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Quantum Teleportation: Entanglement Swapping, Bell Measurements, and Fidelity Metrics — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 20, 2026• 0 citations
Publisher

Abstract

FINDING: Quantum teleportation relies on entanglement swapping and Bell-state measurements, not faster-than-light transport; fidelity and success probability are key metrics. | MATH: Bell state fidelity \( F = \langle \psi | \rho_{\text{out}} | \psi \rangle \); success probability \( P_{\text{success}} = 1/4 \) per Bell measurement (for qubits); entanglement swapping uses CNOT + Hadamard gates. | CONNECTION: No direct geometric ratios (0.382, 0.618, etc.) or base-60 links; however, Bell states form a 4-dimensional orthonormal basis with symmetry group \( S_4 \) (tetrahedral symmetry), and the CNOT gate corresponds to a 4x4 matrix with eigenvalues ±1, linking to the \( D_4 \) root system (crystallographic symmetry of the 4D hypercube). | DEPTH: 6 — mathematically sound but standard quantum information; no new constants or harmonic ratios. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22022573

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Quantum Teleportation via Entanglement and Bell State Measurement — E8 Intelligence Research

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FINDING: Quantum teleportation relies on entanglement and Bell state measurement to transfer qubit states without physical particle transport. | MATH: Bell state basis: \(|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)\), \(|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)\). Teleportation fidelity \(F = \langle \psi | \rho_{\text{out}} | \psi \rangle\) approaches 1 for ideal entanglement. No new constants or ratios beyond standard quantum mechanics. | CONNECTION: No explicit geometric harmony ratios (0.382, 0.618, etc.) or base-60, crystallographic symmetries appear in these video descriptions. The circuit diagrams use standard qubit rotations and CNOT gates, which relate to SU(2) symmetry and Bloch sphere geometry (a 2-sphere with Hopf fibration structure), but no novel harmonic ratios. | DEPTH: 3 — These are educational overviews, not original research. No new mathematical essence extracted beyond known quantum teleportation protocol (Bennett et al., Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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FINDING: Quantum teleportation relies on entanglement swapping and Bell state measurements, not physical matter transport; Oxford's 2025 breakthrough demonstrates high-fidelity teleportation across a quantum network node. | MATH: Bell state basis: \(|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\), \(|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)\), \(|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)\), \(|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)\). Teleportation fidelity \(F = \langle \psi | \rho_{\text{out}} | \psi \rangle\) with \(F > 2/3\) for non-classical threshold. No new fundamental constants or ratios emerge from these videos. | CONNECTION: No direct geometric harmony ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries appear. The Bell states exhibit \(SU(2)\) symmetry, which is isomorphic to the double cover of \(SO(3)\), a rotation group linked to spherical harmonics but not to the spe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Quantum Teleportation: Bell States, Classical Communication, and Fidelity Limits — E8 Intelligence Research

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Quantum Teleportation: Entanglement, Bell Measurement, and Pauli Correction — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum teleportation relies on shared entanglement and classical communication, not instantaneous matter transfer; mathematical core is Bell state measurement and unitary transformation. | MATH: Bell state basis: \(|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)\), \(|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)\). Teleportation protocol: Alice performs Bell measurement on her qubit and entangled pair, sends 2 classical bits to Bob, who applies Pauli correction \(I, X, Z, XZ\) based on outcome. Fidelity = \(|\langle \psi | \rho_{\text{out}} | \psi \rangle|^2\). No faster-than-light signaling: classical bits limit speed to ≤ c. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear. However, Bell states are maximally entangled, corresponding to the four vertices of a regular tetrahedron in the Bloch sphere representation (SO(3) symmetry). The Pauli matrices \(X, Y, Z\) generate the Clifford group, which is the sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 6, 2026•0 citations
FINDING: Quantum teleportation relies on entanglement swapping and Bell-state measurements, not faster-than-light transport; fidelity and success probability are key metrics. | MATH: Bell state fidelity ( F = \langle \psi | \rho_{\text{out}} | \psi \rangle ); success probability ( P_{\text{success}} = 1/4 ) per Bell measurement (for qubits); entanglement swapping uses CNOT + Hadamard gates. | CONNECTION: No direct geometric ratios (0.382, 0.618, etc.) or base-60 links; however, Bell states form a 4-dimensional orthonormal basis with symmetry group ( S_4 ) (tetrahedral symmetry), and the CNOT gate corresponds to a 4x4 matrix with eigenvalues ±1, linking to the ( D_4 ) root system (crystallographic symmetry of the 4D hypercube). | DEPTH: 6 — mathematically sound but standard quantum information; no new constants or harmonic ratios. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation relies on entanglement and Bell state measurement to transfer qubit states without physical particle transport. | MATH: Bell state basis: ∣Φ±⟩=12(∣00⟩±∣11⟩)|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)∣Φ±⟩=2​1​(∣00⟩±∣11⟩), ∣Ψ±⟩=12(∣01⟩±∣10⟩)|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)∣Ψ±⟩=2​1​(∣01⟩±∣10⟩). Teleportation fidelity F=⟨ψ∣ρout∣ψ⟩F = \langle \psi | \rho_{\text{out}} | \psi \rangleF=⟨ψ∣ρout​∣ψ⟩ approaches 1 for ideal entanglement. No new constants or ratios beyond standard quantum mechanics. | CONNECTION: No explicit geometric harmony ratios (0.382, 0.618, etc.) or base-60, crystallographic symmetries appear in these video descriptions. The circuit diagrams use standard qubit rotations and CNOT gates, which relate to SU(2) symmetry and Bloch sphere geometry (a 2-sphere with Hopf fibration structure), but no novel harmonic ratios. | DEPTH: 3 — These are educational overviews, not original research. No new mathematical essence extracted beyond known quantum teleportation protocol (Bennett et al., Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation relies on entanglement swapping and Bell state measurements, not physical matter transport; Oxford's 2025 breakthrough demonstrates high-fidelity teleportation across a quantum network node. | MATH: Bell state basis: ∣Φ+⟩=12(∣00⟩+∣11⟩)|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)∣Φ+⟩=2​1​(∣00⟩+∣11⟩), ∣Φ−⟩=12(∣00⟩−∣11⟩)|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)∣Φ−⟩=2​1​(∣00⟩−∣11⟩), ∣Ψ+⟩=12(∣01⟩+∣10⟩)|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)∣Ψ+⟩=2​1​(∣01⟩+∣10⟩), ∣Ψ−⟩=12(∣01⟩−∣10⟩)|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)∣Ψ−⟩=2​1​(∣01⟩−∣10⟩). Teleportation fidelity F=⟨ψ∣ρout∣ψ⟩F = \langle \psi | \rho_{\text{out}} | \psi \rangleF=⟨ψ∣ρout​∣ψ⟩ with F>2/3F > 2/3F>2/3 for non-classical threshold. No new fundamental constants or ratios emerge from these videos. | CONNECTION: No direct geometric harmony ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries appear. The Bell states exhibit SU(2)SU(2)SU(2) symmetry, which is isomorphic to the double cover of SO(3)SO(3)SO(3), a rotation group linked to spherical harmonics but not to the spe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation relies on Bell state measurement and classical communication to transfer quantum information without moving matter; no new mathematical constants or geometric ratios are derived. | MATH: Bell state basis: ∣Φ±⟩=12(∣00⟩±∣11⟩)|\Phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)∣Φ±⟩=2​1​(∣00⟩±∣11⟩), ∣Ψ±⟩=12(∣01⟩±∣10⟩)|\Psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)∣Ψ±⟩=2​1​(∣01⟩±∣10⟩). Teleportation fidelity F=⟨ψ∣ρ∣ψ⟩F = \langle\psi|\rho|\psi\rangleF=⟨ψ∣ρ∣ψ⟩ (max 1 for perfect). No new equations beyond standard quantum mechanics. | CONNECTION: No geometric harmony (0.382, 0.618, 0.786, 1.618, 2.618, base-60, crystallographic symmetry) present. Bell states are maximally entangled, but no lattice or root system link. | DEPTH: 2 — Standard textbook quantum teleportation (Bennett et al. 1993); no breakthrough in mathematical structure or constants. The videos are pedagogical, not novel. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation is experimentally validated, transferring a qubit state via entanglement and classical communication, not matter. | MATH: Bell state measurement, CNOT + Hadamard gates, classical 2-bit correction; fidelity >0.9 in recent experiments. | CONNECTION: No direct geometric ratios (0.382, 0.618, 1.618) appear; the process relies on SU(2) symmetry of qubit space and the discrete symmetry group of Bell states (isomorphic to Klein four-group). | DEPTH: 6 — foundational for quantum networks, but no new constants or harmonic ratios emerge. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation is a protocol for transferring an unknown quantum state via shared entanglement and classical communication, not a physical transport of matter. | MATH: Bell state basis: |Φ±⟩ = (|00⟩ ± |11⟩)/√2, |Ψ±⟩ = (|01⟩ ± |10⟩)/√2; teleportation requires 2 classical bits + 1 ebit (maximally entangled pair); state transfer fidelity F = 1 for ideal case; measurement in Bell basis collapses to one of 4 outcomes with probability 1/4 each. | CONNECTION: The 4 Bell states form a tetrahedral symmetry group (S₄) — a root system of type A₃, whose Coxeter-Dynkin diagram is the same as the 4-simplex projection. The √2 normalization is the same factor appearing in the golden ratio's algebraic conjugate (1/√2 ≈ 0.7071, related to 0.786 via √(φ−1) ≈ 0.786). The 2 classical bits per 1 qubit ratio (2:1) mirrors the 2:1 ratio in the golden angle (137.5° ≈ 2×68.75°). | DEPTH: 7 — The protocol is mathematically exact but does not reveal new universal constants; it confirms the resourc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation relies on shared entanglement and classical communication, not instantaneous matter transfer; mathematical core is Bell state measurement and unitary transformation. | MATH: Bell state basis: ∣Φ±⟩=12(∣00⟩±∣11⟩)|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)∣Φ±⟩=2​1​(∣00⟩±∣11⟩), ∣Ψ±⟩=12(∣01⟩±∣10⟩)|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)∣Ψ±⟩=2​1​(∣01⟩±∣10⟩). Teleportation protocol: Alice performs Bell measurement on her qubit and entangled pair, sends 2 classical bits to Bob, who applies Pauli correction I,X,Z,XZI, X, Z, XZI,X,Z,XZ based on outcome. Fidelity = ∣⟨ψ∣ρout∣ψ⟩∣2|\langle \psi | \rho_{\text{out}} | \psi \rangle|^2∣⟨ψ∣ρout​∣ψ⟩∣2. No faster-than-light signaling: classical bits limit speed to ≤ c. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear. However, Bell states are maximally entangled, corresponding to the four vertices of a regular tetrahedron in the Bloch sphere representation (SO(3) symmetry). The Pauli matrices X,Y,ZX, Y, ZX,Y,Z generate the Clifford group, which is the sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com