[Paper Review] Categorical Quantum Dynamics

Stefano Gogioso|arXiv (Cornell University)|Sep 28, 2017

Quantum Mechanics and Applications83 references3 citations

TL;DR

This paper introduces a categorical framework for quantum dynamics using dagger compact categories, unifying quantum processes, states, and effects through graphical calculus. It establishes a correspondence between quantum operations and morphisms in a symmetric monoidal category, with key results including the derivation of Frobenius and speciality laws for quantum observables and the characterization of pure states and effects via duality and adjunction.

ABSTRACT

We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.

Motivation & Objective

• To develop a unified categorical framework for quantum dynamics using symmetric monoidal categories with duality.
• To model quantum processes, including states, effects, and transformations, using graphical calculus and morphisms.
• To establish the role of duality and adjunction in characterizing quantum observables and measurements.
• To derive structural laws—such as Frobenius and speciality—governing quantum systems in the categorical setting.
• To connect the categorical structure to standard quantum mechanics via the Choi-Jamiołkowski isomorphism and self-adjoint idempotents.

Proposed method

• Uses dagger compact categories to model quantum systems, with objects representing Hilbert spaces and morphisms representing quantum operations.
• Applies graphical calculus (string diagrams) to represent composition and tensoring of morphisms, including unit, counit, and duality maps.
• Defines the cup and cap maps (η_A and ε_A) as dualizing isomorphisms, with σ_{A,B} for swapping systems.
• Introduces the adjoint operation f† via duality, relating f: A → B to f*: B* → A*, with f* = f† for self-adjoint maps.
• Derives the Frobenius law and speciality condition from the categorical axioms, linking them to quantum measurement and copying.
• Characterizes pure states and effects via the Choi-Jamiołkowski isomorphism, using ∑|e_j⟩⊗|e_j⟩ and its dual for state preparation and measurement.

Experimental results

Research questions

• RQ1 How can quantum dynamics be axiomatized within a symmetric monoidal category with duality?
• RQ2 What categorical structures underlie the ability to copy and delete quantum information?
• RQ3 How do the Frobenius and speciality laws emerge from categorical duality and adjunction?
• RQ4 What is the role of the adjoint operation f† in relating processes to their duals in the category?
• RQ5 How do pure states and effects correspond to morphisms in the category, particularly via the Choi-Jamiołkowski isomorphism?

Key findings

• The category of completely positive maps (CPM[C]) is shown to be isomorphic to the original category C when C is dagger compact and self-adjoint.
• The Frobenius law holds for all objects A in the category, ensuring consistency in the composition of copying and deletion operations.
• The speciality condition is derived from the duality structure, implying that the trace of the identity map on A is dim(A), linking to quantum dimension.
• The map ∑|e_j⟩⊗|e_j⟩ defines a canonical cup (entangled state) and its dual ∑⟨e_j|⊗⟨e_j| defines the cap, enabling state preparation and measurement.
• The adjoint of a morphism f: A → B is given by f†: B* → A*, with f* = f†, establishing a self-adjoint structure on the category.
• Central self-adjoint idempotents p satisfy p† = p and p∘p = p, corresponding to projections in the category, with p = ∑|e_j⟩⟨e_j| for orthonormal bases.