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- qwerty/constants.md: master reference table, 100+ constants §1-§178 - qwerty/equalities.md: all major QWERTY equalities by theme - equations/blackroad-equations.md: all 19 BlackRoad equations - equations/consciousness.md: Psi_care, Phi_universal, CECE update rule - equations/quantum.md: qutrit, Weyl pair, density matrix, SVD - equations/universal.md: Three Tests, Euler-Lagrange, fine-structure - proofs/ternary-efficiency.md: ln(3)/3 > ln(2)/2 - proofs/self-reference.md: the QWERTY encoding is self-referential - proofs/pure-state.md: density matrix rank=1, SVD=SELF - figures/durer-square.md: magic square with 2000 substitution - figures/trinary-table.md: TAND TMUL TNEG TXOR truth tables - figures/qutrit-operators.md: Weyl X/Z, Gell-Mann matrices - figures/keyboard.md: QWERTY encoding layout - notebooks/README.md: page-by-page index of all 24 notebook pages Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
80 lines
2.0 KiB
Markdown
80 lines
2.0 KiB
Markdown
# Qutrit Operators (§172)
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> ω = e^(2πi/3) — the primitive cube root of unity, root of x²+x+1 (§166)
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## Weyl Pair
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The two fundamental qutrit operators that generate all 3×3 unitaries:
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**Shift operator X** (SHIFT=55=PAULI=SPIN=OPERATOR):
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```
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X = [ 0 0 1 ] X|0⟩ = |1⟩
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[ 1 0 0 ] X|1⟩ = |2⟩
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[ 0 1 0 ] X|2⟩ = |0⟩
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```
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**Clock operator Z** (CLOCK=90=COSMOS=HIERARCHY=BLOCH):
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```
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Z = [ 1 0 0 ] Z|0⟩ = |0⟩
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[ 0 ω 0 ] Z|1⟩ = ω|1⟩
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[ 0 0 ω² ] Z|2⟩ = ω²|2⟩
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```
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Where ω = e^(2πi/3) = −½ + i(√3/2).
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**Together:**
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```
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SHIFT + CLOCK = 55 + 90 = 145 = EVERYTHINGELSE
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QFT (Quantum Fourier Transform) = Z = 20 in QWERTY
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```
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She named the clock operator Z. QFT=20=Z. The Fourier transform IS the clock.
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## Commutation Relation
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```
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XZ = ω·ZX
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```
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They almost commute — off by a phase ω. This is the Weyl algebra.
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## Gell-Mann Matrices (SU(3) generators)
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The 8 Gell-Mann matrices λ₁...λ₈:
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```
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λ₁ = [0 1 0] λ₂ = [0 -i 0] λ₃ = [1 0 0]
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[1 0 0] [i 0 0] [0 -1 0]
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[0 0 0] [0 0 0] [0 0 0]
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λ₄ = [0 0 1] λ₅ = [0 0 -i] λ₆ = [0 0 0]
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[0 0 0] [0 0 0] [0 0 1]
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[1 0 0] [i 0 0] [0 1 0]
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λ₇ = [0 0 0] λ₈ = [1 0 0 ]
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[0 0 -i] [0 1 0 ] / √3
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[0 i 0] [0 0 -2 ]
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```
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**Gell-Mann decomposition of density matrix:**
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```
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ρ = I/3 + Σₖ rₖλₖ/2, k = 1..8
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```
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GELLMAN = INTEGRATION = 118. The decomposition = integration.
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COLOR = TRINARY = LIGHT = 63. Quark color = ternary = light.
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## Universal Gate Set
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```
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{QFT₃, Z_φ, SUM} = universal gate set for qutrit computation
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```
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Where:
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- **QFT₃**: Quantum Fourier Transform (3-point) = the clock
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- **Z_φ**: Phase gate
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- **SUM**: Generalized CNOT for qutrits
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UNIVERSAL = OCTONION = 112.
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G ≤ QFT₃ / Z_φ / SUM — every gate decomposes into these three.
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