Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective ... more Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective Learning, Feedback
Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lect... more Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lecturer Interaction
The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pi... more The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pimsner and S.Popa for conditional expectations. As an application, we describe a new strategy for com- puting the Jones index of the range of certain endomorphisms. We use extended transition operators to associate to an endomorphism a unital completely positive map acting on a
Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Appro... more Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Approach 1.1.3 Computations 1.1.4 Parametrization of the Set of Extensions 1.2 An Extension Problem 1.2.1 The Set Z(S,f</font >B){\cal Z}(S,{\phi_{{\cal B}}}) of Extensions 1.2.2 Z{\cal Z} as a Convex Set 1.2.3 Discussion 1.3 Weak Tensor Dilations 1.3.1 The Definition 1.3.2 Representations 1.3.3 Construction of Examples 1.3.4 The Associated Isometry 1.3.5 The Minimal Version of an Associated Isometry 1.3.6 Minimal Part of the Stinespring Representation 1.4 Equivalence of Weak Tensor Dilations 1.4.1 An Equivalence Relation 1.4.2 Equivalence and Unitary Equivalence 1.5 Duality 1.5.1 Dual Stochastic Maps 1.5.2 From Dilation to Extension 1.5.3 From Extension to Dilation 1.5.4 One-to-One Correspondence 1.5.5 Discussion 1.6 The Automorphic Case 1.6.1 Conditional Expectations 1.6.2 Adjoints 1.6.3 Automorphic Tensor Dilations 1.6.4 Duality for Automorphic Tensor Dilations 1.7 Examples 1.7.1 Example 1.1 Revisited 1.7.2 Further Discussion 1.7.3 A Class of Maps on M 2 1.7.4 Maps on M n
4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forwa... more 4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forward Transition 4.1.3 Classification of Stationary Extensions 4.1.4 Transition Operators 4.1.5 Adapted Endomorphisms 4.1.6 Counterexample 4.1.7 Stationary Extensions and Extending Factors 4.1.8 Elementary Tensor Representations 4.1.9 Construction of Extending Factors by Partitions 4.1.10 Discussion 4.1.11 Two-Valued Processes 4.1.12 Markov Processes 4.1.13 Markovian Extensions and Entropy 4.2 Prediction Errors for Commutative Processes 4.2.1 The Problem 4.2.2 Prediction for Finite-Valued Processes 4.2.3 A Guessing Game 4.2.4 A Combinatorial Formula for Prediction Errors 4.2.5 Asymptotics 4.2.6 Example 4.3 Low-Dimensional Examples 4.3.1 Qubits 4.3.2 Associated Stochastic Maps 4.3.3 Determinism and Unitarity 4.3.4 Complete Invariants 4.3.5 Probabilistic Interpretations 4.4 Clifford Algebras and Generalizations 4.4.1 Clifford Algebras 4.4.2 The Clifford Functor 4.4.3 Generalized Clifford Algebras 4.5 Tensor Products of Matrices 4.5.1 Matrix Filtrations 4.5.2 LPR Is Automatic 4.5.3 Associated Stochastic Maps 4.5.4 Non-surjectivity 4.5.5 Adapted Endomorphisms in the Literature 4.6 Noncommutative Extension of Adaptedness 4.6.1 Extending Adaptedness 4.6.2 Three Points of Time 4.6.3 Criteria for the Extendability of Adaptedness 4.6.4 An Example: The Fredkin Gate 4.6.5 Discussion
Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective ... more Key Words: Science, IMAPS, UG Year 3, Lecture Based Learning, Problem-Based Learning, Reflective Learning, Feedback
Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lect... more Key Words: Science, IMAPS, UG Year 3, Problem-Based Learning, Small Group, Feedback, Student-Lecturer Interaction
The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pi... more The probabilistic index of a completely positive map is defined in analogy with a formula of M.Pimsner and S.Popa for conditional expectations. As an application, we describe a new strategy for com- puting the Jones index of the range of certain endomorphisms. We use extended transition operators to associate to an endomorphism a unital completely positive map acting on a
Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Appro... more Contents. 1.1 An Example with 2 ×22 \times 2 - Matrices 1.1.1 A Stochastic Map 1.1.2 Direct Approach 1.1.3 Computations 1.1.4 Parametrization of the Set of Extensions 1.2 An Extension Problem 1.2.1 The Set Z(S,f</font >B){\cal Z}(S,{\phi_{{\cal B}}}) of Extensions 1.2.2 Z{\cal Z} as a Convex Set 1.2.3 Discussion 1.3 Weak Tensor Dilations 1.3.1 The Definition 1.3.2 Representations 1.3.3 Construction of Examples 1.3.4 The Associated Isometry 1.3.5 The Minimal Version of an Associated Isometry 1.3.6 Minimal Part of the Stinespring Representation 1.4 Equivalence of Weak Tensor Dilations 1.4.1 An Equivalence Relation 1.4.2 Equivalence and Unitary Equivalence 1.5 Duality 1.5.1 Dual Stochastic Maps 1.5.2 From Dilation to Extension 1.5.3 From Extension to Dilation 1.5.4 One-to-One Correspondence 1.5.5 Discussion 1.6 The Automorphic Case 1.6.1 Conditional Expectations 1.6.2 Adjoints 1.6.3 Automorphic Tensor Dilations 1.6.4 Duality for Automorphic Tensor Dilations 1.7 Examples 1.7.1 Example 1.1 Revisited 1.7.2 Further Discussion 1.7.3 A Class of Maps on M 2 1.7.4 Maps on M n
4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forwa... more 4.1 Commutative Stationarity 4.1.1 Stationarity in Classical Probability 4.1.2 Backward and Forward Transition 4.1.3 Classification of Stationary Extensions 4.1.4 Transition Operators 4.1.5 Adapted Endomorphisms 4.1.6 Counterexample 4.1.7 Stationary Extensions and Extending Factors 4.1.8 Elementary Tensor Representations 4.1.9 Construction of Extending Factors by Partitions 4.1.10 Discussion 4.1.11 Two-Valued Processes 4.1.12 Markov Processes 4.1.13 Markovian Extensions and Entropy 4.2 Prediction Errors for Commutative Processes 4.2.1 The Problem 4.2.2 Prediction for Finite-Valued Processes 4.2.3 A Guessing Game 4.2.4 A Combinatorial Formula for Prediction Errors 4.2.5 Asymptotics 4.2.6 Example 4.3 Low-Dimensional Examples 4.3.1 Qubits 4.3.2 Associated Stochastic Maps 4.3.3 Determinism and Unitarity 4.3.4 Complete Invariants 4.3.5 Probabilistic Interpretations 4.4 Clifford Algebras and Generalizations 4.4.1 Clifford Algebras 4.4.2 The Clifford Functor 4.4.3 Generalized Clifford Algebras 4.5 Tensor Products of Matrices 4.5.1 Matrix Filtrations 4.5.2 LPR Is Automatic 4.5.3 Associated Stochastic Maps 4.5.4 Non-surjectivity 4.5.5 Adapted Endomorphisms in the Literature 4.6 Noncommutative Extension of Adaptedness 4.6.1 Extending Adaptedness 4.6.2 Three Points of Time 4.6.3 Criteria for the Extendability of Adaptedness 4.6.4 An Example: The Fredkin Gate 4.6.5 Discussion
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