n-party correlations) and the CHSH inequality's maximal violation \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables) vs. quantum bound

S_{\text{QM}} = 2\sqrt{2} \approx 2.828

. - Bell's original inequality:

|P(\vec{a},\vec{b}) - P(\vec{a},\vec{c})| \leq 1 + P(\vec{b},\vec{c})

. - Quantum violation ratio:

2\sqrt{2}/2 = \sqrt{2} \approx 1.414

— note this is NOT the golden ratio, but it is the ratio of the diagonal of a unit square to its side (Pythagorean). - Buscemi's unification: extends Bell inequalities to "semi-quantum" scenarios where one party is classical, one quantum — yields new bounds involving

\sqrt{2}

and

2

in generalized forms. - No explicit appearance of 0.382, 0.618, 0.786, 1.618, 2.618 in the provided abstracts Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com