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projects:quantum:category-qc [2025/04/10 19:44] – [Early Algebraic Approaches to Quantum Mechanics] kymkiprojects:quantum:category-qc [2025/04/16 15:41] (current) – kymki
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-====== Categorical Quantum Computing: A Historical and Theoretical Overview ======+<html> 
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 +====== Categorical Quantum Computing Series: A Historical and Theoretical Overview ======
  
 ===== Early Algebraic Approaches to Quantum Mechanics ===== ===== Early Algebraic Approaches to Quantum Mechanics =====
-BLA[(Knuth68)] 
  
 Quantum theory was originally formulated in algebraic terms using Hilbert spaces and linear operators. In the **Dirac–von Neumann** framework, a quantum system’s state is a vector (or ray) in a complex Hilbert space, observable quantities correspond to self-adjoint operators, and time evolution is given by unitary transformations on the state space ([[https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#:~:text=transformations,states%2C%20and%20physical%20symmetries%20are|Mathematical formulation of quantum mechanics - Wikipedia]]). The combination of two independent systems is described by the tensor product of their Hilbert spaces ([[https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#:~:text=match%20at%20L572%20,systems%20are%20the%20individual%20particles|Mathematical formulation of quantum mechanics - Wikipedia]]). This operator-theoretic formalism, developed in the 1930s, provided a rigorous mathematical foundation for quantum mechanics and remains standard. However, it inherently encodes composition (e.g. forming joint systems via tensor products) within the Hilbert space formalism itself, lacking a high-level language for **composing processes**. Early attempts to capture the logical structure of quantum theory in abstract terms include **quantum logic** – notably the work of Birkhoff and von Neumann (1936), who identified the lattice of projective subspaces of a Hilbert space with a logic of quantum propositions ([[https://en.wikipedia.org/wiki/Quantum_logic#:~:text=In%20his%20classic%201932%20treatise,3|Quantum logic - Wikipedia]]). Subsequent axiomatizations, such as Mackey’s 1963 postulates, sought to derive quantum mechanics from order-theoretic or probabilistic axioms ([[https://en.wikipedia.org/wiki/Quantum_logic#:~:text=logic%20by%20von%C2%A0Neumann%20and%20Birkhoff,3|Quantum logic - Wikipedia]]). These approaches were algebraic and logical, but not yet **categorical**. They treated quantum states and observables as static mathematical objects rather than **processes**, and composition was implicit. By the late 20th century, category theory had matured as a framework for composition in mathematics, setting the stage for a new semantic approach. Quantum theory was originally formulated in algebraic terms using Hilbert spaces and linear operators. In the **Dirac–von Neumann** framework, a quantum system’s state is a vector (or ray) in a complex Hilbert space, observable quantities correspond to self-adjoint operators, and time evolution is given by unitary transformations on the state space ([[https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#:~:text=transformations,states%2C%20and%20physical%20symmetries%20are|Mathematical formulation of quantum mechanics - Wikipedia]]). The combination of two independent systems is described by the tensor product of their Hilbert spaces ([[https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#:~:text=match%20at%20L572%20,systems%20are%20the%20individual%20particles|Mathematical formulation of quantum mechanics - Wikipedia]]). This operator-theoretic formalism, developed in the 1930s, provided a rigorous mathematical foundation for quantum mechanics and remains standard. However, it inherently encodes composition (e.g. forming joint systems via tensor products) within the Hilbert space formalism itself, lacking a high-level language for **composing processes**. Early attempts to capture the logical structure of quantum theory in abstract terms include **quantum logic** – notably the work of Birkhoff and von Neumann (1936), who identified the lattice of projective subspaces of a Hilbert space with a logic of quantum propositions ([[https://en.wikipedia.org/wiki/Quantum_logic#:~:text=In%20his%20classic%201932%20treatise,3|Quantum logic - Wikipedia]]). Subsequent axiomatizations, such as Mackey’s 1963 postulates, sought to derive quantum mechanics from order-theoretic or probabilistic axioms ([[https://en.wikipedia.org/wiki/Quantum_logic#:~:text=logic%20by%20von%C2%A0Neumann%20and%20Birkhoff,3|Quantum logic - Wikipedia]]). These approaches were algebraic and logical, but not yet **categorical**. They treated quantum states and observables as static mathematical objects rather than **processes**, and composition was implicit. By the late 20th century, category theory had matured as a framework for composition in mathematics, setting the stage for a new semantic approach.