Operational Quantum Formalism

• Operational Quantum Formalism is a framework that defines quantum theory in terms of experimental primitives such as preparations, measurements, and transformations while bypassing conventional probabilistic rules.
• It establishes a rigorous state-space structure using axioms like inf-semilattice organization, atomicity, and continuity to characterize pure and mixed quantum states.
• The approach reconstructs quantum logic by recovering the Hilbert lattice and orthomodular structure through Chu semantics and symmetry-preserving mappings.

Operational Quantum Formalism

Operational quantum formalism refers to a family of frameworks that reconstruct or represent the structure of quantum theory directly in terms of operationally defined primitives—preparations, measurements, and transformations—eschewing probabilities (in certain approaches) or supplementing them with minimal algebraic or order-theoretic structure. These frameworks specify how empirical processes and outcomes relate and investigate how core features of quantum theory, such as superposition, orthogonality, pure and mixed states, and symmetry transformations, emerge from operational constraints. Among the distinctive research achievements is a full reconstruction of the Hilbert lattice structure and its symmetries from minimal, non-probabilistic semantics, as presented in the possibilistic formalism (Buffenoir, 2020).

1. Three-Valued Chu Semantics: Foundations

The operational approach formalizes the experimental setup as a Chu space $(P, T, e)$, where $P$ is the set of preparation processes, $T$ the set of yes/no measurement tests, and

$e : P \times T \longrightarrow B,$

with $B = \{\bot, Y, N\}$ ordered by $\bot \le Y$, $\bot \le N$, $Y$ and $N$ incomparable. Here:

• $e(p, t) = Y$ means $P$0 always yields "yes" for $P$1.
• $P$2 means "no" holds with certainty.
• $P$3 means the outcome is indeterminate.

This tripartite valuation encodes certainty, falsity, and indeterminacy in an explicitly informationally interpreted setting.

2. Axiomatic Structure of the State Space

Preparations are equivalent if they generate identical statistics over all tests, giving rise to a quotient set $P$4 of operational mixed states. The state space $P$5 is organized as a complete projective domain under a series of axioms:

• Inf-semilattice structure: Arbitrary mixtures (meets) exist: $P$6.
• Existence of bottom element: There is a minimal state $P$7 such that $P$8.
• Directed completeness/continuity: For $P$9, $T$0-chains have suprema; the evaluation maps $T$1 are Scott-continuous.
• Atomicity and strong discreteness: If $T$2, there exists an atom $T$3 such that $T$4 with no intermediate states.
• Relative complements/modularity: For $T$5, uniquely determined "differences" and semi-modular conditions exist.
• No Type-II atoms: All completely meet-irreducible elements are atoms (i.e., maximal).

Under these axioms, $T$6 is a bounded, directed-complete, atomistic, relatively complemented, modular, meet-continuous projective domain (Buffenoir, 2020).

3. Pure States, Properties, and Measurements

A state is pure iff it is completely meet-irreducible (i.e., no nontrivial decomposition); these form the minimal generating set for $T$7. Every mixed state is a meet (mixture) of pure states dominating it. Properties (tests modulo operational equivalence) generate "actuality" regions $T$8, with $T$9 the associated "potentiality" region.

The formalism defines measurement update via a partial map $e : P \times T \longrightarrow B,$0:

• Monotone: $e : P \times T \longrightarrow B,$1 implies $e : P \times T \longrightarrow B,$2.
• Scott-continuous.
• Special classes include first-kind tests (non-disturbing if repeated), ideal tests (compatibility-preserving), and minimally disturbing tests (minimize disturbance over their domain), which are characterized by local Scott-ideal structure.

4. Orthogonality, Discriminating Tests, and Ortho-Complementation

To embed analogs of quantum orthogonality, the formalism postulates the existence of a closed, complete, irredundant scheme $e : P \times T \longrightarrow B,$3 of discriminating tests $e : P \times T \longrightarrow B,$4 indexed by pairs $e : P \times T \longrightarrow B,$5 such that:

• There is a unique involutive order-reversing map $e : P \times T \longrightarrow B,$6 with $e : P \times T \longrightarrow B,$7.
• Orthogonals are defined: $e : P \times T \longrightarrow B,$8, and De Morgan laws hold.

Orthogonality is then: $e : P \times T \longrightarrow B,$9 This relation is symmetric, irreflexive, and order-reversing. Closure under biorthogonality and compatibility with the projective domain structure are ensured, enforcing a fully ortho-complemented structure (Buffenoir, 2020).

5. Hilbert Lattice: Pure States and the Quantum Logic Structure

In restriction to pure states $B = \{\bot, Y, N\}$0, the ortho-poset of ortho-closed sets

$B = \{\bot, Y, N\}$1

with join $B = \{\bot, Y, N\}$2, meet $B = \{\bot, Y, N\}$3, and orthocomplement $B = \{\bot, Y, N\}$4, forms a complete atomic orthomodular lattice satisfying the covering law, i.e., a Hilbert lattice in the sense of Piron (Buffenoir, 2020). This is the canonical quantum logic structure, recovering the logical basis of projective Hilbert space.

6. Symmetries: Chu Morphisms and Ortho-Morphisms

The symmetry group in this formalism comprises Chu morphisms $B = \{\bot, Y, N\}$5 where:

• $B = \{\bot, Y, N\}$6 injective and Scott-structure preserving.
• $B = \{\bot, Y, N\}$7 surjective and compatible with $B = \{\bot, Y, N\}$8 (e.g., $B = \{\bot, Y, N\}$9).
• $\bot \le Y$0 preserves measurement structures; $\bot \le Y$1 preserves orthogonality and carries minimally disturbing tests to minimally disturbing tests.

At the Hilbert lattice level, $\bot \le Y$2 induces an ortho-morphism $\bot \le Y$3 preserving suprema and atoms, and commuting with orthocomplementation. Thus, this reconstructs the class of lattice automorphisms analogous to unitary and antiunitary symmetries in conventional quantum theory (Buffenoir, 2020).

7. Comparison, Generality, and Significance

The possibilistic operational quantum formalism demonstrates that the essential features of quantum theory—projective structure, orthomodularity, Hilbert lattice, and the algebraic structure of symmetries—arise without recourse to probability, amplitudes, or the Born rule. It provides a model-independent, order-theoretic foundation from which quantum logic and the main structural features of Hilbert space emerge. The framework admits further generalization to probabilistic cases, categorical and compositional semantics, and can be linked to powerful duality principles such as Chu duality, clarifying the deep operational roots of quantum kinematics (Buffenoir, 2020).