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Interpretation and informational aspects of non-Kolmogorovian probability theory

Federico Holik Purdue Winer Memorial Lectures 2018 11-10-2018

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Outline

1

Why generalized theories?

2

Mathematical framework and the problem of interpretation

3

Informational aspects

4

Conclusions

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Outline

1

Why generalized theories?

2

Mathematical framework and the problem of interpretation

3

Informational aspects

4

Conclusions

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Why looking beyond standard QM?

To understand better what we already know: compare QM with other alternative theories. As a fabric of new physical theories: the quest for axioms, principles and candidates formalisms for quantum gravity and rigorous formulations of QFT. To find applications of non-Kolmogorovian probability theory outside the standard quantum domain: cognition, social sciences, etc.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Why looking beyond standard QM?

To understand better what we already know: compare QM with other alternative theories. As a fabric of new physical theories: the quest for axioms, principles and candidates formalisms for quantum gravity and rigorous formulations of QFT. To find applications of non-Kolmogorovian probability theory outside the standard quantum domain: cognition, social sciences, etc.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Why looking beyond standard QM?

To understand better what we already know: compare QM with other alternative theories. As a fabric of new physical theories: the quest for axioms, principles and candidates formalisms for quantum gravity and rigorous formulations of QFT. To find applications of non-Kolmogorovian probability theory outside the standard quantum domain: cognition, social sciences, etc.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Kolmogorov’s axioms

Probility measures

µ : Σ → [0, 1] (1) such that: 1 µ(∅) = 0 2 µ(Ac) = 1 − µ(A) 3 For any denumerable family of pairwise disjoint sets {Ai}i∈I µ(

• i∈I

Ai) =

• i

µ(Ai)

Classical case

σ : Γ − → [0; 1], such that

• Γ σ(p, q)d3pd3q = 1.

F =

• Γ F(p, q)σ(p, q)d3pd3q

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Boolean algebra

Figure: Hasse diagrams for B2 and B3

B2 ∅ {1} {2} {1, 2} B3 ∅ {1} {2} {3} {2, 3} {1, 3} {1, 2} {1, 2, 3}

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Boolean algebra

Boolean algebras are examples of lattices. A lattice is a partially ordered set: P ≤ Q (not totally ordered). For any pair of elements P, Q there exists an infimum P ∧ Q and a supremum P ∨ Q. Boolean algebras also have an orthocomplementation: ¬P. Boolean algebras are distributive: P ∧ (Q ∨ ¬Q) = (P ∧ Q) ∨ (P ∧ ¬Q). Boolean algebras are particular cases of orthomodular lattices. These satisfy the weaker condition: P ≤ Q = ⇒ Q = P ∨ (Q ∧ ¬P).

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Boolean algebra

Boolean algebras are examples of lattices. A lattice is a partially ordered set: P ≤ Q (not totally ordered). For any pair of elements P, Q there exists an infimum P ∧ Q and a supremum P ∨ Q. Boolean algebras also have an orthocomplementation: ¬P. Boolean algebras are distributive: P ∧ (Q ∨ ¬Q) = (P ∧ Q) ∨ (P ∧ ¬Q). Boolean algebras are particular cases of orthomodular lattices. These satisfy the weaker condition: P ≤ Q = ⇒ Q = P ∨ (Q ∧ ¬P).

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Boolean algebra

Boolean algebras are examples of lattices. A lattice is a partially ordered set: P ≤ Q (not totally ordered). For any pair of elements P, Q there exists an infimum P ∧ Q and a supremum P ∨ Q. Boolean algebras also have an orthocomplementation: ¬P. Boolean algebras are distributive: P ∧ (Q ∨ ¬Q) = (P ∧ Q) ∨ (P ∧ ¬Q). Boolean algebras are particular cases of orthomodular lattices. These satisfy the weaker condition: P ≤ Q = ⇒ Q = P ∨ (Q ∧ ¬P).

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Boolean algebra

Boolean algebras are examples of lattices. A lattice is a partially ordered set: P ≤ Q (not totally ordered). For any pair of elements P, Q there exists an infimum P ∧ Q and a supremum P ∨ Q. Boolean algebras also have an orthocomplementation: ¬P. Boolean algebras are distributive: P ∧ (Q ∨ ¬Q) = (P ∧ Q) ∨ (P ∧ ¬Q). Boolean algebras are particular cases of orthomodular lattices. These satisfy the weaker condition: P ≤ Q = ⇒ Q = P ∨ (Q ∧ ¬P).

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Boolean algebra

Boolean algebras are examples of lattices. A lattice is a partially ordered set: P ≤ Q (not totally ordered). For any pair of elements P, Q there exists an infimum P ∧ Q and a supremum P ∨ Q. Boolean algebras also have an orthocomplementation: ¬P. Boolean algebras are distributive: P ∧ (Q ∨ ¬Q) = (P ∧ Q) ∨ (P ∧ ¬Q). Boolean algebras are particular cases of orthomodular lattices. These satisfy the weaker condition: P ≤ Q = ⇒ Q = P ∨ (Q ∧ ¬P).

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Kolmogorov axioms

States of classical statistical theories can be considered as Kolmogorovian measures. Observables are random variables. Notice that the axiomatic formulation of classical information theory rests (from a logical point of view) on the notions of probability and random variable.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Kolmogorov axioms

States of classical statistical theories can be considered as Kolmogorovian measures. Observables are random variables. Notice that the axiomatic formulation of classical information theory rests (from a logical point of view) on the notions of probability and random variable.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Kolmogorov axioms

States of classical statistical theories can be considered as Kolmogorovian measures. Observables are random variables. Notice that the axiomatic formulation of classical information theory rests (from a logical point of view) on the notions of probability and random variable.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Quantum probability (Born’s rule)

Probability measures

s : LvN − → [0; 1] (2) such that: 1 s(0) = 0 (0 is the null subspace). 2 s(P⊥) = 1 − s(P) 3 For any denumerable family of pairwise orthogonal projectors we have (Pj), s(

j Pj) = j s(Pj)

Teorema (Gleason)

sρ(P) = tr(ρP) (3)

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Quantum probability (Born’s rule)

Probability measures

s : LvN − → [0; 1] (2) such that: 1 s(0) = 0 (0 is the null subspace). 2 s(P⊥) = 1 − s(P) 3 For any denumerable family of pairwise orthogonal projectors we have (Pj), s(

j Pj) = j s(Pj)

Teorema (Gleason)

Gleason’s theorem grants that there exists a density operator for the above measures (for dim(H) ≥ 3).

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Lattice of projection operators acting on a Hilbert space

LvN is an orthomodular lattice. LvN 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

Lattice of projection operators acting on a Hilbert space

LvN is an orthomodular lattice. LvN 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. P(C2) = ⇒ {0, P, ¬P⊥, 1C2} with P = |ϕϕ| for some unit norm vector |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. P(C2) = ⇒ {0, P, ¬P⊥, 1C2} with P = |ϕϕ| for some unit norm vector |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(C2) 1 . . . ¬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-trit

Qtrit-contextuality

P(C3) = ⇒ P({a, b, c}) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} Given |ϕ1, |ϕ2 and |ϕ3 = ⇒ {0, P1, P2, P3, P12, P13, P23, 1C3} Pi = |ϕiϕi| (i = 1, 2, 3) and Pij := |ϕiϕi| + |ϕjϕj| (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(C3) = ⇒ P({a, b, c}) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} Given |ϕ1, |ϕ2 and |ϕ3 = ⇒ {0, P1, P2, P3, P12, P13, P23, 1C3} Pi = |ϕiϕi| (i = 1, 2, 3) and Pij := |ϕiϕi| + |ϕjϕj| (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 C3

B3 ∅ {1} {2} {3} {2, 3} {1, 3} {1, 2} {1, 2, 3} B3 ∅ P1 P2 P3 P23 P13 P12 1C3

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Figure: Skeleton of C3

P(C3) · · · · · · · · · · · · ∅ P1 P2 P3 P23 P13 P12 1C3

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Outline

1

Why generalized theories?

2

Mathematical framework and the problem of interpretation

3

Informational aspects

4

Conclusions

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

• n 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

• n 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

• n 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

• n 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

• n 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-

• usly 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

• rthomodular lattice. Classical models can be described as commutative

algebras. The models of standard quantum mechanics can be described by using Type I factors (Type In 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

• rthomodular lattice. Classical models can be described as commutative

algebras. The models of standard quantum mechanics can be described by using Type I factors (Type In 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

• rthomodular lattice. Classical models can be described as commutative

algebras. The models of standard quantum mechanics can be described by using Type I factors (Type In 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

• rthomodular lattice. Classical models can be described as commutative

algebras. The models of standard quantum mechanics can be described by using Type I factors (Type In 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

• f theories). [A. D¨
• ring, 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

• f theories). [A. D¨
• ring, 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

• f theories). [A. D¨
• ring, 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

• f theories). [A. D¨
• ring, 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 Ej s(

• j

Ej) =

• j

s(Ej). 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: sB(. . .) := 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: sB(. . .) := 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: sB(. . .) := 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(

• j

Bj) =

• j

f −1(Bj) (5d) for any disjoint denumerable family Bj. Also, f −1(Bc) = (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(

• j

(Bj)) =

• j

M(Bj), (6d) for any disjoint denumerable family Bj. Also, M(Bc) = 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(

• j

(Bj)) =

• j

M(Bj), (7d) for any disjoint denumerable family Bj. Also, M(Bc) = 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

CLASSICAL QUANTUM GENERAL Lattice P(Γ) P(H) L 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

• perators. A. D¨
• ring, 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

• perators. A. D¨
• ring, 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

1

Why generalized theories?

2

Mathematical framework and the problem of interpretation

3

Informational aspects

4

Conclusions

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

NAME ENTROPIC FUNCTIONAL ENTROPY Shannon h(x) = x, φ(x) = −x ln x H(p) = −

i pi ln pi

R´ enyi h(x) = ln(x)

1−α ,

φ(x) = xα Rα(p) =

1 1−α ln ( i p α i )

Tsallis h(x) = x−1

1−α,

φ(x) = xα Tα(p) =

1 1−α ( i p α i − 1)

Unified h(x) =

xs−1 (1−r)s,

φ(x) = xr Es

r(p) = 1 (1−r)s [( i p r i )s − 1]

Kaniadakis h(x) = x, φ(x) = xκ+1−x−κ+1

2κ

Sκ(p) = −

i pκ+1

i

−p−κ+1

i

2κ

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

• f 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

• f 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 λi |eiei| with eigenvalues λi ≥ 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

• i=1

pi ≥

k

• i=1

qi ∀ k = 1, . . . , d − 1. (9)

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Scheme

CLASSICAL QUANTUM GENERAL LATTICE P(Γ) P(H) L ENTROPY −

i p(i) ln(p(i))

−trρ ln(ρ) infF∈E HF(µ)

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 {νi}, such that for any ν it can be written as: ν =

• i

piνi 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: ω(a) =

• dµ(ω′)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¨

• dinger 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 =

• i

λiνi =

• i

piτi then (λ1, λ2, . . . , λn) (p1, p2, . . . , pn) 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 Mν be the set of probability vectors associated to all possible convex decompositions

• f a state ν in terms of pure states (i.e., extreme points of C):

Mν := {p(ν) = {pi} | ν =

• i

piνi for pure νi}

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 Mν (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 =

• i

¯ pi¯ νi S(ν) = −

• i

¯ pi ln(¯ pi)

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

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

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

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

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

Generalized majorization

Definition

Given two states µ and ν, one has that µ is majorized by ν, and we denote it by µ ≺ ν, if and only if: ¯ p(µ) ≺ ¯ p(ν) where ¯ p(µ) and ¯ p(ν) are the corresponding spectra.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Funciones de estados

Our definition can be also used to define a function of a generalized state φ. For any possible mixture {pi, νi} of ν, we define the application of a functional φ to the state as: φ(ν)|{pi,νi} :=

• i

φ(pi)νi In particular, we are interested in the mixture {¯ pi, ¯ νi}, that leads naturally to the definition: φ(ν) := φ(ν)|{¯

pi,¯ νi}

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

(h, φ)-entropies in generalized models

In the generalized formalism, a functional uC plays the role of a trace. This allows us to define generalized (h, φ)-entropies.

Definition

• H(h,φ)(ν) = h (uC (φ(ν)))

They satisfy:

• H(h,φ)(ν) = H(h,φ)(¯

p(ν))

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Outline

1

Why generalized theories?

2

Mathematical framework and the problem of interpretation

3

Informational aspects

4

Conclusions

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Conclusions

There are many examples of physically meaningful contextual theories. The KS theorem is valid for many models of interest (for example, von Neumann algebras). We are looking for new physical contextual models as candidates for developing new physics. In that quest, Information Theory can be very useful. In particular, the study of information measures and other informational principles and quantities, such as majorization and MaxEnt.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Conclusions

There are many examples of physically meaningful contextual theories. The KS theorem is valid for many models of interest (for example, von Neumann algebras). We are looking for new physical contextual models as candidates for developing new physics. In that quest, Information Theory can be very useful. In particular, the study of information measures and other informational principles and quantities, such as majorization and MaxEnt.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Conclusions

There are many examples of physically meaningful contextual theories. The KS theorem is valid for many models of interest (for example, von Neumann algebras). We are looking for new physical contextual models as candidates for developing new physics. In that quest, Information Theory can be very useful. In particular, the study of information measures and other informational principles and quantities, such as majorization and MaxEnt.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Conclusions

Due to the existence of “generalized” versions of the KS theorem (VN algebras), we find that for many models the description of systems as bundles of actual properties will be problematic. A quick alternative, could be to postulate hidden variables. This can be useful in many examples. But these hidden variables should be contextual, or highly non-local in physical theories (as is the case in standard quantum mechanics). We reviewed an approach in which states are regarded as functions measuring the degree of belief of a rational agent, assuming that (possibly) contextual phenomena is accepted as a starting point.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Conclusions

Due to the existence of “generalized” versions of the KS theorem (VN algebras), we find that for many models the description of systems as bundles of actual properties will be problematic. A quick alternative, could be to postulate hidden variables. This can be useful in many examples. But these hidden variables should be contextual, or highly non-local in physical theories (as is the case in standard quantum mechanics). We reviewed an approach in which states are regarded as functions measuring the degree of belief of a rational agent, assuming that (possibly) contextual phenomena is accepted as a starting point.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Conclusions

Due to the existence of “generalized” versions of the KS theorem (VN algebras), we find that for many models the description of systems as bundles of actual properties will be problematic. A quick alternative, could be to postulate hidden variables. This can be useful in many examples. But these hidden variables should be contextual, or highly non-local in physical theories (as is the case in standard quantum mechanics). We reviewed an approach in which states are regarded as functions measuring the degree of belief of a rational agent, assuming that (possibly) contextual phenomena is accepted as a starting point.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018

Many thanks for your attention!

• F. Holik, A. Plastino and M. S´
• aenz. “A discussion in the origin of

quantum probabilities”, Annals Of Physics, Volume 340, Issue 1, 293-310, (2014).

• F. Holik, A. Plastino, and M. S´
• aenz. “Natural information measures for

contextual probabilistic models”, Quantum Information & Computation, 16 (1 & 2) 0115-0133 (2016).

• 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.

• F. Holik, C. Massri, and A. Plastino. “Geometric probability theory and

Jaynes’s methodology”, Int. J. Geom. Methods Mod. Phys. 13, 1650025 (2016). 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

Conference in Argentina

Figure: Conference in Argentina: https://sites.google.com/site/viijornadasfundamentoscuantica/.

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theory Purdue Winer Memorial Lectures 201811-10-2018