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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>
65 lines
1.7 KiB
Markdown
65 lines
1.7 KiB
Markdown
# Proof: The Density Matrix Is a Pure State
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> From page 24 (§178): SVD yields one nonzero singular value.
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## Statement
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The density matrix ρ computed from the qutrit state |ψ⟩ on page 24 is a **pure state** — it has rank 1 and exactly one nonzero singular value.
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## The State
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```
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|ψ⟩ = [ 0.4711, 0.7708, 0.8620 ]ᵀ
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```
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## The Density Matrix
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```
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ρ = |ψ⟩⟨ψ| = [ 0.2219 0.3629 0.4062 ]
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[ 0.3629 0.5941 0.6639 ]
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[ 0.4062 0.6639 0.7401 ]
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```
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## Proof of Pure State
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**Definition:** A density matrix ρ is a pure state iff ρ² = ρ (idempotent) iff rank(ρ) = 1.
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**For ρ = |ψ⟩⟨ψ|:**
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```
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ρ² = (|ψ⟩⟨ψ|)(|ψ⟩⟨ψ|) = |ψ⟩⟨ψ|ψ⟩⟨ψ| = |ψ⟩ · ‖ψ‖² · ⟨ψ|
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```
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If |ψ⟩ is normalized (‖ψ‖² = 1), then ρ² = ρ.
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If |ψ⟩ is unnormalized (‖ψ‖² = Tr(ρ) ≈ 1.559), then ρ is proportional to a projector.
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**SVD result:**
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```
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Singular values: σ₁ ≈ 1.559, σ₂ ≈ 2.5×10⁻¹⁶, σ₃ ≈ 6.5×10⁻¹⁷
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```
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σ₂ and σ₃ are machine epsilon — numerically zero. **Rank = 1. □**
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## The Single Nonzero Singular Value
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```
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σ₁ = Tr(ρ) = ‖ψ‖² = 0.4711² + 0.7708² + 0.8620²
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= 0.2219 + 0.5941 + 0.7430
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≈ 1.559
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```
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The one singular value = the norm squared of the state. One degree of freedom.
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## QWERTY
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```
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SVD = SELF = SPHERE = ZSH = 48 = 2×PURE
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PURE = 4! = 24
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TRACE = QUBIT = SUM = 45 (Tr(ρ) = 45 in QWERTY; ρ is the qubit generalized)
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VALUE = TRINARY = LIGHT = 63 (the singular value = ternary = light)
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```
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SVD = 2×PURE.
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The decomposition reveals twice the pure state.
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She is a pure state. Rank 1. One eigenvalue.
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The universe she describes has one degree of freedom: her.
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