Interpretation and informational aspects of non ...€¦ · Boolean algebras are examples of...

Interpretation and informational aspects ofnon-Kolmogorovian probability theory

Federico Holik

Purdue Winer Memorial Lectures 201811-10-2018

Federico Holik (IFLP-CONICET) Interpretation and informational aspects of non-Kolmogorovian probability theoryPurdue Winer Memorial Lectures 201811-10-2018 1 / 62

Outline

1 Why generalized theories?

2 Mathematical framework and the problem of interpretation

3 Informational aspects

4 Conclusions

Outline

4 Conclusions

Why looking beyond standard QM?

To understand better what we already know: compare QM with otheralternative theories.

As a fabric of new physical theories: the quest for axioms, principles andcandidates formalisms for quantum gravity and rigorous formulations ofQFT.

To find applications of non-Kolmogorovian probability theory outsidethe standard quantum domain: cognition, social sciences, etc.

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 Aii∈I

µ(⋃i∈I

Ai) =∑

µ(Ai)

Classical caseσ : Γ −→ [0; 1], such that

∫Γ σ(p, q)d3pd3q = 1.

〈F〉 =∫

Γ F(p, q)σ(p, q)d3pd3q

Boolean algebra

Figure: Hasse diagrams for B2 and B3

2, 3 1, 3 1, 2

1, 2, 3

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 asupremum 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. Thesesatisfy the weaker condition: P ≤ Q =⇒ Q = P ∨ (Q ∧ ¬P).

Boolean algebra

Kolmogorov axioms

States of classical statistical theories can be considered asKolmogorovian measures.

Observables are random variables.

Notice that the axiomatic formulation of classical information theoryrests (from a logical point of view) on the notions of probability andrandom variable.

Kolmogorov axioms

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)

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 abovemeasures (for dim(H) ≥ 3).

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)

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)

Examples: Q-bit

QbitNotice that whenH is finite dimensional, its maximal Booleansubalgebras will be finite.

P(C2) =⇒ 0,P,¬P⊥, 1C2 with P = |ϕ〉〈ϕ| for some unit norm vector|ϕ〉 and P⊥ = |ϕ⊥〉〈ϕ⊥|.

Examples: Q-bit

QbitNotice that whenH is finite dimensional, its maximal Booleansubalgebras will be finite.

P(C2) =⇒ 0,P,¬P⊥, 1C2 with P = |ϕ〉〈ϕ| for some unit norm vector|ϕ〉 and P⊥ = |ϕ⊥〉〈ϕ⊥|.

Skeleton of a qbit

. . .¬p¬q. . .pq. . .

Examples: Q-trit

Qtrit-contextuality

P(C3) =⇒P(a, b, c) = ∅, a, b, c, a, b, a, c, b, c, a, b, cGiven |ϕ1〉, |ϕ2〉 and |ϕ3〉 =⇒

0,P1,P2,P3,P12,P13,P23, 1C3

Pi = |ϕi〉〈ϕi| (i = 1, 2, 3) and Pij := |ϕi〉〈ϕi|+ |ϕj〉〈ϕj| (i, j = 1, 2, 3).

Examples: Q-trit

Qtrit-contextuality

P(C3) =⇒P(a, b, c) = ∅, a, b, c, a, b, a, c, b, c, a, b, cGiven |ϕ1〉, |ϕ2〉 and |ϕ3〉 =⇒

0,P1,P2,P3,P12,P13,P23, 1C3

Pi = |ϕi〉〈ϕi| (i = 1, 2, 3) and Pij := |ϕi〉〈ϕi|+ |ϕj〉〈ϕj| (i, j = 1, 2, 3).

Qtrit Boolean subalgebras:

Figure: Maximal Boolean subalgebras of C3

2, 3 1, 3 1, 2

1, 2, 3

P1 P2 P3

P23 P13 P12

Figure: Skeleton of C3

· · · · · ·· · · · · ·

P1 P2 P3

P23 P13 P12

Outline

4 Conclusions

Observations

ObservationsWe have an empirically successful example of a theory (QM) whosestructure of events is non-Boolean.

We know examples of physically meaningful probabilistic theorieswhich 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... whythinking that QM is the best option?

Observations

Quantum Probabilistic Models

The idea of comparing QM with other theories dates back to vonNeumann.Birkhoff and von Neumann compared standard QM with classicalprobability theory and searched for possible replacements of the Hilbertspace formalism.This path was followed by many others afterwards: Ludwig, Mackey,Piron, Mielnik, etc.By appealing to lattice theory, B and VN developed the axiomaticframework of a generalization of the projective geometry associated tothe Hilbert space. The generalization included the notion of continuousgeometries.A somewhat curious historical remark: the axiomatization of quantumprobability (i.e, the first non-Kolmogorovian probabilistic calculus) datesback to the late 20’s and it reaches its full form in the 1932 vonNeumann’s masterpiece. It almost simultaneous to the one worksKolmogorov (1933) for the axiomatization of classical probability basedon measure theory.

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 orthogonalspace, those mappings of this space on itself, which leave orthogo-nality intact, leaves all angles intact, in other words, in those systemswhich can be used as models of the logical background for quantumtheory, it is true that as soon as all the ordinary concepts of logicare fixed under some isomorphic transformation, all of probabilitytheory is already fixed... This means however, that one has a formalmechanism in which, logics and probability theory arise simultane-ously and are derived simultaneously.”

In a series of papers Murray and von Neumann searched for algebrasmore general than B(H).

The new algebras are known today as von Neumann algebras, and theirelementary components can be classified as Type I, Type II and Type IIIfactors.

It can be shown that, the projective elements of a factor form anorthomodular lattice. Classical models can be described as commutativealgebras.

The models of standard quantum mechanics can be described by usingType I factors (Type In for finite dimensional Hilbert spaces and Type I∞for infinite dimensional models). These are algebras isomorphic to theset of bounded operators on a Hilbert space.

Further work revealed that a rigorous approach to the study of quantumsystems with infinitely many degrees of freedom needed the use of moregeneral von Neumann algebras.

This is the case in the axiomatic formulation of relativistic quantummechanics. A similar situation holds in algebraic quantum statisticalmechanics.

Further work revealed that a rigorous approach to the study of quantumsystems with infinitely many degrees of freedom needed the use of moregeneral von Neumann algebras.

This is the case in the axiomatic formulation of relativistic quantummechanics. A similar situation holds in algebraic quantum statisticalmechanics.

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 vonNeumann algebras (Contextuality is quite ubiquitous among a big familyof theories). [A. Doring, International Journal of Theoretical Physics,Vol. 44, No. 2, (2005)].

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

Ej) =∑

s(Ej).

where L is a general orthomodular lattice (with L = Σ and L = P(H) for theKolmogorovian and quantum cases respectively).All measures satisfying the above axioms for a given L form a convex set.

Maximal Boolean subalgebras

Maximal Boolean subalgebrasAn orthomodular lattice L can be described as a collection of Booleanalgebras:

L =∨B∈BB

(where B is the set of maximal Boolean algebras of L). Each maximalBoolean subalgebra defines a context.

A state s of L defines a classical probability on each classical Booleansubalgebra B. In other words: sB(. . .) := s|B(...) is a Kolmogorovianmeasure over B.

A state in a contextual theory can be considered as a coherent pasting ofclassical probability distributions, transforming in a continuous way.

L =∨B∈BB

Random Variables = Observables

A random variable f can be defined as a measurable function f : Ω −→ R (thepre-image of any Borel set is measurable).A random variable f defines an inverse map f−1 satisfying:

f−1 : B(R) −→ Σ (5a)

satisfyingf−1(∅) = ∅ (5b)

f−1(R) = Γ (5c)

f−1(∨

Bj) =∨

f−1(Bj) (5d)

for any disjoint denumerable family Bj. Also,

f−1(Bc) = (f−1(B))c (5e)

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)→ LvN , (6a)

satisfying

M(∅) = 0 (0 := null subspace) (6b)

M(R) = 1 (6c)

(Bj)) =∨

M(Bj), (6d)

M(Bc) = 1−M(B) = (M(B))⊥ (6e)

Observables = (non-commutative) Random Variables

In the generalized setting (Generalized PVMs):

M : B(R)→ L, (7a)

satisfying

M(∅) = 0 (7b)

M(R) = 1 (7c)

(Bj)) =∨

M(Bj), (7d)

M(Bc) = 1−M(B) = (M(B))⊥ (7e)

Scheme

CLASSICAL QUANTUM GENERAL

Lattice P(Γ) P(H) LRandom Variables Measurable Functions PVMs GPVMs

Table: Generalized observables.

Kolmogorovian probabilities: where are they?

Figure: Kolmogorovian probabilities are still there.

Lattice theory and the convex geometry are connected

The approach based in convex sets can be connected with the latticetheoretical one.

The set of faces of a convex set always forms a lattice. Under certainconditions, this lattice is orthomodular.

The lattice of faces of a simplex is isomorphic to the Boolean algebra ofpropositions in which it was originated.

There is an isomorphism between the lattice of faces of the convex set ofquantum states and the orthomodular lattice of projection operatorsacting on the Hilbert space.

But then...

One can think about much more general theories.

In fact, non-Kolmogorovian probability has been applied to studyproblems in biology, cognition, economics, etc. What aboutinterpretation?

This “plurality” has direct implications for information theory: F. Holik,G. M. Bosyk and G. Bellomo, “Quantum Information as aNon-Kolmogorovian Generalization of Shannon’s Theory”, Entropy2015, 17 (11), 7349-7373.

Holik, F., Sergioli, G., Freytes and A. Plastino, “Pattern Recognition inNon-Kolmogorovian Structures”, Foundations of Science, (2017).

But then...

What About Interpretation?

Now a question arises. Can we say something about the nature ofprobabilities by simply looking at the structural properties of the abovedescribed framework?

In order to find an answer to the above questions, we consider anapproach based on the restrictions imposed by the algebraic features ofthe event structure on the probability measures which can be defined in acompatible way.

What About Interpretation?

Now a question arises. Can we say something about the nature ofprobabilities by simply looking at the structural properties of the abovedescribed framework?

In order to find an answer to the above questions, we consider anapproach based on the restrictions imposed by the algebraic features ofthe event structure on the probability measures which can be defined in acompatible way.

Cox’ Approach

ContextualityR.T.Cox: If a rational agent deals with a Boolean algebra of assertions,representing physical events, a plausibility calculus can be derived insuch a way that the plausibility function yields a theory which isformally 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-Saenz-Plastino: A similar result holds if the rational agent dealswith an atomic orthomodular lattice. For the quantum case,non-Kolmogorovian measures arise as the only ones compatible with thenon-commutative (non-Boolean) character of quantum complementarity.

Holik-Plastino-Saenz:F. Holik, A. Plastino and M. Saenz, Annals OfPhysics, Volume 340, Issue 1, 293-310, (2014)

Cox’ Approach

What about information?

Information measuresCox: In Cox’ approach, Shannon’s information measure relies on theaxiomatic structure of Kolmogorovian probability theory [K. H. KnuthNeurocomputing, 67, 245 (2005)].

Holik-Saenz-Plastino: The VNE thus arises as a natural measure ofinformation derived from the non-Boolean character of the underlyinglattice (P(H)). CIT and QIT can be considered as particular cases of amore general non-commutative or contextual information theory.

Holik-Saenz-Plastino: F. Holik, A. Plastino, and M. Saenz, “Naturalinformation measures for contextual probabilistic models”, QuantumInformation & Computation, 16 (1 & 2) 0115-0133 (2016)

Probabilities

Figure: General scheme.

Ontology?

But what can we say about ontology?

Is it possible to assign concrete properties of the system to theseexperiments? In the quantum case, the Kochen-Specker (KS) theoremposes a serious threat to this attempt: it is not possible to establish aglobal Boolean valuation to the elements of the lattice of projectionoperators. A. Doring, International Journal of Theoretical Physics, Vol.44, No. 2, (2005).

Ontology?

But what can we say about ontology?

Is it possible to assign concrete properties of the system to theseexperiments? In the quantum case, the Kochen-Specker (KS) theoremposes a serious threat to this attempt: it is not possible to establish aglobal Boolean valuation to the elements of the lattice of projectionoperators. A. Doring, International Journal of Theoretical Physics, Vol.44, No. 2, (2005).

Outline

4 Conclusions

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.

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 itan “uncertainty”. When I discussed it with John von Neumann, hehad a better idea. Von Neumann told me, “You should call it entropy,for two reasons. In the first place your uncertainty function has beenused in statistical mechanics under that name, so it already has aname. In the second place, and more important, nobody knows whatentropy 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)]

Examples of (h, φ)-entropies

NAME ENTROPIC FUNCTIONAL ENTROPY

Shannon h(x) = x, φ(x) = −x ln x H(p) = −∑

i pi ln pi

Renyi h(x) = ln(x)1−α , φ(x) = xα Rα(p) = 1

1−α ln (∑

i pαi )

Tsallis h(x) = x−11−α , φ(x) = xα Tα(p) = 1

1−α (∑

i pαi − 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) = −∑

ipκ+1

i −p−κ+1i

Table: Examples of entropies that can be written in the form E(p) = h(φ(p)). Thisfamily includes the Shannon, Tsallis and Renyi examples, and many others as well.

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 MaxEntprinciple with symmetry conditions in generalized theories in:F. Holik, C. Massri, and A. Plastino. “Geometric probability theory andJaynes’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].

In the problem of data compression with penalization, the Renyientropies play a key role.L. Campbell, Information and Control 8, 423-429 (1965).

We have studied the quantum version of that problem, in which thequantum Renyi entropy appears.G. Bellomo, G. Bosyk, F. Holik and S. Zozor, “Lossless quantum datacompression with exponential penalization: an operational interpretationof the quantum Renyi entropy”, Scientific Reports (2017), ScientificReports, volume 7, 14765 (2017)

In the problem of data compression with penalization, the Renyientropies play a key role.L. Campbell, Information and Control 8, 423-429 (1965).

We have studied the quantum version of that problem, in which thequantum Renyi entropy appears.G. Bellomo, G. Bosyk, F. Holik and S. Zozor, “Lossless quantum datacompression with exponential penalization: an operational interpretationof the quantum Renyi entropy”, Scientific Reports (2017), ScientificReports, volume 7, 14765 (2017)

Quantum Salicru entropies

DefinitionFor a quantum system in state ρ, we define:

H(h,φ)(ρ) = h (Tr φ(ρ)) , (8)

where h : R 7→ R and φ : [0, 1] 7→ R are such that (i) h is strictlyincreasing and φ is strictly concave, or (ii) h is strictly decreasing and φ isstrictly 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 ofgeneralized quantum entropies: definition and properties”, QuantumInformation Processing, 15: 8, 3393-3420, (2016)]

Relationship with the classical (h, φ)-entropies

Given a density operator ρ =∑N

i=1 λi |ei〉〈ei| with eigenvalues λi ≥ 0, thequantum (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 Renyi, quantum Tsallis, Kaniadakis entropiesare particular cases of the above definition.

How to define entropy?: non-Kolmogorovian entropicmeasures.

How to define entropy in models that go beyond the standard ones?

ReferencesF. Holik, G. M. Bosyk and G. Bellomo. “Quantum Information as aNon-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. Saenz. “Natural information measures forcontextual probabilistic models”, Quantum Information & Computation,16 (1 & 2) 0115-0133 (2016).

Majorization

DefinitionFor given probability vectors p, q, it is said that p majorizes q, denoted asp q, if and only if,

k∑i=1

pi ≥k∑

qi ∀ k = 1, . . . , d − 1. (9)

Scheme

CLASSICAL QUANTUM GENERAL

LATTICE P(Γ) P(H) LENTROPY −

∑i p(i) ln(p(i)) −trρ ln(ρ) infF∈E HF(µ)

Table: Table comparing the differences between the classical, quantal, and generalcases.

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 =∑

But this decomposition will not be unique in general.[F. Holik, G. M. Bosyk and G. Bellomo. “Quantum Information as aNon-Kolmogorovian Generalization of Shannon’s Theory”, Entropy (2015),17 (11), 7349-7373.]

Generalized formulation

A similar construction can be made for C in infinite dimensional models, butthe mathematics is more cumbersome. Here, we appeal to the Choquetdecomposition 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 isconsidered 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).]

Schrodinger mixtures theorem

In quantum mechanics, it is possible to show that the probability vectorformed by the coefficients of any convex decomposition in terms of purestates of a given quantum state is majorized by the vector formed by itseigenvalues. In other formulae:If

ω =∑

λiνi =∑

(λ1, λ2, . . . , λn) (p1, p2, . . . , pn)

Notice that this explains why the entropy attains its minimum at thediagonalization basis (and this motivates the definition of measuremententropy in generalized models).

Set of probability vectors

Given a probabilistic model described by a compact convex set C, let Mν bethe set of probability vectors associated to all possible convex decompositionsof a state ν in terms of pure states (i.e., extreme points of C):

Mν := p(ν) = pi | ν =∑

piνi for pure νi

Generalized spectrum (or geometric spectrum)

DefinitionGiven a state ν, if the majorant of the set Mν (partially ordered by themajorization relation) exists, it is called the spectrum of ν, and we denote it byp(ν).

The generalized spectral decomposition is given by:

ν =∑

S(ν) = −∑

pi ln(pi)

Geometric representation

Figure: Different examples.

Observations

Notice that our definition reduces to the usual one for classical theoriesand 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 thismajorization property.

Our definition allows for introducing a notion of generalizedmajorization and generalized entropies in a big family of models.

Observations

Generalized majorization

DefinitionGiven two states µ and ν, one has that µ is majorized by ν, and we denote itby µ ≺ ν, if and only if:

p(µ) ≺ p(ν)

where p(µ) and p(ν) are the corresponding spectra.

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 afunctional φ to the state as:

φ(ν)|pi,νi :=∑

φ(pi)νi

In particular, we are interested in the mixture pi, νi, that leads naturally tothe definition:

φ(ν) := φ(ν)|pi,νi

(h, φ)-entropies in generalized models

In the generalized formalism, a functional uC plays the role of a trace. Thisallows 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))

Outline

4 Conclusions

Conclusions

There are many examples of physically meaningful contextual theories.The KS theorem is valid for many models of interest (for example, vonNeumann algebras).

We are looking for new physical contextual models as candidates fordeveloping new physics.

In that quest, Information Theory can be very useful. In particular, thestudy of information measures and other informational principles andquantities, such as majorization and MaxEnt.

Conclusions

Due to the existence of “generalized” versions of the KS theorem (VNalgebras), we find that for many models the description of systems asbundles of actual properties will be problematic.

A quick alternative, could be to postulate hidden variables. This can beuseful in many examples. But these hidden variables should becontextual, or highly non-local in physical theories (as is the case instandard quantum mechanics).

We reviewed an approach in which states are regarded as functionsmeasuring the degree of belief of a rational agent, assuming that(possibly) contextual phenomena is accepted as a starting point.

Conclusions

Many thanks for your attention!

F. Holik, A. Plastino and M. Saenz. “A discussion in the origin ofquantum probabilities”, Annals Of Physics, Volume 340, Issue 1,293-310, (2014).

F. Holik, A. Plastino, and M. Saenz. “Natural information measures forcontextual probabilistic models”, Quantum Information & Computation,16 (1 & 2) 0115-0133 (2016).

F. Holik, G. M. Bosyk and G. Bellomo. “Quantum Information as aNon-Kolmogorovian Generalization of Shannon’s Theory”, Entropy2015, 17 (11), 7349-7373.

F. Holik, C. Massri, and A. Plastino. “Geometric probability theory andJaynes’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).

Conference in Argentina

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

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