Dedicated Operational Context in Categorical QM

• Dedicated operational context is a precisely delineated framework that specifies empirical primitives—preparations, transformations, and measurements—with explicit evaluation rules.
• It utilizes categorical quantum mechanics with diagrammatic and monoidal structures to unify and compare different physical theories.
• This approach enables rigorous, theory-independent analysis of quantum, classical, and no-signaling models, guiding both foundational research and practical protocols.

A dedicated operational context refers to a precisely delimited and structured set of empirical and compositional elements within which a physical theory or information-processing framework is implemented, interpreted, or compared. In the context of quantum theory and its categorical generalizations, as detailed in “Operational theories and Categorical quantum mechanics” (Abramsky et al., 2012), this notion captures the exact operational primitives—such as preparations, transformations, and measurements—along with their probabilistic or structural evaluation rules. Through the categorical treatment, one arrives at a unified platform for not only defining and realizing quantum mechanics operationally but also for comparing it against alternative or “foil” theories using invariant structural terms. Dedicated operational contexts thus underwrite rigorous, theory-independent analysis of properties like nonlocality, empirical model classes, and information-processing capabilities.

1. Operational Theories and Categorical Quantum Mechanics

Operational theories specify physical scenarios in terms of the full suite of performable actions and their outcome statistics: preparations (states), transformations (dynamics), and measurements (observables), together with an explicit evaluation rule assigning probabilities to observed outcomes. In categorical quantum mechanics (CQM), these primitives are encoded diagrammatically within a symmetric monoidal (dagger) category sometimes equipped with zero morphisms or trace ideals.

The paper demonstrates a categorical construction wherein:

• Each system type $A$ is associated with sets of preparations ($P_A$) and measurements ($M_A$)—e.g., density operators and projector families in finite-dimensional Hilbert spaces (FHilb).
• The evaluation rule is given by a dinatural transformation $d$ that relates preparations and measurements. In standard cases, the evaluation takes the form:

$d_A(s, m)(o) = \mathrm{Tr}_A(s \circ P_o)$

where $s$ is a state, $m$ a measurement (as a collection of projectors $\{P_o\}$), and $\mathrm{Tr}_A$ the trace in the symmetric monoidal category.

The operational representation preserves the monoidal structure: for composite systems,

$$d_{A \otimes B}(p \otimes p', m \otimes m') = d_A(p, m)\; d_B(p', m')$$

guaranteeing probabilistic independence for independently prepared subsystems.

2. Nonlocality: Formulation and Structural Examples

Nonlocality is abstracted as a property of empirical models arising in an operational category. For a composite system $A_1 \otimes \cdots \otimes A_n$,

• Fix a global state $s$ and tuples of local measurements $m_i \in M_{A_i}$.
• The empirical model is a conditional probability distribution:

$p(\mathbf{o}\mid \mathbf{m}) = d_A(s, m)(o)$

where $\mathbf{o} = (o_1, \ldots, o_n)$, $\mathbf{m} = (m_1, \ldots, m_n)$.

A local model is realized if there exists a hidden variable space $\Lambda$ with distribution $d$, such that

$(o \mid m)_\lambda = \prod_{i=1}^n (o_i | m_i)$

and

$$p(o|m) = \sum_{\lambda \in \Lambda} (o|m)_\lambda \cdot d(\lambda).$$

Failure of such a decomposition characterizes nonlocality in the operational category.

Examples:

• In FHilb, with standard quantum states and projective measurements, operational models reproduce the experimental statistics of Bell-type scenarios, demonstrating nonlocality.
• In $\mathrm{Rel}(\Omega)$ (relations valued in a locale), all mixed states are operationally pure, and explicit hidden-variable models can be constructed—precluding nonlocality.
• In Stoch (classical probability), operational models correspond to classical probability tables, again precluding nonlocality.
• SStoch (signed stochastic maps) allows negative “probabilities” and recovers the full class of no-signalling empirical models, including those with super-quantum correlations (e.g., PR box).

3. Model Taxonomy and Operational Characterization

The operational framework distinguishes three classes of models via their empirical structure:

• Local Models: Those admitting hidden-variable decompositions; i.e., empirical tables that factorize as independent marginals and are recovered by averaging over hidden variables.
• Quantum Models: Those realized in FHilb or C*-category instantiations; states as density operators, measurements as projectors, and evaluation as the canonical

$d_A(s, m)(o) = \mathrm{Tr}(s \circ P_o).$

These exhibit quantum nonlocality (Bell inequality violation).

• No-Signalling Models: In SStoch, the set of achievable empirical statistics coincides with the set of no-signalling models, exceeding the quantum set.

Critical structural formulas include:

• Independence of Simple Tensors:

$$d_{A \otimes B}(p \otimes p', m \otimes m') = d_A(p, m) \cdot d_B(p', m')$$

• Chu Morphism Condition:

For a process $f : A \to B$,

$d_B(f_+(p), m) = d_A(p, f^*(m))$

connecting the Schrödinger and Heisenberg pictures categorically.

4. Implications for Dedicated Operational Contexts

Adopting a categorical operational framework enables:

• Transparent comparison of physical theories (quantum, classical, no-signalling, and super-quantum) in structurally unified terms, making explicit what features differentiate quantum mechanics.
• Identification and axiomatic analysis of critical properties (e.g., nonlocality, contextuality, no-broadcasting, teleportation) that demarcate operational boundaries between theories.
• Systematic construction of alternative (foil) theories via modification of the underlying categorical scaffolding (e.g., switching from Hilb to Rel or SStoch), illuminating the operational landscape.

The operational context so defined is “dedicated” in the sense that it fixes not only the set of possible operations and their composition rules, but also the precise evaluation rule—yielding an explicit delineation of empirical predictions and the boundaries of the physical theory.

A key consequence is that quantum mechanics can be positioned as a sharply defined region in this operational “space of theories,” inviting generalizations, reconstructions, and detailed exploration of physical principles at play.

5. Abstract Representation and Theoretical Extensions

The use of symmetric monoidal dagger categories, trace ideals, and dinatural transformations in representing operational content:

• Ensures that composition, duality, and evaluation are fully abstract and diagrammatic.
• Provides a setting for general theorems—both for quantum information phenomena and for alternative models that could inform foundational questions or suggest new protocols.
• Facilitates the exploration of modifications (e.g., idempotent scalars in Rel, negative probabilities in SStoch) that alter the operational properties, e.g., generating no-signalling correlations unattainable in quantum theory.

This categorical abstraction is not simply formal: it is the tool by which the “dedicated operational context” of a theory is identified, parameterized, and investigated.

6. Comparative Table of Model Types

Category	States/Measurements	Nonlocality?	Distinguishing Feature
FHilb (Quantum)	Density ops/Projectors	Yes	Violates Bell inequalities
Rel	Subsets/Δ_S projectors	No	All mixed states are pure; local HV
Stoch	Classical probabilities	No	Classical marginals/hv decomposition
SStoch	Signed maps	No-signalling only	Superquantum (PR box) correlations

7. Summary and Outlook

The rigorous connection established between operational theories and categorical quantum mechanics lays the groundwork for a precise, structural notion of dedicated operational context. This context encompasses all preparable states, measurable observables, allowable transformations, and their empirical relationships, encoded in categorical data preserved under composition and evaluation. By framing operational contexts in this way, the framework supports not only the robust analysis of quantum theory but also the systematic investigation of alternative possible theories and their experimental predictions, aiding both foundational understanding and the development of new information protocols.