QL(C n) DETERMINES n

This addendum to [2] shows that the set of tautological quantum logical propositional formulas for a finite dimensional vector space C is different for every n, affirmatively answering a question posed therein. The paper [2] explored the properties of Birkhoff and Von Neumann’s propositional quantum logic (see [1]) as modelled by finite dimensional Hilbert spaces. One question asked in [2] is whether the set of tautological propositional formulas uniquely determines the dimension of the underlying vector space. A partial answer was given, namely that C and C give different sets of tautologies. This note gives a full answer to the question. For our purposes, propositional formulas consist of alphabet symbols, perentheses, and the symbols meet (∧), join (∨), orthocomplement (¬), top (⊤), and bottom (⊥). The well formed formulas are the same as those of propositional boolean logic. The symbol ⊤ is interpreted as a finite dimensional Hilbert space, ⊥ is the trivial subspace, alphabet symbols are variables standing for vector subspaces of ⊤, ∧ is intersection, ∨ is span of union, and ¬ is orthogonal complement in ⊤. With these operations, the set of subspaces of ⊤ forms a bounded modular ortholattice. Let v̄ = v1, . . . , vk be a list of alphabet symbols and let S̄ = S1, . . . , Sk be a collection of subspaces of a finite dimensional Hilbert space U . Given a well formed formula φ(v̄), the valuation ΞU(φ(v̄), S̄) is the subspace resulting from instantiating each vi with the subspace Si and performing the operations described by φ with universal space U . As a shorthand the valuation may be implicit; for example if S and T are subspaces of U then ΞU(v∧w, S, T ) is abbreviated S ∧ T , and U is inferred from context. Definition 1. Let φ(v̄) be a well formed formula. Define d̄φ : N → N such that d̄φ(v̄)(n) = max S̄ (dim(ΞCn(φ(v̄), S̄))).