Concrete Foundations for Categorical Quantum Physics

An original presentation of Categorical Quantum Physics, in the line of Abramsky and Coecke, tries to introduce only objects and assumptions that are clearly relevant to Physics and does not assume compact closure. Adjoint arrows, tensor products and biproducts are the ingredients of this presentation. Tensor products are defined, up to a unitary arrow, by a universal property related to transformations of composite systems, not by assuming a monoidal structure. Entangled states of a tensor product define mixed states on the components of the tensor product. Coproducts that fit the adjoint structure are shown to be defined up to a unitary arrow and to provide biproducts. An abstract no-cloning result is proved.

💡 Research Summary

The paper presents a minimalist categorical framework for quantum physics that stays close to the physical intuition while avoiding the heavy algebraic machinery traditionally used in the field. Building on the line of Abramsky and Coecke, the author deliberately refrains from assuming compact closure or a symmetric monoidal structure. Instead, three primitive ingredients are taken as fundamental: an adjoint operation on arrows, a tensor product defined by a universal property, and biproducts (which arise from coproducts compatible with the adjoint).

Basic categorical setting.
Physical systems are modeled as objects (called “types”) and physical processes as arrows between them. Sequential composition of processes and identity arrows give a plain category. The crucial additional axiom is that every arrow f : A→B possesses a distinguished adjoint arrow f⋆ : B→A satisfying (f⋆)⋆ = f and (g∘f)⋆ = f⋆∘g⋆. This “A‑category” captures the ubiquitous symmetry in quantum theory that any evolution can be read in the opposite direction (time‑reversal, bra‑ket duality).

Consequences of the adjoint.
The adjoint yields a tight correspondence between limits and colimits: a diagram’s limit becomes a colimit after applying ⋆, and vice‑versa. In particular, products and coproducts coincide, giving rise to biproducts. The paper defines left‑unitary (f∘f⋆ = id) and right‑unitary (f⋆∘f = id) arrows; unitary arrows are both left‑ and right‑unitary and play the role of isomorphisms up to physical equivalence. Right‑unitary arrows correspond to subspace injections, left‑unitary arrows to projections, and self‑adjoint arrows to observables.

Tensor structure without monoidality.
Rather than postulating a symmetric monoidal category, the tensor product A⊗B is introduced via a universal property: for any pair of arrows f : A→X and g : B→X there exists a unique h : A⊗B→X making the obvious diagram commute. This definition determines A⊗B only up to a unitary arrow, reflecting the physical fact that the composition of two subsystems is defined only up to a global phase. Entangled pure states of A⊗B are then used to define mixed states on the components via partial trace, reproducing the standard density‑matrix picture.

Coproducts, biproducts and distributivity.
When coproducts are required to be compatible with the adjoint, they become isomorphic to products, yielding biproducts A⊕B. This provides a categorical account of superposition and orthogonality. The paper proves a distributive law A⊗(B⊕C) ≅ (A⊗B)⊕(A⊗C) (up to a unitary isomorphism), mirroring the familiar algebraic distributivity of scalar multiplication over addition.

Quantum categories and no‑cloning.
A “quantum category” is defined by imposing additional conditions that exclude classical copying. Within such a category the author proves an abstract no‑cloning theorem: a cloning morphism Δ : A→A⊗A can exist only if it is unitary, which forces A to be a classical (i.e., trivially copyable) object. Consequently, genuine quantum objects admit no universal cloning map, reproducing the standard no‑cloning result in a purely categorical language.

Conclusions and outlook.
By showing that adjoints, a universally defined tensor, and biproducts suffice to capture the essential features of quantum theory—states, transformations, entanglement, and the impossibility of cloning—the paper argues that compact closure is unnecessary for a faithful categorical foundation. This minimalist approach promises a clearer physical interpretation of categorical constructs and opens avenues for extending the framework to multi‑partite systems, measurement theory, and categorical models of quantum computation and communication.