Source
- Coecke, Pavlovic & Vicary, LIT-274, Thm 5.1 and §6, read in full in NOTE-247 (its claims C1 and C2, “strong (proof)”).
- Abramsky & Heunen, LIT-275, Lemma 3 and Thm 22, read in NOTE-248.
- Parts (a) and (c) are elementary and written out below.
What was actually shown
(a) No basis or factorisation is intrinsic. Let H be a Hilbert space of finite dimension n. If {e_i} and {f_i} are orthonormal bases, the map U e_i = f_i is unitary, so the unitary group acts transitively on orthonormal bases. If U₁, U₂ : H_A ⊗ H_B → H are two unitary identifications with factors of fixed dimensions, U₂U₁⁻¹ is a unitary of H that carries the first factorisation to the second. So nothing in H alone picks out a basis or a factorisation. A non-degenerate self-adjoint operator does fix a basis, its eigenbasis, up to a phase on each vector. Under unitary conjugation, though, all that survives is its list of eigenvalues.
(b) Bases are commutative special †-Frobenius algebras, over ℂ. On a finite-dimensional complex Hilbert space, every commutative †-Frobenius monoid determines an orthogonal basis, namely its copyable elements, and is the copy-and-delete structure of that basis (LIT-274, Thm 5.1). It is special, m∘m† = id, exactly when the basis is orthonormal (§6). Without a unit the correspondence extends to every dimension. On a complex Hilbert space of any dimension, a commutative special †-Frobenius algebra comes from an orthonormal basis exactly when it satisfies Ambrose’s H*-axiom, equivalently when it is semisimple (LIT-275, Thm 22). A unital Frobenius algebra exists in Hilb only in finite dimension (Lemma 3).
(c) Over ℝ it fails. Take ℂ as the real Hilbert space ℝ² with orthonormal basis {1, i} and inner product ⟨a, b⟩ = Re(a b̄). Let m be complex multiplication, so m(1⊗1) = 1, m(1⊗i) = m(i⊗1) = i and m(i⊗i) = −1.
- Special. m m†(1) = ⟨1, 1⟩·1 + ⟨−1, 1⟩·(−1) = 2, and likewise m m†(i) = 2i. So μ = m/√2 satisfies μμ† = id.
- Commutative and associative, as complex multiplication is.
- Frobenius. m†(z) is the bilinear form (a, b) ↦ Re(ab z̄). Since the adjoint of multiplication by x is multiplication by x̄, the form attached to (x ⊗ 1)m†(z) is (a, b) ↦ Re(x̄a·b·z̄), which is Re(ab·conj(xz)), the form of m†(xz). So m† is a module map, which is the Frobenius law.
- No copyable vector. A vector x is copyable when μ†(x) = x ⊗ x. The form μ†(x) has matrix (1/√2)[[p, −q], [−q, −p]] in {1, i}, where x̄ = p + qi. Its determinant is −|x|²/2, so it has rank 2 for x ≠ 0, while x ⊗ x has rank 1. So no nonzero x is copyable.
So over the reals a basis still induces a commutative special †-Frobenius algebra, but such an algebra need not come from a basis. NOTE-248 adds that this ℂ, as a real algebra, satisfies the H*-axiom, so the axiom does not rescue the real case in any dimension.
What this does not say
- That basis-dependent models are wrong, or which basis a model should use. That, and where the record’s Hilbert-space works get their basis, is THEORY-017’s synthesis, which is Proposed.
- Anything about continuous observables. They have no eigenbasis and fall outside Thm 22 (LIT-275, §6).
- That a tensor factorisation is a basis. It is coarser: many bases respect it. Part (a) treats both only as structure the space does not supply.
- Whether the H-axiom is automatic.* LIT-275 leaves this open (Prop. 23).