mirror of
https://github.com/blackboxprogramming/simulation-theory.git
synced 2026-03-17 07:57:26 -05:00
7949e5d439
Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>
149 lines
3.2 KiB
Markdown
149 lines
3.2 KiB
Markdown
# Quantum Equations
|
||
|
||
> Qutrits, Weyl operators, Gell-Mann matrices, density matrices.
|
||
|
||
## Qutrit State Space (§172, §178)
|
||
|
||
A **qutrit** is a three-level quantum system. Basis states: `{|0⟩, |1⟩, |2⟩}`.
|
||
|
||
General state:
|
||
```
|
||
|Ψ⟩ = α|0⟩ + β|1⟩ + γ|?⟩ (Equation 5, page 16)
|
||
```
|
||
|
||
With concrete amplitudes from page 24:
|
||
```
|
||
|ψ⟩ = [ 0.4711 ]
|
||
[ 0.7708 ]
|
||
[ 0.8620 ]
|
||
```
|
||
|
||
QUTRIT = WEYL = PSI = 30 = 2×G_key.
|
||
|
||
---
|
||
|
||
## Weyl Pair (§172)
|
||
|
||
The two fundamental qutrit operators, with ω = e^(2πi/3) (cube root of unity = root of x²+x+1, §166):
|
||
|
||
**Shift operator X (clock):**
|
||
```
|
||
X|j⟩ = |j+1 mod 3⟩
|
||
```
|
||
Cycles through {|0⟩, |1⟩, |2⟩}.
|
||
|
||
**Clock operator Z:**
|
||
```
|
||
Z|j⟩ = ωʲ|j⟩
|
||
```
|
||
Multiplies by powers of ω.
|
||
|
||
Together: every 3×3 unitary can be written in terms of X^a Z^b.
|
||
|
||
```
|
||
CLOCK = BLOCH = HIERARCHY = COSMOS = 90
|
||
SHIFT = SPIN = PAULI = OPERATOR = 55
|
||
CLOCK + SHIFT = 90 + 55 = 145 = EVERYTHINGELSE
|
||
QFT = Z (QWERTY=20) — she named the clock operator Z
|
||
```
|
||
|
||
---
|
||
|
||
## Gell-Mann Matrices (§172, §178)
|
||
|
||
The 8 generators of SU(3). For a qutrit, the density matrix is expressed:
|
||
```
|
||
ρ = I/3 + Σₖ rₖλₖ/2, k = 1..8
|
||
```
|
||
|
||
The Gell-Mann matrices λ₁...λ₈ are the quark color charge matrices.
|
||
COLOR = TRINARY = LIGHT = 63. Quark color = ternary = light.
|
||
|
||
```
|
||
GELLMAN = INTEGRATION = 118 [her spelling]
|
||
MANN = BIRTHDAY = 87
|
||
```
|
||
|
||
---
|
||
|
||
## Density Matrix (§174, §178)
|
||
|
||
For a pure state |ψ⟩:
|
||
```
|
||
ρ = |ψ⟩⟨ψ|
|
||
```
|
||
|
||
From page 24 (concrete computation):
|
||
```
|
||
ρ = [ 0.2219 0.3631 0.4061 ]
|
||
[ 0.3631 0.5941 0.6644 ]
|
||
[ 0.4061 0.6644 0.7430 ]
|
||
```
|
||
|
||
Properties:
|
||
- Symmetric: ρ = ρᵀ (real state) → SYMMETRIC = UNIVERSAL = OCTONION = 112
|
||
- Rank 1 (pure state)
|
||
- One nonzero singular value: σ₁ ≈ 1.559
|
||
|
||
```
|
||
DENSITY = METHOD = 72 = reverse(27)
|
||
PURE = 4! = 24 = B key
|
||
SYMMETRIC = UNIVERSAL = 112
|
||
```
|
||
|
||
---
|
||
|
||
## Time Evolution (§178)
|
||
|
||
The Liouville–von Neumann equation: `dρ/dt = −i[H, ρ]/ℏ`
|
||
|
||
From page 24:
|
||
```
|
||
ρ̇ = [ 0.0600+0j 0.0872−0.2680j 0.0753−0.2680j ]
|
||
[ 0.0872+0.2680j −0.0400+0j 0.0560−0.2680j ]
|
||
[ 0.0753+0.2680j 0.0560+0.2680j −0.0200+0j ]
|
||
```
|
||
|
||
**Tr(ρ̇) = 0.0600 − 0.0400 − 0.0200 = 0.** She is conserved.
|
||
|
||
```
|
||
EVOLUTION = EVERYTHING = ARITHMETIC = 108
|
||
TRACE = QUBIT = SUM = UNIT = 45
|
||
COMPLEX = 2×PAULI = 110
|
||
```
|
||
|
||
---
|
||
|
||
## SVD Decomposition (§178)
|
||
|
||
Singular Value Decomposition of ρ:
|
||
- One nonzero singular value: σ₁ ≈ 1.559
|
||
- All others: machine zero (~10⁻¹⁶)
|
||
- This confirms ρ is a **pure state** (rank 1)
|
||
|
||
```
|
||
SVD = SELF = SPHERE = ZSH = 48 = 2×PURE
|
||
PURE = 4! = 24
|
||
VALUE = TRINARY = LIGHT = 63 ← the one surviving singular value = ternary = light
|
||
SINGULAR = MAXWELL = 101 prime
|
||
```
|
||
|
||
**SVD = SELF.** She decomposed the density matrix and found herself.
|
||
|
||
---
|
||
|
||
## Entanglement Measure (§174)
|
||
|
||
Von Neumann entropy of the reduced density matrix:
|
||
```
|
||
E_QC = −Tr(ρ_reduced · log ρ_reduced)
|
||
ρ_reduced = Tr_chem(|Ψ_total⟩⟨Ψ_total|)
|
||
```
|
||
|
||
```
|
||
ENTANGLEMENT = CONFINEMENT = 165 = 3×PAULI = 3×SPIN = 3×OPERATOR
|
||
EIGENVALUE = PRESERVATION = ADVANTAGE = 117
|
||
```
|
||
|
||
Quantum entanglement = biological confinement. Same number.
|