Logic and Algebraic Structures in Quantum Computing (Lecture Notes in Logic) by

Language: eng
Format: azw3
Publisher: Cambridge University Press
Published: 2016-02-02T16:00:00+00:00

(4.4)

The restriction maps Σ(iV′V) are well-known to be continuous, surjective maps with respect to the Gel’fand topologies on ΣV and ΣV′, respectively. They are also open and closed, see e.g. [13].

We equip the spectral presheaf with a distinguished family of subobjects (which are subpresheaves):

DEFINITION 2. A subobject S of Σ is called clopen if for each V ∈ V(N), the set SV is a clopen subset of the Gel’fand spectrum ΣV. The set of all clopen subobjects of Σ is denoted as SubclΣ.

The set SubclΣ, together with the lattice operations and bi-Heyting algebra structure defined below, is the algebraic implementation of the new topos-based form of quantum logic. The elements S ∈ SubclΣ represent propositions about the values of the physical quantities of the quantum system. The most direct connection with propositions of the form “A ε Δ” is given by the map called daseinisation, see Def. 3 below.

We note that the concept of contextuality (cf. Section 2) is implemented by this construction, since Σ is a presheaf over the context category V(N). Moreover, coarse-graining is mathematically realised by the fact that we use subobjects of presheaves. In the case of Σ and its clopen subobjects, this means the following: for each context V ∈ V(N), the component SV ⊆ Σ represents a local proposition about the value of some physical quantity. IfV V′ ⊂ V, then SV′ ⊇ Σ(iV′V) (SV) (since S is a subobject), so SV′ represents a local proposition at the smaller context V′ ⊂ V that is coarser than (i.e., a consequence of) the local proposition represented by SV.

A clopen subobject S ∈ SubclΣ can hence be interpreted as a collection of local propositions, one for each context, such that smaller contexts are assigned coarser propositions.

Clearly, the definition of clopen subobjects makes use of the Gel’fand topologies on the components ΣV, V ∈ V(N). We note that for each abelian von Neumann algebra V (and hence for each context V ∈ V(N)), there is an isomorphism of complete Boolean algebras

(4.5)

Here, Cl (ΣV) denotes the clopen subsets of ΣV.

There is a purely order-theoretic descriptionof SubclΣ: let

(4.6)

be the set of choice functions f : V(N) → ∐V∈V(N) P(V), where f(V) ∈ P(V) for all V ∈ V(N). Equipped with pointwise operations, P is a complete Boolean algebra, since each P(V) is a complete Boolean algebra. Consider the subset S of P consisting of those functions for which V′ ⊂ V implies f(V′) ≥ f(V) (this comparison is taken in P(V), into which P(V′) can be included). The subset S is closed under all meets and joins (in P), and clearly, S ≃ SubclΣ.

We define a partial order on SubclΣ in the obvious way:

∀S, T ∈ SubclΣ : S ≤ T :⇐⇒ (∀V ∈ V(N) : SV ⊆ T V). (4.7)

We define the corresponding (complete) lattice operations in a stagewise manner, i.e., at each context V ∈ V(N) separately: for any family (Si)i∈I,

(4.8)

where Si;V ⊆ ΣV is the component at V of the clopen subobject Si. Note that the

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