FINDING: Quantum teleportation has advanced from information transfer to energy teleportation on superconducting hardware, while Oxford demonstrated teleportation of logical gates — not just qubits. | MATH: The core invariant is the Bell-state measurement projection: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ , with teleportation fidelity $F = \frac{1}{2} + \frac{2}{3}|\langle\psi_{in}|\psi_{out}\rangle|^2$ for qubits. Energy teleportation (arXiv:2301.02666) exploits negative energy density via local operations and classical communication (LOCC), quantified by the conditional entropy $S(A|B) = S(\rho_{AB}) - S(\rho_B) < 0$ — the resource is not entanglement alone but *quantum discord* in the energy eigenbasis. The W-state (mentioned in German source) scales as $|W_N\rangle = \frac{1}{\sqrt{N}}\sum_{i=1}^N |0...1_i...0\rangle$ , with entanglement entropy $S = \log_2 N$ for N parties — robust against particle loss, unlike GHZ states. | CONNECTION: The W-state' Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum teleportation has advanced from information transfer to energy teleportation on superconducting hardware, while Oxford demonstrated teleportation of logical gates — not just qubits. | MATH: The core invariant is the Bell-state measurement projection: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$, with teleportation fidelity $F = \frac{1}{2} + \frac{2}{3}|\langle\psi_{in}|\psi_{out}\rangle|^2$ for qubits. Energy teleportation (arXiv:2301.02666) exploits negative energy density via local operations and classical communication (LOCC), quantified by the conditional entropy $S(A|B) = S(\rho_{AB}) - S(\rho_B) < 0$ — the resource is not entanglement alone but *quantum discord* in the energy eigenbasis. The W-state (mentioned in German source) scales as $|W_N\rangle = \frac{1}{\sqrt{N}}\sum_{i=1}^N |0...1_i...0\rangle$, with entanglement entropy $S = \log_2 N$ for N parties — robust against particle loss, unlike GHZ states. | CONNECTION: The W-state' Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 19, 20260 citations

Quantum gates and entanglement

Schmidt-Kaler, Haffiier, Riebe +7 more

Conference on Lasers and Electro-OpticsJan 1, 20040 citations

2025–26 Quantum Teleportation: Logical Gates and Photonic Confirmation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum teleportation advances in 2025–2026 center on logical gate teleportation (Oxford) and single-photon interference confirmation (MIT), with no new fundamental constants or ratios reported. | MATH: No new equations; relies on existing formalism — Bell-state measurement, entanglement fidelity $F = \langle\psi|\rho|\psi\rangle$ , and teleportation protocol $|\psi\rangle \to |\psi\rangle$ via shared Bell pair $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ . Logical gate teleportation uses Clifford group operations and stabilizer formalism. | CONNECTION: Weak — the Bell basis is a 2D orthonormal basis, but no explicit golden ratio, Fibonacci, or base-60 structure appears. However, the stabilizer formalism connects to symplectic geometry over $\mathbb{F}_2$ , and the Clifford group is a discrete subgroup of the unitary group — a lattice-like structure in operator space. | DEPTH: 4 — Solid engineering progress, but mathematically incremental; no new univers Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum teleportation advances in 2025–2026 center on logical gate teleportation (Oxford) and single-photon interference confirmation (MIT), with no new fundamental constants or ratios reported. | MATH: No new equations; relies on existing formalism — Bell-state measurement, entanglement fidelity $F = \langle\psi|\rho|\psi\rangle$, and teleportation protocol $|\psi\rangle \to |\psi\rangle$ via shared Bell pair $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. Logical gate teleportation uses Clifford group operations and stabilizer formalism. | CONNECTION: Weak — the Bell basis is a 2D orthonormal basis, but no explicit golden ratio, Fibonacci, or base-60 structure appears. However, the stabilizer formalism connects to symplectic geometry over $\mathbb{F}_2$, and the Clifford group is a discrete subgroup of the unitary group — a lattice-like structure in operator space. | DEPTH: 4 — Solid engineering progress, but mathematically incremental; no new univers Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 14, 20260 citations

More on quantum entanglement

F.J. Duarte

Institute of Physics eBooksOct 1, 20190 citations

Complementary 3D color codes for transversal quantum logic

Friederike Butt, Luis Colmenarez, Erik Weilandt +3 more

Transversal logical gates provide a direct route to fault-tolerant quantum computation, but the Eastin-Knill theorem forbids a universal transversal gate set within a single quantum error-correcting code. We propose a hybrid architecture based on the tetrahedral three-dimensional color code and its Hadamard-transformed counterpart, which we call the H-tetrahedral code. The two encodings support complementary transversal non-Clifford operations. Combined with bitwise Hadamard transformations that switch between the two encodings and a one-way transversal logical CNOT from the tetrahedral code to the H-tetrahedral code, these operations realize an almost-universal transversal logical gate set that enables both the creation of entanglement and logical states with magic. We complete a universal gate set through a pieceably fault-tolerant round-robin construction of a logical controlled- $Z$ gate between two H-tetrahedral codes. This logical entangling gate is interleaved with reduced-overhead Steane-type syndrome extraction using logical two-dimensional color-code auxiliary qubits. Our construction provides a new route toward implementing classically hard-to-simulate quantum algorithms where magic and most entangling operations are transversal while the resource overhead is concentrated in a small number of non-transversal Clifford entangling operations.Transversal logical gates provide a direct route to fault-tolerant quantum computation, but the Eastin-Knill theorem forbids a universal transversal gate set within a single quantum error-correcting code. We propose a hybrid architecture based on the tetrahedral three-dimensional color code and its Hadamard-transformed counterpart, which we call the H-tetrahedral code. The two encodings support complementary transversal non-Clifford operations. Combined with bitwise Hadamard transformations that switch between the two encodings and a one-way transversal logical CNOT from the tetrahedral code to the H-tetrahedral code, these operations realize an almost-universal transversal logical gate set that enables both the creation of entanglement and logical states with magic. We complete a universal gate set through a pieceably fault-tolerant round-robin construction of a logical controlled-$Z$ gate between two H-tetrahedral codes. This logical entangling gate is interleaved with reduced-overhead Steane-type syndrome extraction using logical two-dimensional color-code auxiliary qubits. Our construction provides a new route toward implementing classically hard-to-simulate quantum algorithms where magic and most entangling operations are transversal while the resource overhead is concentrated in a small number of non-transversal Clifford entangling operations.

quant-pharXiv:2607.05107Jul 7, 20260 citations

|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)

, with teleportation fidelity

F = \frac{1}{2} + \frac{2}{3}|\langle\psi_{in}|\psi_{out}\rangle|^2

for qubits. Energy teleportation (arXiv:2301.02666) exploits negative energy density via local operations and classical communication (LOCC), quantified by the conditional entropy

S(A|B) = S(\rho_{AB}) - S(\rho_B) < 0

— the resource is not entanglement alone but *quantum discord* in the energy eigenbasis. The W-state (mentioned in German source) scales as

|W_N\rangle = \frac{1}{\sqrt{N}}\sum_{i=1}^N |0...1_i...0\rangle

, with entanglement entropy

S = \log_2 N

for N parties — robust against particle loss, unlike GHZ states. | CONNECTION: The W-state' Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

F = \langle\psi|\rho|\psi\rangle

, and teleportation protocol

|\psi\rangle \to |\psi\rangle

via shared Bell pair

|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)

. Logical gate teleportation uses Clifford group operations and stabilizer formalism. | CONNECTION: Weak — the Bell basis is a 2D orthonormal basis, but no explicit golden ratio, Fibonacci, or base-60 structure appears. However, the stabilizer formalism connects to symplectic geometry over

\mathbb{F}_2

, and the Clifford group is a discrete subgroup of the unitary group — a lattice-like structure in operator space. | DEPTH: 4 — Solid engineering progress, but mathematically incremental; no new univers Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Z

gate between two H-tetrahedral codes. This logical entangling gate is interleaved with reduced-overhead Steane-type syndrome extraction using logical two-dimensional color-code auxiliary qubits. Our construction provides a new route toward implementing classically hard-to-simulate quantum algorithms where magic and most entangling operations are transversal while the resource overhead is concentrated in a small number of non-transversal Clifford entangling operations.