C. Bra-Ket Notation Reference

Dirac notation, decoded. Ket |ψ⟩: a column vector — a state. Bra ⟨φ|: its conjugate transpose — a row vector, an unfired question. Bra-ket ⟨φ|ψ⟩: inner product — "how much φ in ψ"; 0 ⇒ orthogonal (perfectly distinguishable); 1 ⇒ identical. Ket-bra |φ⟩⟨ψ|: outer product — an operator; |φ⟩⟨φ| is the projector onto φ (measurement's mathematical action: ρ → |φ⟩⟨φ|ρ|φ⟩⟨φ|). Computational basis: |0⟩=(1,0)ᵀ, |1⟩=(0,1)ᵀ; n-qubit |bₙ₋₁…b₀⟩ = tensor of bits; index = the integer the bits encode. Common named states: |+⟩ = (|0⟩+|1⟩)/√2, |−⟩ = (|0⟩−|1⟩)/√2, |i⟩=(|0⟩+i|1⟩)/√2, Bell states (|00⟩±|11⟩)/√2, GHZ (|0…0⟩+|1…1⟩)/√2. Operator elements: ⟨φ|A|ψ⟩ — "the amplitude of φ after A acts on ψ"; ⟨0|X|1⟩ = 1 reads "X maps 1 into 0, amplitude 1." Subtleties: ⟨ψ|ψ⟩ ≠ |ψ⟩⟨ψ| (number vs. operator); tensor order is convention (Ch. 16.11's endianness); phases: |ψ⟩ and −|ψ⟩ identical, |0⟩+|1⟩ vs |0⟩−|1⟩ maximally different.