DISTINCTION

Quantum Mechanical Grounding

Quantum mechanics is fundamentally a theory of distinguishability:

Orthogonality principle: Two states |ψ⟩ and |φ⟩ are perfectly distinguishable iff ⟨ψ|φ⟩ = 0

Measurement postulate: Observable A has distinct eigenvalues a_i corresponding to distinguishable outcomes:

• A|ψ_i⟩ = a_i|ψ_i⟩
• ⟨ψ_i|ψ_j⟩ = δ_ij (orthonormality = distinguishability)

Quantum distinguishability theorem: For identical particles, the symmetrization postulate (bosons/fermions) determines which states are physically distinguishable. Even "indistinguishable" particles have distinguishable quantum numbers (spin, position, momentum).

Pauli Exclusion: No two fermions can occupy the same quantum state → fermions MUST be distinguished by at least one quantum number.

Formal Definition

Let U be a universal domain. A distinction is a partition of U into non-empty subsets:

U = A \cup A^c, \quad A \neq \emptyset, \quad A^c \neq \emptyset

Minimal distinction: |A| = 1 (one element vs. the rest) → the bit ([[005_D1.2_Bit-Definition|D1.2]])

• Holevo Bound

Bridge Count: 7