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The Future of Now · UChicago QSNC companion
🪙 The 3-Sided Quantum Coin
MODULE 1 · EXERCISE 1
The Future of Now — Quantum Campus · UChicago QSNC companion
🎯 The big idea (for everyone)
A normal quantum coin has 2 sides: 0 and 1.
This one has 3 sides: 0, 1, and 2. We call it a qutrit — a 3-sided quantum spinner!
We get an even mix of all three sides. We call that mix Phi (say "fie").
Then we shine a flashlight from one special direction called psi (say "sigh").
Phi casts a shadow on psi. We split Phi into two pieces:
① the piece that lines up with psi, and ② the leftover piece.
Executive / UChicago layer: This is an orthogonal decomposition in the
complex vector space ℂ³. With Φ = (1/√3)(|0⟩+|1⟩+|2⟩) and the unit vector
ψ = (1/√3)(|0⟩ + ω|1⟩ − |2⟩), we write Φ = α·ψ + w, where
α = ⟨ψ|Φ⟩ is the projection coefficient and w ∈ V⊥ is the component orthogonal to ψ.
The inner product is antilinear in the first slot: ⟨ψ|Φ⟩ = Σ ψi* Φi.
🎮 Play with it
Slide the twist (the phase ω). Watch the shadow and the two energy
bars. No matter the twist, the split is always the same: 1/9 on psi, 8/9 leftover. That's the punchline!
twist angle (ω)60°
α = ⟨ψ|Φ⟩—
|α|² (on psi)0.111
|w|² (leftover)0.889
|α|²+|w|²1.000
energy on psi
leftover energy
🧠 Quick quiz — earn your stars
1) When we change the twist ω, the leftover energy |w|²…
2) The leftover piece w always points…
3) α = ⟨ψ|Φ⟩ measures…
🏆 You got it! You just performed an orthogonal decomposition in ℂ³ — graduate-level quantum math, understood in plain words.