Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Connecting the categorical and the modal logic approaches to Quantum Mechanics Giovanni Cin` a based on MSc thesis supervised by A. Baltag Institute for Logic, Language and Computation University of Amsterdam 30.11.2013 Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Introduction The development of Quantum Computation and Information has caused a new wave of studies in Quantum Mechanics. In particular, we seek to develop formal models to increase our understanding of quantum processes. Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Introduction The development of Quantum Computation and Information has caused a new wave of studies in Quantum Mechanics. In particular, we seek to develop formal models to increase our understanding of quantum processes. We examine two research programs. They have a common goal: crafting a formalism that captures the features of quantum processes the same intended application: a formal system capable of proving the correctness of quantum algorithms Question: can we connect them? Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Outline Categorical Quantum Mechanics 1 Dynamic Quantum Logic 2 LQP LQP n Modal logics for small categories 3 Examples Modal logics for locally small categories Logics for FdHil 4 Logics for H and S Logics for H and F Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Categorical Quantum Mechanics The first approach, initiated by Abramsky and Coecke, recasts the concepts of Hilbert space Quantum Mechanics in the abstract language of Category Theory. Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Categorical Quantum Mechanics The first approach, initiated by Abramsky and Coecke, recasts the concepts of Hilbert space Quantum Mechanics in the abstract language of Category Theory. Definition A category C is made of objects A , B , C , . . . arrows f : A → B , g : A → C , . . . Arrows are closed under composition (when target and source match) and composition of arrows is associative. Every object has an identity arrow that works as the unit of the composition. Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Definition A functor F : C → D is a pair of maps ( F 1 , F 2 ) such that F 1 maps object of C in objects of D F 2 maps arrows of C in arrows of D and also preserves sources and targets, identities and compositions. Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil Definition A functor F : C → D is a pair of maps ( F 1 , F 2 ) such that F 1 maps object of C in objects of D F 2 maps arrows of C in arrows of D and also preserves sources and targets, identities and compositions. Definition Given two functors F , G : C → D , a natural transformation η : F → G is a family of arrows in D indexed by the objects of C such that, for every arrow f : C → B in C , in D we have F ( C ) F ( B ) F ( f ) η C η B G ( f ) G ( C ) G ( B ) Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil The target of this study is FdHil , the category having as objects finite-dimensional Hilbert spaces over the field of complex numbers and as morphisms linear maps. Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic Modal logics for small categories Logics for FdHil The target of this study is FdHil , the category having as objects finite-dimensional Hilbert spaces over the field of complex numbers and as morphisms linear maps. Theorem (Abramsky and Coecke, [1]) The category FdHil is a dagger compact closed category with biproducts. This in particular means that FdHil is a symmetric monoidal category 1 a compact closed category 2 a dagger category 3 a category with biproducts 4 Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic LQP LQPn Modal logics for small categories Logics for FdHil Dynamic Quantum Logic The second approach, proposed by Baltag and Smets, exploits the formalism of Propositional Dynamic Logic to design a Logic of Quantum Programs , abbreviated in LQP . The core ideas behind this logic are two: we can see the states of a physical system as states of a Modal 1 Logic frame the dynamics of the system can be captured by representing 2 measurements as tests and unitary maps as actions This leads to an abstraction from Hilbert spaces to labelled transition systems. Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic LQP LQPn Modal logics for small categories Logics for FdHil Quantum dynamic frames Definition Given a Hilbert space H , a quantum dynamic frame is a tuple P a ? U � Σ H , { − − →} a ∈ L H , { − →} U ∈U � such that: Σ H is the set of all one-dimensional linear subspaces of H 1 Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic LQP LQPn Modal logics for small categories Logics for FdHil Quantum dynamic frames Definition Given a Hilbert space H , a quantum dynamic frame is a tuple P a ? U � Σ H , { − − →} a ∈ L H , { − →} U ∈U � such that: Σ H is the set of all one-dimensional linear subspaces of H 1 P a ? { − − →} a ∈ L H is a family of quantum tests , partial maps from Σ H into 2 Σ H associated to the projectors of the Hilbert space H . Given P a ? v ∈ Σ H , they are defined as − − → ( v ) = P a ( v ). Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic LQP LQPn Modal logics for small categories Logics for FdHil Quantum dynamic frames Definition Given a Hilbert space H , a quantum dynamic frame is a tuple P a ? U � Σ H , { − − →} a ∈ L H , { − →} U ∈U � such that: Σ H is the set of all one-dimensional linear subspaces of H 1 P a ? { − − →} a ∈ L H is a family of quantum tests , partial maps from Σ H into 2 Σ H associated to the projectors of the Hilbert space H . Given P a ? v ∈ Σ H , they are defined as − − → ( v ) = P a ( v ). U { − →} U ∈U is a collection of partial maps from Σ H into Σ H associated 3 to the unitary maps from H into H . Their definition is U − → ( v ) = U ( v ). Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic LQP LQPn Modal logics for small categories Logics for FdHil LQP Given a set of atomic propositions At and a set of atomic actions AtAct , the set of formulas F LQP and the set of actions Act are built by mutual recursion as follows: ψ ::= p | ¬ ψ | ψ ∧ φ | [ π ] ψ π ::= U | π † | π ∪ π ′ | π ; π ′ | ψ ? where p ∈ At and U ∈ AtAct . Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic LQP LQPn Modal logics for small categories Logics for FdHil LQP n Unfortunately LQP is not enough, we need to express locality . Consider as semantics only the quantum dynamic frames given by n -th tensor products of 2-dimensional Hilbert spaces (systems of n qubits). Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
Categorical Quantum Mechanics Dynamic Quantum Logic LQP LQPn Modal logics for small categories Logics for FdHil LQP n Unfortunately LQP is not enough, we need to express locality . Consider as semantics only the quantum dynamic frames given by n -th tensor products of 2-dimensional Hilbert spaces (systems of n qubits). Enrich the language with ψ ::= ⊤ I | 1 | + | p | ¬ ψ | ψ ∧ φ | [ π ] ψ π ::= triv I | U | π † | π ∪ π ′ | π ; π ′ | ψ ? where I ⊆ { 1 , . . . , n } . Theorem (Baltag and Smets, [2]) There is a proof system for LQP n which is sound and proves the correctness of some quantum protocols. Giovanni Cin` a Connecting the categorical and the modal logic approaches to Quantum Mechanics
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