A. Linear Algebra Reference

The working minimum, in the book's notation. Vectors: a state of n qubits is a vector ψ in ℂ²ⁿ; column by convention; ⟨ψ| is the conjugate-transpose (row). Inner product ⟨φ|ψ⟩ = Σ φᵢ*ψᵢ — measures overlap; states are normalized ⟨ψ|ψ⟩=1; orthogonal states are perfectly distinguishable. Outer product |φ⟩⟨ψ| is an operator (maps |ψ⟩-direction into |φ⟩-direction). Matrices as operators: (AB)|ψ⟩ = A(B|ψ⟩); non-commutative AB≠BA in general — the source of uncertainty and much else. Unitary: U†U = I — preserves inner products (lengths, angles, probabilities); all quantum gates. Hermitian: A = A† — real eigenvalues; all observables. Eigen-decomposition: A|vᵢ⟩ = λᵢ|vᵢ⟩; for Hermitian A the eigenvectors form an orthonormal basis — measurement returns eigenvalues, collapses onto eigenvectors. Tensor product: (A⊗B)(|φ⟩⊗|ψ⟩) = A|φ⟩⊗B|ψ⟩; dimension multiplies: n-qubit operators are 2ⁿ×2ⁿ. Trace: tr(A) = Σ Aᵢᵢ; tr(|φ⟩⟨ψ|) = ⟨ψ|φ⟩; density matrices live on traces (Appendix H). Spectral norm / fidelity: ‖A‖ = max eigenvalue of √(A†A); used in error bounds. A one-line numpy check for every claim above: instantiate at dimension 2–4 and assert.