Lattice of projection operators acting on a Hilbert space L v N is an orthomodular lattice. L v N is not distributive: P ∧ ( Q ∨ ¬ Q ) = ( P ∧ Q ) ∨ ( P ∧ ¬ Q ) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Examples: Q-bit Qbit Notice that when H is finite dimensional, its maximal Boolean subalgebras will be finite. ⇒ { 0 , P , ¬ P ⊥ , 1 C 2 } with P = | ϕ �� ϕ | for some unit norm vector P ( C 2 ) = | ϕ � and P ⊥ = | ϕ ⊥ �� ϕ ⊥ | . Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Examples: Q-bit Qbit Notice that when H is finite dimensional, its maximal Boolean subalgebras will be finite. ⇒ { 0 , P , ¬ P ⊥ , 1 C 2 } with P = | ϕ �� ϕ | for some unit norm vector P ( C 2 ) = | ϕ � and P ⊥ = | ϕ ⊥ �� ϕ ⊥ | . Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Skeleton of a qbit P ( C 2 ) 1 ¬ q ¬ p . . . q p . . . . . . 0 Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Examples: Q-trit Qtrit-contextuality P ( C 3 ) = ⇒ P ( { a , b , c } ) = {∅ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c }} Given | ϕ 1 � , | ϕ 2 � and | ϕ 3 � = ⇒ { 0 , P 1 , P 2 , P 3 , P 12 , P 13 , P 23 , 1 C 3 } P i = | ϕ i �� ϕ i | ( i = 1 , 2 , 3) and P ij := | ϕ i �� ϕ i | + | ϕ j �� ϕ j | ( i , j = 1 , 2 , 3). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Examples: Q-trit Qtrit-contextuality P ( C 3 ) = ⇒ P ( { a , b , c } ) = {∅ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c }} Given | ϕ 1 � , | ϕ 2 � and | ϕ 3 � = ⇒ { 0 , P 1 , P 2 , P 3 , P 12 , P 13 , P 23 , 1 C 3 } P i = | ϕ i �� ϕ i | ( i = 1 , 2 , 3) and P ij := | ϕ i �� ϕ i | + | ϕ j �� ϕ j | ( i , j = 1 , 2 , 3). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Qtrit Boolean subalgebras: Figure: Maximal Boolean subalgebras of C 3 { 1 , 2 , 3 } 1 C 3 { 2 , 3 } { 1 , 3 } { 1 , 2 } P 23 P 13 P 12 { 1 } { 2 } { 3 } P 1 P 2 P 3 ∅ ∅ B 3 B 3 Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Figure: Skeleton of C 3 1 C 3 · · · · · · · · · · · · P 23 P 13 P 12 P 1 P 2 P 3 ∅ P ( C 3 ) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Outline Why generalized theories? 1 Mathematical framework and the problem of interpretation 2 Informational aspects 3 Conclusions 4 Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Observations Observations We have an empirically successful example of a theory (QM) whose structure of events is non-Boolean. We know examples of physically meaningful probabilistic theories which are not quantum, nor classical either. In physics: why expecting that standard QM is the end of the story? People applies the QM formalism outside of the QM domain... why thinking that QM is the best option? Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Observations Observations We have an empirically successful example of a theory (QM) whose structure of events is non-Boolean. We know examples of physically meaningful probabilistic theories which are not quantum, nor classical either. In physics: why expecting that standard QM is the end of the story? People applies the QM formalism outside of the QM domain... why thinking that QM is the best option? Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Observations Observations We have an empirically successful example of a theory (QM) whose structure of events is non-Boolean. We know examples of physically meaningful probabilistic theories which are not quantum, nor classical either. In physics: why expecting that standard QM is the end of the story? People applies the QM formalism outside of the QM domain... why thinking that QM is the best option? Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Observations Observations We have an empirically successful example of a theory (QM) whose structure of events is non-Boolean. We know examples of physically meaningful probabilistic theories which are not quantum, nor classical either. In physics: why expecting that standard QM is the end of the story? People applies the QM formalism outside of the QM domain... why thinking that QM is the best option? Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models The idea of comparing QM with other theories dates back to von Neumann. Birkhoff and von Neumann compared standard QM with classical probability theory and searched for possible replacements of the Hilbert space formalism. This path was followed by many others afterwards: Ludwig, Mackey, Piron, Mielnik, etc. By appealing to lattice theory, B and VN developed the axiomatic framework of a generalization of the projective geometry associated to the Hilbert space. The generalization included the notion of continuous geometries . A somewhat curious historical remark: the axiomatization of quantum probability (i.e, the first non-Kolmogorovian probabilistic calculus) dates back to the late 20’s and it reaches its full form in the 1932 von Neumann’s masterpiece. It almost simultaneous to the one works Kolmogorov (1933) for the axiomatization of classical probability based on measure theory. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models The idea of comparing QM with other theories dates back to von Neumann. Birkhoff and von Neumann compared standard QM with classical probability theory and searched for possible replacements of the Hilbert space formalism. This path was followed by many others afterwards: Ludwig, Mackey, Piron, Mielnik, etc. By appealing to lattice theory, B and VN developed the axiomatic framework of a generalization of the projective geometry associated to the Hilbert space. The generalization included the notion of continuous geometries . A somewhat curious historical remark: the axiomatization of quantum probability (i.e, the first non-Kolmogorovian probabilistic calculus) dates back to the late 20’s and it reaches its full form in the 1932 von Neumann’s masterpiece. It almost simultaneous to the one works Kolmogorov (1933) for the axiomatization of classical probability based on measure theory. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models The idea of comparing QM with other theories dates back to von Neumann. Birkhoff and von Neumann compared standard QM with classical probability theory and searched for possible replacements of the Hilbert space formalism. This path was followed by many others afterwards: Ludwig, Mackey, Piron, Mielnik, etc. By appealing to lattice theory, B and VN developed the axiomatic framework of a generalization of the projective geometry associated to the Hilbert space. The generalization included the notion of continuous geometries . A somewhat curious historical remark: the axiomatization of quantum probability (i.e, the first non-Kolmogorovian probabilistic calculus) dates back to the late 20’s and it reaches its full form in the 1932 von Neumann’s masterpiece. It almost simultaneous to the one works Kolmogorov (1933) for the axiomatization of classical probability based on measure theory. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models The idea of comparing QM with other theories dates back to von Neumann. Birkhoff and von Neumann compared standard QM with classical probability theory and searched for possible replacements of the Hilbert space formalism. This path was followed by many others afterwards: Ludwig, Mackey, Piron, Mielnik, etc. By appealing to lattice theory, B and VN developed the axiomatic framework of a generalization of the projective geometry associated to the Hilbert space. The generalization included the notion of continuous geometries . A somewhat curious historical remark: the axiomatization of quantum probability (i.e, the first non-Kolmogorovian probabilistic calculus) dates back to the late 20’s and it reaches its full form in the 1932 von Neumann’s masterpiece. It almost simultaneous to the one works Kolmogorov (1933) for the axiomatization of classical probability based on measure theory. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models The idea of comparing QM with other theories dates back to von Neumann. Birkhoff and von Neumann compared standard QM with classical probability theory and searched for possible replacements of the Hilbert space formalism. This path was followed by many others afterwards: Ludwig, Mackey, Piron, Mielnik, etc. By appealing to lattice theory, B and VN developed the axiomatic framework of a generalization of the projective geometry associated to the Hilbert space. The generalization included the notion of continuous geometries . A somewhat curious historical remark: the axiomatization of quantum probability (i.e, the first non-Kolmogorovian probabilistic calculus) dates back to the late 20’s and it reaches its full form in the 1932 von Neumann’s masterpiece. It almost simultaneous to the one works Kolmogorov (1933) for the axiomatization of classical probability based on measure theory. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
In a sense, von Neumann was looking for a connection between logic, geometry and probability theory: “In order to have probability all you need is a concept of all angles, I mean, other than 90. Now it is perfectly quite true that in geometry, as soon as you can define the right angle, you can define all angles. Another way to put it is that if you take the case of an orthogonal space, those mappings of this space on itself, which leave orthogo- nality intact, leaves all angles intact, in other words, in those systems which can be used as models of the logical background for quantum theory, it is true that as soon as all the ordinary concepts of logic are fixed under some isomorphic transformation, all of probability theory is already fixed... This means however, that one has a formal mechanism in which, logics and probability theory arise simultane- ously and are derived simultaneously.” Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models In a series of papers Murray and von Neumann searched for algebras more general than B ( H ) . The new algebras are known today as von Neumann algebras, and their elementary components can be classified as Type I, Type II and Type III factors. It can be shown that, the projective elements of a factor form an orthomodular lattice. Classical models can be described as commutative algebras. The models of standard quantum mechanics can be described by using Type I factors (Type I n for finite dimensional Hilbert spaces and Type I ∞ for infinite dimensional models). These are algebras isomorphic to the set of bounded operators on a Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models In a series of papers Murray and von Neumann searched for algebras more general than B ( H ) . The new algebras are known today as von Neumann algebras, and their elementary components can be classified as Type I, Type II and Type III factors. It can be shown that, the projective elements of a factor form an orthomodular lattice. Classical models can be described as commutative algebras. The models of standard quantum mechanics can be described by using Type I factors (Type I n for finite dimensional Hilbert spaces and Type I ∞ for infinite dimensional models). These are algebras isomorphic to the set of bounded operators on a Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models In a series of papers Murray and von Neumann searched for algebras more general than B ( H ) . The new algebras are known today as von Neumann algebras, and their elementary components can be classified as Type I, Type II and Type III factors. It can be shown that, the projective elements of a factor form an orthomodular lattice. Classical models can be described as commutative algebras. The models of standard quantum mechanics can be described by using Type I factors (Type I n for finite dimensional Hilbert spaces and Type I ∞ for infinite dimensional models). These are algebras isomorphic to the set of bounded operators on a Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models In a series of papers Murray and von Neumann searched for algebras more general than B ( H ) . The new algebras are known today as von Neumann algebras, and their elementary components can be classified as Type I, Type II and Type III factors. It can be shown that, the projective elements of a factor form an orthomodular lattice. Classical models can be described as commutative algebras. The models of standard quantum mechanics can be described by using Type I factors (Type I n for finite dimensional Hilbert spaces and Type I ∞ for infinite dimensional models). These are algebras isomorphic to the set of bounded operators on a Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models Further work revealed that a rigorous approach to the study of quantum systems with infinitely many degrees of freedom needed the use of more general von Neumann algebras. This is the case in the axiomatic formulation of relativistic quantum mechanics. A similar situation holds in algebraic quantum statistical mechanics. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Probabilistic Models Further work revealed that a rigorous approach to the study of quantum systems with infinitely many degrees of freedom needed the use of more general von Neumann algebras. This is the case in the axiomatic formulation of relativistic quantum mechanics. A similar situation holds in algebraic quantum statistical mechanics. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Facts Fact 1: states spaces of von Neumann algebras are convex . Fact 2: states in VN algebras define measures over orthomodular lattices. Fact 3: they give place to non-equivalent probabilistic models. Fact 4: The Kochen-Specker (KS) theorem is valid for all factor von Neumann algebras (Contextuality is quite ubiquitous among a big family of theories). [A. D¨ oring, International Journal of Theoretical Physics , Vol. 44 , No. 2 , (2005)]. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Facts Fact 1: states spaces of von Neumann algebras are convex . Fact 2: states in VN algebras define measures over orthomodular lattices. Fact 3: they give place to non-equivalent probabilistic models. Fact 4: The Kochen-Specker (KS) theorem is valid for all factor von Neumann algebras (Contextuality is quite ubiquitous among a big family of theories). [A. D¨ oring, International Journal of Theoretical Physics , Vol. 44 , No. 2 , (2005)]. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Facts Fact 1: states spaces of von Neumann algebras are convex . Fact 2: states in VN algebras define measures over orthomodular lattices. Fact 3: they give place to non-equivalent probabilistic models. Fact 4: The Kochen-Specker (KS) theorem is valid for all factor von Neumann algebras (Contextuality is quite ubiquitous among a big family of theories). [A. D¨ oring, International Journal of Theoretical Physics , Vol. 44 , No. 2 , (2005)]. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Facts Fact 1: states spaces of von Neumann algebras are convex . Fact 2: states in VN algebras define measures over orthomodular lattices. Fact 3: they give place to non-equivalent probabilistic models. Fact 4: The Kochen-Specker (KS) theorem is valid for all factor von Neumann algebras (Contextuality is quite ubiquitous among a big family of theories). [A. D¨ oring, International Journal of Theoretical Physics , Vol. 44 , No. 2 , (2005)]. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Generalized probabilities Let L be an orthomodular lattice. Then, define s : L → [ 0 ; 1 ] , ( L standing for the lattice of all events) such that: s ( 0 ) = 0 . (4) s ( E ⊥ ) = 1 − s ( E ) , and, for a denumerable and pairwise orthogonal family of events E j � � s ( E j ) = s ( E j ) . j j where L is a general orthomodular lattice (with L = Σ and L = P ( H ) for the Kolmogorovian and quantum cases respectively). All measures satisfying the above axioms for a given L form a convex set. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Maximal Boolean subalgebras Maximal Boolean subalgebras An orthomodular lattice L can be described as a collection of Boolean algebras: � L = B B∈ B (where B is the set of maximal Boolean algebras of L ). Each maximal Boolean subalgebra defines a context . A state s of L defines a classical probability on each classical Boolean subalgebra B . In other words: s B ( . . . ) := s | B ( ... ) is a Kolmogorovian measure over B . A state in a contextual theory can be considered as a coherent pasting of classical probability distributions, transforming in a continuous way. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Maximal Boolean subalgebras Maximal Boolean subalgebras An orthomodular lattice L can be described as a collection of Boolean algebras: � L = B B∈ B (where B is the set of maximal Boolean algebras of L ). Each maximal Boolean subalgebra defines a context . A state s of L defines a classical probability on each classical Boolean subalgebra B . In other words: s B ( . . . ) := s | B ( ... ) is a Kolmogorovian measure over B . A state in a contextual theory can be considered as a coherent pasting of classical probability distributions, transforming in a continuous way. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Maximal Boolean subalgebras Maximal Boolean subalgebras An orthomodular lattice L can be described as a collection of Boolean algebras: � L = B B∈ B (where B is the set of maximal Boolean algebras of L ). Each maximal Boolean subalgebra defines a context . A state s of L defines a classical probability on each classical Boolean subalgebra B . In other words: s B ( . . . ) := s | B ( ... ) is a Kolmogorovian measure over B . A state in a contextual theory can be considered as a coherent pasting of classical probability distributions, transforming in a continuous way. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Random Variables = Observables A random variable f can be defined as a measurable function f : Ω − → R (the pre-image of any Borel set is measurable). A random variable f defines an inverse map f − 1 satisfying: f − 1 : B ( R ) − → Σ (5a) satisfying f − 1 ( ∅ ) = ∅ (5b) f − 1 ( R ) = Γ (5c) � � f − 1 ( f − 1 ( B j ) B j ) = (5d) j j for any disjoint denumerable family B j . Also, f − 1 ( B c ) = ( f − 1 ( B )) c (5e) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Observables = (non-commutative) Random Variables In a formal way, a PVM is a map M defined over the Borel sets as follows M : B ( R ) → L ⊑N , (6a) satisfying M ( ∅ ) = 0 ( 0 := null subspace ) (6b) M ( R ) = 1 (6c) � � M ( ( B j )) = M ( B j ) , (6d) j j for any disjoint denumerable family B j . Also, M ( B c ) = 1 − M ( B ) = ( M ( B )) ⊥ (6e) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Observables = (non-commutative) Random Variables In the generalized setting (Generalized PVMs): M : B ( R ) → L , (7a) satisfying M ( ∅ ) = 0 (7b) M ( R ) = 1 (7c) � � M ( ( B j )) = M ( B j ) , (7d) j j for any disjoint denumerable family B j . Also, M ( B c ) = 1 − M ( B ) = ( M ( B )) ⊥ (7e) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Scheme C LASSICAL Q UANTUM G ENERAL P (Γ) P ( H ) L Lattice Random Variables Measurable Functions PVMs GPVMs Table: Generalized observables. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Kolmogorovian probabilities: where are they? Figure: Kolmogorovian probabilities are still there. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Lattice theory and the convex geometry are connected The approach based in convex sets can be connected with the lattice theoretical one. The set of faces of a convex set always forms a lattice . Under certain conditions, this lattice is orthomodular. The lattice of faces of a simplex is isomorphic to the Boolean algebra of propositions in which it was originated. There is an isomorphism between the lattice of faces of the convex set of quantum states and the orthomodular lattice of projection operators acting on the Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Lattice theory and the convex geometry are connected The approach based in convex sets can be connected with the lattice theoretical one. The set of faces of a convex set always forms a lattice . Under certain conditions, this lattice is orthomodular. The lattice of faces of a simplex is isomorphic to the Boolean algebra of propositions in which it was originated. There is an isomorphism between the lattice of faces of the convex set of quantum states and the orthomodular lattice of projection operators acting on the Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Lattice theory and the convex geometry are connected The approach based in convex sets can be connected with the lattice theoretical one. The set of faces of a convex set always forms a lattice . Under certain conditions, this lattice is orthomodular. The lattice of faces of a simplex is isomorphic to the Boolean algebra of propositions in which it was originated. There is an isomorphism between the lattice of faces of the convex set of quantum states and the orthomodular lattice of projection operators acting on the Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Lattice theory and the convex geometry are connected The approach based in convex sets can be connected with the lattice theoretical one. The set of faces of a convex set always forms a lattice . Under certain conditions, this lattice is orthomodular. The lattice of faces of a simplex is isomorphic to the Boolean algebra of propositions in which it was originated. There is an isomorphism between the lattice of faces of the convex set of quantum states and the orthomodular lattice of projection operators acting on the Hilbert space. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
But then... One can think about much more general theories. In fact, non-Kolmogorovian probability has been applied to study problems in biology, cognition, economics, etc. What about interpretation? This “plurality” has direct implications for information theory: F. Holik, G. M. Bosyk and G. Bellomo, “Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory”, Entropy 2015, 17 (11), 7349-7373. Holik, F., Sergioli, G., Freytes and A. Plastino, “Pattern Recognition in Non-Kolmogorovian Structures”, Foundations of Science , (2017). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
But then... One can think about much more general theories. In fact, non-Kolmogorovian probability has been applied to study problems in biology, cognition, economics, etc. What about interpretation? This “plurality” has direct implications for information theory: F. Holik, G. M. Bosyk and G. Bellomo, “Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory”, Entropy 2015, 17 (11), 7349-7373. Holik, F., Sergioli, G., Freytes and A. Plastino, “Pattern Recognition in Non-Kolmogorovian Structures”, Foundations of Science , (2017). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
But then... One can think about much more general theories. In fact, non-Kolmogorovian probability has been applied to study problems in biology, cognition, economics, etc. What about interpretation? This “plurality” has direct implications for information theory: F. Holik, G. M. Bosyk and G. Bellomo, “Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory”, Entropy 2015, 17 (11), 7349-7373. Holik, F., Sergioli, G., Freytes and A. Plastino, “Pattern Recognition in Non-Kolmogorovian Structures”, Foundations of Science , (2017). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
But then... One can think about much more general theories. In fact, non-Kolmogorovian probability has been applied to study problems in biology, cognition, economics, etc. What about interpretation? This “plurality” has direct implications for information theory: F. Holik, G. M. Bosyk and G. Bellomo, “Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory”, Entropy 2015, 17 (11), 7349-7373. Holik, F., Sergioli, G., Freytes and A. Plastino, “Pattern Recognition in Non-Kolmogorovian Structures”, Foundations of Science , (2017). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
What About Interpretation? Now a question arises. Can we say something about the nature of probabilities by simply looking at the structural properties of the above described framework? In order to find an answer to the above questions, we consider an approach based on the restrictions imposed by the algebraic features of the event structure on the probability measures which can be defined in a compatible way. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
What About Interpretation? Now a question arises. Can we say something about the nature of probabilities by simply looking at the structural properties of the above described framework? In order to find an answer to the above questions, we consider an approach based on the restrictions imposed by the algebraic features of the event structure on the probability measures which can be defined in a compatible way. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Cox’ Approach Contextuality R.T.Cox: If a rational agent deals with a Boolean algebra of assertions, representing physical events, a plausibility calculus can be derived in such a way that the plausibility function yields a theory which is formally equivalent to that of Kolmogorov.Cox, R.T. Probability, frequency, and reasonable expectation. Am. J. Phys. 14 , (1946) 1-13. Knuth, K.H. “Lattice duality: The origin of probability and entropy”, Neurocomputing 67 C, (2005) 245-274. Holik-S´ aenz-Plastino: A similar result holds if the rational agent deals with an atomic orthomodular lattice. For the quantum case, non-Kolmogorovian measures arise as the only ones compatible with the non-commutative (non-Boolean) character of quantum complementarity. Holik-Plastino-S´ aenz:F. Holik, A. Plastino and M. S´ aenz, Annals Of Physics , Volume 340 , Issue 1 , 293-310, (2014) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Cox’ Approach Contextuality R.T.Cox: If a rational agent deals with a Boolean algebra of assertions, representing physical events, a plausibility calculus can be derived in such a way that the plausibility function yields a theory which is formally equivalent to that of Kolmogorov.Cox, R.T. Probability, frequency, and reasonable expectation. Am. J. Phys. 14 , (1946) 1-13. Knuth, K.H. “Lattice duality: The origin of probability and entropy”, Neurocomputing 67 C, (2005) 245-274. Holik-S´ aenz-Plastino: A similar result holds if the rational agent deals with an atomic orthomodular lattice. For the quantum case, non-Kolmogorovian measures arise as the only ones compatible with the non-commutative (non-Boolean) character of quantum complementarity. Holik-Plastino-S´ aenz:F. Holik, A. Plastino and M. S´ aenz, Annals Of Physics , Volume 340 , Issue 1 , 293-310, (2014) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Cox’ Approach Contextuality R.T.Cox: If a rational agent deals with a Boolean algebra of assertions, representing physical events, a plausibility calculus can be derived in such a way that the plausibility function yields a theory which is formally equivalent to that of Kolmogorov.Cox, R.T. Probability, frequency, and reasonable expectation. Am. J. Phys. 14 , (1946) 1-13. Knuth, K.H. “Lattice duality: The origin of probability and entropy”, Neurocomputing 67 C, (2005) 245-274. Holik-S´ aenz-Plastino: A similar result holds if the rational agent deals with an atomic orthomodular lattice. For the quantum case, non-Kolmogorovian measures arise as the only ones compatible with the non-commutative (non-Boolean) character of quantum complementarity. Holik-Plastino-S´ aenz:F. Holik, A. Plastino and M. S´ aenz, Annals Of Physics , Volume 340 , Issue 1 , 293-310, (2014) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
What about information? Information measures Cox: In Cox’ approach, Shannon’s information measure relies on the axiomatic structure of Kolmogorovian probability theory [K. H. Knuth Neurocomputing , 67 , 245 (2005)]. Holik-S´ aenz-Plastino: The VNE thus arises as a natural measure of information derived from the non-Boolean character of the underlying lattice ( P ( H ) ). CIT and QIT can be considered as particular cases of a more general non-commutative or contextual information theory . Holik-S´ aenz-Plastino: F. Holik, A. Plastino, and M. S´ aenz, “Natural information measures for contextual probabilistic models”, Quantum Information & Computation , 16 (1 & 2) 0115-0133 (2016) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
What about information? Information measures Cox: In Cox’ approach, Shannon’s information measure relies on the axiomatic structure of Kolmogorovian probability theory [K. H. Knuth Neurocomputing , 67 , 245 (2005)]. Holik-S´ aenz-Plastino: The VNE thus arises as a natural measure of information derived from the non-Boolean character of the underlying lattice ( P ( H ) ). CIT and QIT can be considered as particular cases of a more general non-commutative or contextual information theory . Holik-S´ aenz-Plastino: F. Holik, A. Plastino, and M. S´ aenz, “Natural information measures for contextual probabilistic models”, Quantum Information & Computation , 16 (1 & 2) 0115-0133 (2016) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
What about information? Information measures Cox: In Cox’ approach, Shannon’s information measure relies on the axiomatic structure of Kolmogorovian probability theory [K. H. Knuth Neurocomputing , 67 , 245 (2005)]. Holik-S´ aenz-Plastino: The VNE thus arises as a natural measure of information derived from the non-Boolean character of the underlying lattice ( P ( H ) ). CIT and QIT can be considered as particular cases of a more general non-commutative or contextual information theory . Holik-S´ aenz-Plastino: F. Holik, A. Plastino, and M. S´ aenz, “Natural information measures for contextual probabilistic models”, Quantum Information & Computation , 16 (1 & 2) 0115-0133 (2016) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Probabilities Figure: General scheme. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Ontology? But what can we say about ontology? Is it possible to assign concrete properties of the system to these experiments? In the quantum case, the Kochen-Specker (KS) theorem poses a serious threat to this attempt: it is not possible to establish a global Boolean valuation to the elements of the lattice of projection operators. A. D¨ oring, International Journal of Theoretical Physics , Vol. 44 , No. 2 , (2005). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Ontology? But what can we say about ontology? Is it possible to assign concrete properties of the system to these experiments? In the quantum case, the Kochen-Specker (KS) theorem poses a serious threat to this attempt: it is not possible to establish a global Boolean valuation to the elements of the lattice of projection operators. A. D¨ oring, International Journal of Theoretical Physics , Vol. 44 , No. 2 , (2005). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Outline Why generalized theories? 1 Mathematical framework and the problem of interpretation 2 Informational aspects 3 Conclusions 4 Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Entropic measures in physics and information theory The notion of entropy plays a key role in many areas of physics. But it is also a key concept in information theory... The relationship between information theory and physics is a fruitful one. One of the most important examples is quantum information theory. Figure: Distinguished users of entropic measures. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Entropic measures in physics and information theory The notion of entropy plays a key role in many areas of physics. But it is also a key concept in information theory... The relationship between information theory and physics is a fruitful one. One of the most important examples is quantum information theory. Figure: Distinguished users of entropic measures. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Entropic measures in physics and information theory The notion of entropy plays a key role in many areas of physics. But it is also a key concept in information theory... The relationship between information theory and physics is a fruitful one. One of the most important examples is quantum information theory. Figure: Distinguished users of entropic measures. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Entropic measures in physics and information theory The notion of entropy plays a key role in many areas of physics. But it is also a key concept in information theory... The relationship between information theory and physics is a fruitful one. One of the most important examples is quantum information theory. Figure: Distinguished users of entropic measures. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Anecdote Shannon dixit: My greatest concern was what to call it. I thought of calling it an “information”, but the word was overly used, so I decided to call it an “uncertainty”. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, “You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have an advantage”. [M. Tribus and E. C. Mcirvine. Energy and Information, Sci. Am. , 225 (3):179-188, (1971)] Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Examples of ( h , φ ) -entropies N AME E NTROPIC FUNCTIONAL E NTROPY H ( p ) = − � φ ( x ) = − x ln x Shannon h ( x ) = x , i p i ln p i 1 − α ln ( � h ( x ) = ln( x ) φ ( x ) = x α 1 i p α R´ enyi 1 − α , R α ( p ) = i ) 1 − α ( � h ( x ) = x − 1 φ ( x ) = x α 1 i p α T α ( p ) = i − 1 ) Tsallis 1 − α , ( 1 − r ) s [( � i ) s − 1 ] x s − 1 φ ( x ) = x r E s 1 i p r h ( x ) = r ( p ) = Unified ( 1 − r ) s , S κ ( p ) = − � p κ + 1 − p − κ + 1 h ( x ) = x , φ ( x ) = x κ + 1 − x − κ + 1 Kaniadakis i i i 2 κ 2 κ Table: Examples of entropies that can be written in the form E ( p ) = h ( φ ( p )) . This family includes the Shannon, Tsallis and R´ enyi examples, and many others as well. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Why looking for quantum entropies? Schumacher’s theorem [B. Schumacher. Phys Rev A , (1995); 51 (4):2738-2747]. Maximum Entropy principle (E.T. Jaynes). We have studied the MaxEnt principle with symmetry conditions in generalized theories in: F. Holik, C. Massri, and A. Plastino. “Geometric probability theory and Jaynes’s methodology”, Int. J. Geom. Methods Mod. Phys. 13 , 1650025 (2016). Entropic uncertainty relations [G. Bosyk, M. Portesi, F. Holik and A. Plastino. Phys. Scr. 87 (2013) 065002]. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Why looking for quantum entropies? Schumacher’s theorem [B. Schumacher. Phys Rev A , (1995); 51 (4):2738-2747]. Maximum Entropy principle (E.T. Jaynes). We have studied the MaxEnt principle with symmetry conditions in generalized theories in: F. Holik, C. Massri, and A. Plastino. “Geometric probability theory and Jaynes’s methodology”, Int. J. Geom. Methods Mod. Phys. 13 , 1650025 (2016). Entropic uncertainty relations [G. Bosyk, M. Portesi, F. Holik and A. Plastino. Phys. Scr. 87 (2013) 065002]. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Why looking for quantum entropies? Schumacher’s theorem [B. Schumacher. Phys Rev A , (1995); 51 (4):2738-2747]. Maximum Entropy principle (E.T. Jaynes). We have studied the MaxEnt principle with symmetry conditions in generalized theories in: F. Holik, C. Massri, and A. Plastino. “Geometric probability theory and Jaynes’s methodology”, Int. J. Geom. Methods Mod. Phys. 13 , 1650025 (2016). Entropic uncertainty relations [G. Bosyk, M. Portesi, F. Holik and A. Plastino. Phys. Scr. 87 (2013) 065002]. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Why looking for quantum entropies? In the problem of data compression with penalization, the R´ enyi entropies play a key role. L. Campbell, Information and Control 8 , 423-429 (1965). We have studied the quantum version of that problem, in which the quantum R´ enyi entropy appears. G. Bellomo, G. Bosyk, F. Holik and S. Zozor, “Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum R´ enyi entropy”, Scientific Reports (2017), Scientific Reports , volume 7 , 14765 (2017) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Why looking for quantum entropies? In the problem of data compression with penalization, the R´ enyi entropies play a key role. L. Campbell, Information and Control 8 , 423-429 (1965). We have studied the quantum version of that problem, in which the quantum R´ enyi entropy appears. G. Bellomo, G. Bosyk, F. Holik and S. Zozor, “Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum R´ enyi entropy”, Scientific Reports (2017), Scientific Reports , volume 7 , 14765 (2017) Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Quantum Salicru entropies Definition For a quantum system in state ρ , we define: H ( h ,φ ) ( ρ ) = h (Tr φ ( ρ )) , (8) where h : R �→ R and φ : [ 0 , 1 ] �→ R are such that (i) h is strictly increasing and φ is strictly concave, or (ii) h is strictly decreasing and φ is strictly convex. Additionally, we ak that φ ( 0 ) = 0 and h ( φ ( 1 )) = 0. [G. M. Bosyk, S. Zozor, F. Holik, M. Portesi and P. W. Lamberti. “A family of generalized quantum entropies: definition and properties”, Quantum Information Processing , 15 : 8, 3393-3420, (2016)] Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Relationship with the classical ( h , φ ) -entropies Given a density operator ρ = � N i = 1 λ i | e i �� e i | with eigenvalues λ i ≥ 0, the quantum ( h , φ ) -entropies satisfy: H ( h ,φ ) ( ρ ) = H ( h ,φ ) ( λ ) where λ is the probability vector formed by the eigenvalues of ρ . The von Neumann, quantum R´ enyi, quantum Tsallis, Kaniadakis entropies are particular cases of the above definition. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
How to define entropy?: non-Kolmogorovian entropic measures. How to define entropy in models that go beyond the standard ones? References F. Holik, G. M. Bosyk and G. Bellomo. “Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory”, Entropy (2015), 17 (11), 7349-7373. M. Portesi, F. Holik, P.W. Lamberti, G.M. Bosyk, G. Bellomo y S. Zozor, “Generalized entropies in quantum and classical statistical theories”, European Physical Journal-Special Topics, 227 , 335-344 (2018), (2018). F. Holik, A. Plastino, and M. S´ aenz. “Natural information measures for contextual probabilistic models”, Quantum Information & Computation , 16 (1 & 2) 0115-0133 (2016). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Majorization Definition For given probability vectors p , q , it is said that p majorizes q , denoted as p � q , if and only if, k k � � p i ≥ q i ∀ k = 1 , . . . , d − 1 . (9) i = 1 i = 1 Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Scheme C LASSICAL Q UANTUM G ENERAL P (Γ) P ( H ) L L ATTICE − � i p ( i ) ln( p ( i )) − tr ρ ln( ρ ) inf F ∈ E H F ( µ ) E NTROPY Table: Table comparing the differences between the classical, quantal, and general cases. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Generalized formulation Now we restrict to sets of states C (convex, compact and finite dimensional). There are pure states { ν i } , such that for any ν it can be written as: � ν = p i ν i i But this decomposition will not be unique in general. [F. Holik, G. M. Bosyk and G. Bellomo. “Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory”, Entropy (2015), 17 (11), 7349-7373.] Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Generalized formulation A similar construction can be made for C in infinite dimensional models, but the mathematics is more cumbersome. Here, we appeal to the Choquet decomposition theory and write: � d µ ( ω ′ ) ω ′ ( a ) ω ( a ) = where µ is a measure over C supported by the extremal points of C and ω is considered as a functional. [M. Portesi, F. Holik, P.W. Lamberti, G.M. Bosyk, G. Bellomo y S. Zozor, “Generalized entropies in quantum and classical statistical theories”, European Physical Journal-Special Topics , 227 , 335-344 (2018), (2018).] Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Schr¨ odinger mixtures theorem In quantum mechanics, it is possible to show that the probability vector formed by the coefficients of any convex decomposition in terms of pure states of a given quantum state is majorized by the vector formed by its eigenvalues. In other formulae: If � � ω = λ i ν i = p i τ i i i then ( λ 1 , λ 2 , . . . , λ n ) � ( p 1 , p 2 , . . . , p n ) Notice that this explains why the entropy attains its minimum at the diagonalization basis (and this motivates the definition of measurement entropy in generalized models). Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Set of probability vectors Given a probabilistic model described by a compact convex set C , let M ν be the set of probability vectors associated to all possible convex decompositions of a state ν in terms of pure states (i.e., extreme points of C ): � M ν := { p ( ν ) = { p i } | ν = p i ν i for pure ν i } i Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Generalized spectrum (or geometric spectrum ) Definition Given a state ν , if the majorant of the set M ν (partially ordered by the majorization relation) exists, it is called the spectrum of ν , and we denote it by ¯ p ( ν ) . The generalized spectral decomposition is given by: � ν = p i ¯ ¯ ν i i � S ( ν ) = − ¯ p i ln(¯ p i ) i Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Geometric representation Figure: Different examples. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
Observations Notice that our definition reduces to the usual one for classical theories and for quantum mechanics. It relies only on purely geometrical notions: geometrical spectrum. For an arbitrary theory, ¯ p ( ν ) may not exist for some states. Our guess is that only physically meaningful theories possess this majorization property. Our definition allows for introducing a notion of generalized majorization and generalized entropies in a big family of models. Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018
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