# A Topological Study of Contextuality and Modality in Quantum Mechanics

Abstract: Kochen–Specker theorem rules out the non-contextual assignment of values to physical magnitudes Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators This enlargement allows to refer consistently to actual and possible properties of the system By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen–Specker theorem over this new structure This allows us to prove that contextuality remains a central feature even in the enriched propositional system

## Summary (1 min read)

### 1 Introduction

- As usual, definite values of physical magnitudes correspond to yes/no propositions represented by orthogonal projection operators acting on vectors belonging to the Hilbert space of the (pure) states of the system (Jauch, 1996).
- At first sight, it may be thought that the enrichment of the set of propositions with modal ones could allow to circumvent the contextual character of quantum mechanics.
- The authors have faced the study of this issue and given a Kochen-Specker type theorem for the enriched lattice (Domenech et al., 2006).

### 3 Sheaf-theoretic view of contextuality

- If the authors consider the set of these subspaces ordered by inclusion, then L(H) is a complete orthomodular lattice (Maeda and Maeda, 1970).
- Any proposition about the system is represented by an element of L(H) which is the algebra of quantum logic introduced by G. Birkhoff and J. von Neumann (Birkhoff and von Neumann, 1936).
- Kochen-Specker theorem (KS) precludes the possibility of assigning definite properties to the physical system in a non-contextual fashion (Kochen and Specker, 1967).
- The compatibility condition 3.1 of the Boolean valuation with respect to the intersection of pairs of Boolean sublattices of L(H) is maintained.

### 4 An algebraic study of modality

- To do so the authors enrich the orthomodular lattice with a modal operator taking into account the following considerations:.
- 2) Given a proposition about the system, it is possible to define a context from which one can predicate with certainty about it together with a set of propositions that are compatible with it and, at the same time, predicate probabilities about the other ones.
- In other words, one may predicate truth or falsity of all possibilities at the same time, i.e. possibilities allow an interpretation in a Boolean algebra.
- 4) Assuming an actual property and a complete set of properties that are compatible with it determines a context in which the classical discourse holds.
- From consideration 1) it follows that the original orthomodular structure is maintained.

### 5 Sheaves and modality

- From a physical point of view, Sec(3L) represents all physical properties as possible properties.
- The next theorem allows a representation of the Born rule in terms of continuous local sections of sheaves.
- To conclude the authors may say that the addition of modalities to the discourse about the properties of a quantum system enlarges its expressive power.

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