Logic, Geometry And Probability Theory Federico Holik 1 1-Center Leo - - PowerPoint PPT Presentation

Logic, Geometry And Probability Theory

Federico Holik1

1-Center Leo Apostel for Interdisciplinary Studies and, Department of Mathematics, Brussels Free University Krijgskundestraat 33, 1160 Brussels, Belgium

Whither Quantum Structures In The 21th Century? Brussels - 2013

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Outline

Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov

• R. T. Cox

Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Outline

Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov

• R. T. Cox

Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Abstract

◮ We study the origin of quantum probabilities as arising

from non-boolean propositional-operational structures and deduce non-kolmogorvian probability measures of quantum mechanics.

◮ We discuss some problems posed by von Neumann

regarding the development of a quantum probability theory.

◮ We present an alternative perspective to the problem of

compound quantum systems.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Abstract

◮ We study the origin of quantum probabilities as arising

from non-boolean propositional-operational structures and deduce non-kolmogorvian probability measures of quantum mechanics.

◮ We discuss some problems posed by von Neumann

regarding the development of a quantum probability theory.

◮ We present an alternative perspective to the problem of

compound quantum systems.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Abstract

◮ We study the origin of quantum probabilities as arising

from non-boolean propositional-operational structures and deduce non-kolmogorvian probability measures of quantum mechanics.

◮ We discuss some problems posed by von Neumann

regarding the development of a quantum probability theory.

◮ We present an alternative perspective to the problem of

compound quantum systems.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Outline

Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov

• R. T. Cox

Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Regularities

Given a physical theory:

◮ We concentrate on events or states of affairs. Given a

certain state of affairs, we ask for how likely is that another state of affairs occurs. How can we measure this?

◮ How are events of a given theory structured? Are they

structured in some way? Is there a link between the event structure and probability theory?

◮ The existence of well defined events is a necessary

condition in order to have a scientific theory. This is independent of the position that we adopt, realism, empiricism, etc.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Regularities

Given a physical theory:

◮ We concentrate on events or states of affairs. Given a

certain state of affairs, we ask for how likely is that another state of affairs occurs. How can we measure this?

◮ How are events of a given theory structured? Are they

structured in some way? Is there a link between the event structure and probability theory?

◮ The existence of well defined events is a necessary

condition in order to have a scientific theory. This is independent of the position that we adopt, realism, empiricism, etc.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Regularities

Given a physical theory:

◮ We concentrate on events or states of affairs. Given a

certain state of affairs, we ask for how likely is that another state of affairs occurs. How can we measure this?

◮ How are events of a given theory structured? Are they

structured in some way? Is there a link between the event structure and probability theory?

◮ The existence of well defined events is a necessary

condition in order to have a scientific theory. This is independent of the position that we adopt, realism, empiricism, etc.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Different kinds of regularities

Several possibilities:

◮ We can predict with absolute certainty (example: classical

mechanics, determinism). Absolute regularity.

◮ We can only compute probabilities: given two equivalent

preparations, we may have different effects, but with probabilities cogently defined (example: quantum mechanics). Statistical regularity.

◮ There is no regularity at all. Several reasons: ontological,

impossibility of defining equivalent preparations, ect. Even in this case, we can (sometimes) define probabilities (but we will not discuss this here).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Different kinds of regularities

Several possibilities:

◮ We can predict with absolute certainty (example: classical

mechanics, determinism). Absolute regularity.

◮ We can only compute probabilities: given two equivalent

preparations, we may have different effects, but with probabilities cogently defined (example: quantum mechanics). Statistical regularity.

◮ There is no regularity at all. Several reasons: ontological,

impossibility of defining equivalent preparations, ect. Even in this case, we can (sometimes) define probabilities (but we will not discuss this here).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Different kinds of regularities

Several possibilities:

◮ We can predict with absolute certainty (example: classical

mechanics, determinism). Absolute regularity.

◮ We can only compute probabilities: given two equivalent

preparations, we may have different effects, but with probabilities cogently defined (example: quantum mechanics). Statistical regularity.

◮ There is no regularity at all. Several reasons: ontological,

impossibility of defining equivalent preparations, ect. Even in this case, we can (sometimes) define probabilities (but we will not discuss this here).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Events in the QL approach

von Neumann: “And one also has the parallelism that logics corresponds to set theory and probability theory corresponds to measure theory and that a given system of logics, so given a system of sets, if all is right, you can introduce measures, you can introduce probability and you can always do it in very many different ways.”(vN-Amsterdam Talk)

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

In this way, the connection between Logic, Set Theory, and Probability Theory is clear. What does this means? The definition of Cantor of a set reads: “A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of

• ur thought —which are called elements of the set.”

(Cantor)

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Events in the QL approach

Propositional systems:

◮ Operational questions in CM correspond to subsets of

phase space.

◮ Operational questions in QM correspond to subspaces

(projections) in a Hilbert space.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Events in the QL approach

Propositional systems:

◮ Operational questions in CM correspond to subsets of

phase space.

◮ Operational questions in QM correspond to subspaces

(projections) in a Hilbert space.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Projective Measurements

Projective Measures

◮ Let M : B(R) → P(H) ◮ M(0) = 0 ◮ M(R) = 1 ◮ M(∪j(Bj)) = j M(Bj) for any disjoint denumerable family

Bj.

◮ M(Bc) = 1 − M(B) = (M(B))⊥

Connection with observables M is a projective measure. It defines an observable (because of the spectral resolution theorem). P(H) is an event algebra (YES-NO tests).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Projective Measurements

Projective Measures

◮ Let M : B(R) → P(H) ◮ M(0) = 0 ◮ M(R) = 1 ◮ M(∪j(Bj)) = j M(Bj) for any disjoint denumerable family

Bj.

◮ M(Bc) = 1 − M(B) = (M(B))⊥

Connection with observables M is a projective measure. It defines an observable (because of the spectral resolution theorem). P(H) is an event algebra (YES-NO tests).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Projective Measurements

Projective Measures

◮ Let M : B(R) → P(H) ◮ M(0) = 0 ◮ M(R) = 1 ◮ M(∪j(Bj)) = j M(Bj) for any disjoint denumerable family

Bj.

◮ M(Bc) = 1 − M(B) = (M(B))⊥

Connection with observables M is a projective measure. It defines an observable (because of the spectral resolution theorem). P(H) is an event algebra (YES-NO tests).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Projective Measurements

Projective Measures

◮ Let M : B(R) → P(H) ◮ M(0) = 0 ◮ M(R) = 1 ◮ M(∪j(Bj)) = j M(Bj) for any disjoint denumerable family

Bj.

◮ M(Bc) = 1 − M(B) = (M(B))⊥

Connection with observables M is a projective measure. It defines an observable (because of the spectral resolution theorem). P(H) is an event algebra (YES-NO tests).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Projective Measurements

Projective Measures

◮ Let M : B(R) → P(H) ◮ M(0) = 0 ◮ M(R) = 1 ◮ M(∪j(Bj)) = j M(Bj) for any disjoint denumerable family

Bj.

◮ M(Bc) = 1 − M(B) = (M(B))⊥

Connection with observables M is a projective measure. It defines an observable (because of the spectral resolution theorem). P(H) is an event algebra (YES-NO tests).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Projective Measurements

Projective Measures

◮ Let M : B(R) → P(H) ◮ M(0) = 0 ◮ M(R) = 1 ◮ M(∪j(Bj)) = j M(Bj) for any disjoint denumerable family

Bj.

◮ M(Bc) = 1 − M(B) = (M(B))⊥

Connection with observables M is a projective measure. It defines an observable (because of the spectral resolution theorem). P(H) is an event algebra (YES-NO tests).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

How Are Questions Structured?

The set of events of many scientific theories can be endowed with well defined structures:

◮ Events in CM (subsets of phase space) form a boolean

algebra (a classical logic).

◮ Events in QM (closed subspaces of Hilbert space) form an

• rthomodular lattice (a quantum logic).

Connection with observables This is of great generality, because one can conceive more general theories in principle.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

How Are Questions Structured?

The set of events of many scientific theories can be endowed with well defined structures:

◮ Events in CM (subsets of phase space) form a boolean

algebra (a classical logic).

◮ Events in QM (closed subspaces of Hilbert space) form an

• rthomodular lattice (a quantum logic).

Connection with observables This is of great generality, because one can conceive more general theories in principle.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

How Are Questions Structured?

The set of events of many scientific theories can be endowed with well defined structures:

◮ Events in CM (subsets of phase space) form a boolean

algebra (a classical logic).

◮ Events in QM (closed subspaces of Hilbert space) form an

• rthomodular lattice (a quantum logic).

Connection with observables This is of great generality, because one can conceive more general theories in principle.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Distributive lattice

◮ 1 x ∨ x = x, x ∧ x = x (idempotence) ◮ 2 x ∨ y = y ∨ x, x ∧ y = y ∧ x (commutativity) ◮ 3 x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity)

◮ 4 x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absortion) ◮ 5 x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (distributivity 1) ◮ 6 x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (distributivity 2)

Boolean algebra An orthocomplemented distributive lattice is a Boolean algebra.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Distributive lattice

◮ 1 x ∨ x = x, x ∧ x = x (idempotence) ◮ 2 x ∨ y = y ∨ x, x ∧ y = y ∧ x (commutativity) ◮ 3 x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity)

◮ 4 x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absortion) ◮ 5 x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (distributivity 1) ◮ 6 x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (distributivity 2)

Boolean algebra An orthocomplemented distributive lattice is a Boolean algebra.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Distributive lattice

◮ 1 x ∨ x = x, x ∧ x = x (idempotence) ◮ 2 x ∨ y = y ∨ x, x ∧ y = y ∧ x (commutativity) ◮ 3 x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity)

◮ 4 x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absortion) ◮ 5 x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (distributivity 1) ◮ 6 x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (distributivity 2)

Boolean algebra An orthocomplemented distributive lattice is a Boolean algebra.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Distributive lattice

◮ 1 x ∨ x = x, x ∧ x = x (idempotence) ◮ 2 x ∨ y = y ∨ x, x ∧ y = y ∧ x (commutativity) ◮ 3 x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity)

◮ 4 x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absortion) ◮ 5 x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (distributivity 1) ◮ 6 x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (distributivity 2)

Boolean algebra An orthocomplemented distributive lattice is a Boolean algebra.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Distributive lattice

◮ 1 x ∨ x = x, x ∧ x = x (idempotence) ◮ 2 x ∨ y = y ∨ x, x ∧ y = y ∧ x (commutativity) ◮ 3 x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity)

◮ 4 x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absortion) ◮ 5 x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (distributivity 1) ◮ 6 x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (distributivity 2)

Boolean algebra An orthocomplemented distributive lattice is a Boolean algebra.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Distributive lattice

◮ 1 x ∨ x = x, x ∧ x = x (idempotence) ◮ 2 x ∨ y = y ∨ x, x ∧ y = y ∧ x (commutativity) ◮ 3 x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity)

◮ 4 x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absortion) ◮ 5 x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (distributivity 1) ◮ 6 x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (distributivity 2)

Boolean algebra An orthocomplemented distributive lattice is a Boolean algebra.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Distributive lattice

◮ 1 x ∨ x = x, x ∧ x = x (idempotence) ◮ 2 x ∨ y = y ∨ x, x ∧ y = y ∧ x (commutativity) ◮ 3 x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity)

◮ 4 x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absortion) ◮ 5 x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (distributivity 1) ◮ 6 x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) (distributivity 2)

Boolean algebra An orthocomplemented distributive lattice is a Boolean algebra.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Modular Identities

◮ 1 x ≤ b −

→ x ∨ (a ∧ b) = (x ∨ a) ∧ b (Modular identity)

◮ 2 x ≤ b −

→ x ∨ (¬x ∧ b) = b (Orthomodular identity) Ortomodular Lattice An orthocomplemented lattice which satisfies 2 is an

• rthomodular lattice. Hilbert spaces of finite dimension are

always modular and the infinite dimensional ones are

• rthomodular.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Modular Identities

◮ 1 x ≤ b −

→ x ∨ (a ∧ b) = (x ∨ a) ∧ b (Modular identity)

◮ 2 x ≤ b −

→ x ∨ (¬x ∧ b) = b (Orthomodular identity) Ortomodular Lattice An orthocomplemented lattice which satisfies 2 is an

• rthomodular lattice. Hilbert spaces of finite dimension are

always modular and the infinite dimensional ones are

• rthomodular.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Boolean algebra

Definitions

Modular Identities

◮ 1 x ≤ b −

→ x ∨ (a ∧ b) = (x ∨ a) ∧ b (Modular identity)

◮ 2 x ≤ b −

→ x ∨ (¬x ∧ b) = b (Orthomodular identity) Ortomodular Lattice An orthocomplemented lattice which satisfies 2 is an

• rthomodular lattice. Hilbert spaces of finite dimension are

always modular and the infinite dimensional ones are

• rthomodular.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Outline

Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov

• R. T. Cox

Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Kolmogorov

Definitions

Kolmogorov

◮ µ : Σ → [0, 1] ◮ satisfying µ(∅) = 0 ◮ µ(Ac) = 1 − µ(A), where (. . .)c means set theoretical

complement and

◮ for any denumerable and pairwise disjoint collection {Ai}i∈I

µ(

i∈I Ai) = i µ(Ai)

Booleanity And Sum Rule Σ is a boolean algebra and it satisfies µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B) (inclusion-exclusion principle). We also have: µ(A ∪ B) ≤ µ(A) + µ(B)

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Kolmogorov

Definitions

Kolmogorov

◮ µ : Σ → [0, 1] ◮ satisfying µ(∅) = 0 ◮ µ(Ac) = 1 − µ(A), where (. . .)c means set theoretical

complement and

◮ for any denumerable and pairwise disjoint collection {Ai}i∈I

µ(

i∈I Ai) = i µ(Ai)

Booleanity And Sum Rule Σ is a boolean algebra and it satisfies µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B) (inclusion-exclusion principle). We also have: µ(A ∪ B) ≤ µ(A) + µ(B)

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Kolmogorov

Definitions

Kolmogorov

◮ µ : Σ → [0, 1] ◮ satisfying µ(∅) = 0 ◮ µ(Ac) = 1 − µ(A), where (. . .)c means set theoretical

complement and

◮ for any denumerable and pairwise disjoint collection {Ai}i∈I

µ(

i∈I Ai) = i µ(Ai)

Booleanity And Sum Rule Σ is a boolean algebra and it satisfies µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B) (inclusion-exclusion principle). We also have: µ(A ∪ B) ≤ µ(A) + µ(B)

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Kolmogorov

Definitions

Kolmogorov

◮ µ : Σ → [0, 1] ◮ satisfying µ(∅) = 0 ◮ µ(Ac) = 1 − µ(A), where (. . .)c means set theoretical

complement and

◮ for any denumerable and pairwise disjoint collection {Ai}i∈I

µ(

i∈I Ai) = i µ(Ai)

Booleanity And Sum Rule Σ is a boolean algebra and it satisfies µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B) (inclusion-exclusion principle). We also have: µ(A ∪ B) ≤ µ(A) + µ(B)

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Kolmogorov

Definitions

Kolmogorov

◮ µ : Σ → [0, 1] ◮ satisfying µ(∅) = 0 ◮ µ(Ac) = 1 − µ(A), where (. . .)c means set theoretical

complement and

◮ for any denumerable and pairwise disjoint collection {Ai}i∈I

µ(

i∈I Ai) = i µ(Ai)

Booleanity And Sum Rule Σ is a boolean algebra and it satisfies µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B) (inclusion-exclusion principle). We also have: µ(A ∪ B) ≤ µ(A) + µ(B)

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

• R. T. Cox

Cox’s

• R. T. Cox (1946)

◮ Cox starts with a distributive algebra of propositions. ◮ Studying the symmetries of this algebra, he deduces the

general form of the possible measures.

◮ He deduces Kolmogorovian probability theory with this

method (sum rule, inclusion-exclusion principle, product rule, Bayes formulae, etc.).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

• R. T. Cox

Cox’s

• R. T. Cox (1946)

◮ Cox starts with a distributive algebra of propositions. ◮ Studying the symmetries of this algebra, he deduces the

general form of the possible measures.

◮ He deduces Kolmogorovian probability theory with this

method (sum rule, inclusion-exclusion principle, product rule, Bayes formulae, etc.).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

• R. T. Cox

Cox’s

• R. T. Cox (1946)

◮ Cox starts with a distributive algebra of propositions. ◮ Studying the symmetries of this algebra, he deduces the

general form of the possible measures.

◮ He deduces Kolmogorovian probability theory with this

method (sum rule, inclusion-exclusion principle, product rule, Bayes formulae, etc.).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Quantum Probabilities

Kolmogorov is not the end of the story...

Non-kolmogorovian probability

◮ Define a map s : P(H) → [0; 1] such that ◮ s(0) = 0 (0 is the nule subspace) ◮ s(P⊥) = 1 − s(P) ◮ for any denumerable and parwise orthogonal collection of

projections Pj, s(

j Pj) = j s(Pj).

Quantum Probability s is defined on a non-boolean algebra: P(H) is an

• rthomodular lattice. Gleason: =

⇒ ∀s ∃ρ such that s(P) = tr(ρP). s(a ∨ b) ≥ s(a) + s(b).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Quantum Probabilities

Kolmogorov is not the end of the story...

Non-kolmogorovian probability

◮ Define a map s : P(H) → [0; 1] such that ◮ s(0) = 0 (0 is the nule subspace) ◮ s(P⊥) = 1 − s(P) ◮ for any denumerable and parwise orthogonal collection of

projections Pj, s(

j Pj) = j s(Pj).

Quantum Probability s is defined on a non-boolean algebra: P(H) is an

• rthomodular lattice. Gleason: =

⇒ ∀s ∃ρ such that s(P) = tr(ρP). s(a ∨ b) ≥ s(a) + s(b).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Quantum Probabilities

Kolmogorov is not the end of the story...

Non-kolmogorovian probability

◮ Define a map s : P(H) → [0; 1] such that ◮ s(0) = 0 (0 is the nule subspace) ◮ s(P⊥) = 1 − s(P) ◮ for any denumerable and parwise orthogonal collection of

projections Pj, s(

j Pj) = j s(Pj).

Quantum Probability s is defined on a non-boolean algebra: P(H) is an

• rthomodular lattice. Gleason: =

⇒ ∀s ∃ρ such that s(P) = tr(ρP). s(a ∨ b) ≥ s(a) + s(b).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Quantum Probabilities

Kolmogorov is not the end of the story...

Non-kolmogorovian probability

◮ Define a map s : P(H) → [0; 1] such that ◮ s(0) = 0 (0 is the nule subspace) ◮ s(P⊥) = 1 − s(P) ◮ for any denumerable and parwise orthogonal collection of

projections Pj, s(

j Pj) = j s(Pj).

Quantum Probability s is defined on a non-boolean algebra: P(H) is an

• rthomodular lattice. Gleason: =

⇒ ∀s ∃ρ such that s(P) = tr(ρP). s(a ∨ b) ≥ s(a) + s(b).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Quantum Probabilities

Kolmogorov is not the end of the story...

Non-kolmogorovian probability

◮ Define a map s : P(H) → [0; 1] such that ◮ s(0) = 0 (0 is the nule subspace) ◮ s(P⊥) = 1 − s(P) ◮ for any denumerable and parwise orthogonal collection of

projections Pj, s(

j Pj) = j s(Pj).

Quantum Probability s is defined on a non-boolean algebra: P(H) is an

• rthomodular lattice. Gleason: =

⇒ ∀s ∃ρ such that s(P) = tr(ρP). s(a ∨ b) ≥ s(a) + s(b).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Quantum Probabilities

Questions

◮ What happens if the Cox’s method is applied to

non-boolean algebras?

◮ Does the underlying logical structure of a given theory

determines its probability theory? What happens with information?

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions Quantum Probabilities

Questions

◮ What happens if the Cox’s method is applied to

non-boolean algebras?

◮ Does the underlying logical structure of a given theory

determines its probability theory? What happens with information?

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

von Neumann: “In the quantum mechanical machinery the situation is quite different. Namely instead of the sets use the linear sub-sets of a suitable space, say of a Hilbert

• space. The set theoretical situation of logics is

replaced by the machinery of projective geometry, which is in itself quite simple.”

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

“I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more. After all, Hilbert-space (as far as quantum-mechanical things are concerned) was

• btained by generalizing Euclidean space, footing on

the principle of “conserving the validity of all formal rules”. [...] Thus Hilbert-space is the straightforward generalization of Euclidean space, if one considers the vectors as the essential notions. Now we begin to believe, that it is not the vectors which matter but the lattice of all linear (closed)

• subspaces. [...]”

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

von Neumann: “However, all quantum mechanical probabilities are defined by inner products of vectors. Essentially if a state of a system is given by one vector, the transition probability in another state is the inner product of the two which is the square of the cosine of the angle between them. In other words, probability corresponds precisely to introducing the angles geometrically.” (vN

• Amsterdam Talk).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

“In order to have probability all you need is a concept

• f all angles, I mean angles other than 90o. 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

• rthogonal space, those mappings of this space on

itself, which leave orthogonality intact, leave all the 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

• rdinary concepts of logics are fixed under some

isomorphic transformation, all of probability theory is already fixed.” (vN Amsterdam Talk).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

◮ What is the meaning of the connection between Geometry

and Logic in the above quotations?

◮ When confronting with the empirical propositions of QM we

are facing essentially a Geometry, which is at the same time a Logic. In which sense?

◮ But this Geometry is not the geometry of classical

space-time. Quite on the contrary, is the geometrical form in which quantum events are organized. Space-time description is only a feature of it.

◮ And of course, this geometrical form has an internal logical

structuration, which is the quantum logic.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

◮ What is the meaning of the connection between Geometry

and Logic in the above quotations?

◮ When confronting with the empirical propositions of QM we

are facing essentially a Geometry, which is at the same time a Logic. In which sense?

◮ But this Geometry is not the geometry of classical

space-time. Quite on the contrary, is the geometrical form in which quantum events are organized. Space-time description is only a feature of it.

◮ And of course, this geometrical form has an internal logical

structuration, which is the quantum logic.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

◮ What is the meaning of the connection between Geometry

and Logic in the above quotations?

◮ When confronting with the empirical propositions of QM we

are facing essentially a Geometry, which is at the same time a Logic. In which sense?

◮ But this Geometry is not the geometry of classical

space-time. Quite on the contrary, is the geometrical form in which quantum events are organized. Space-time description is only a feature of it.

◮ And of course, this geometrical form has an internal logical

structuration, which is the quantum logic.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

◮ What is the meaning of the connection between Geometry

and Logic in the above quotations?

◮ When confronting with the empirical propositions of QM we

are facing essentially a Geometry, which is at the same time a Logic. In which sense?

◮ But this Geometry is not the geometry of classical

space-time. Quite on the contrary, is the geometrical form in which quantum events are organized. Space-time description is only a feature of it.

◮ And of course, this geometrical form has an internal logical

structuration, which is the quantum logic.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions The von Neumann program

von Neumann: “This means, however, that one has a formal mechanism, in which logics and probability theory arise simultaneously and are derived simultaneously.” (vN - Amsterdam Talk).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Outline

Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov

• R. T. Cox

Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

We proceed as follows:

◮ Let L be an orthomodular lattice. ◮ We assume that L represents the propositional structure of

a given system.

◮ We assume that there is a given state of affairs determined

by the preparation of the system.

◮ We define a function s : L −

→ R such that s(a) ≥ 0∀a ∈ L and it is order preserving (a ≤ b − → s(a) ≤ s(b)).

◮ Under these rather general assumption a probability theory

can be developed.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

We proceed as follows:

◮ Let L be an orthomodular lattice. ◮ We assume that L represents the propositional structure of

a given system.

◮ We assume that there is a given state of affairs determined

by the preparation of the system.

◮ We define a function s : L −

→ R such that s(a) ≥ 0∀a ∈ L and it is order preserving (a ≤ b − → s(a) ≤ s(b)).

◮ Under these rather general assumption a probability theory

can be developed.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

We proceed as follows:

◮ Let L be an orthomodular lattice. ◮ We assume that L represents the propositional structure of

a given system.

◮ We assume that there is a given state of affairs determined

by the preparation of the system.

◮ We define a function s : L −

→ R such that s(a) ≥ 0∀a ∈ L and it is order preserving (a ≤ b − → s(a) ≤ s(b)).

◮ Under these rather general assumption a probability theory

can be developed.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

We proceed as follows:

◮ Let L be an orthomodular lattice. ◮ We assume that L represents the propositional structure of

a given system.

◮ We assume that there is a given state of affairs determined

by the preparation of the system.

◮ We define a function s : L −

→ R such that s(a) ≥ 0∀a ∈ L and it is order preserving (a ≤ b − → s(a) ≤ s(b)).

◮ Under these rather general assumption a probability theory

can be developed.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

We proceed as follows:

◮ Let L be an orthomodular lattice. ◮ We assume that L represents the propositional structure of

a given system.

◮ We assume that there is a given state of affairs determined

by the preparation of the system.

◮ We define a function s : L −

→ R such that s(a) ≥ 0∀a ∈ L and it is order preserving (a ≤ b − → s(a) ≤ s(b)).

◮ Under these rather general assumption a probability theory

can be developed.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

It is possible to show that:

◮ s({ai}i∈N) = ∞ i=1 s(ai) ◮ s(¬a) = 1 − s(a) ◮ s(0) = 0

The probability theory defined by these equations is non classical in the general case. If L is not Boolean, it may happen that (a ∧ b) ∨ (a ∧ ¬b) < a. and then, (using (a ∧ ¬b)⊥(a ∧ b)), s((a ∧ ¬b) ∨ (a ∧ b)) = s(a ∧ ¬b) + s(a ∧ b) ≤ s(a). But any Kolmogorovian probability satisfies s(a) = s(a ∧ b) + s(a ∧ ¬b). F . Holik, A. Plastino and M. S´ aenz, Annals Of Physics, 340, 1, 293-310, January (2014).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

Figure : Schematic diagram of the method.

Quantum Probability But probability measures always define a convex set!!!

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions General Method

Figure : Schematic diagram of the method.

Quantum Probability But probability measures always define a convex set!!!

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Outline

Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov

• R. T. Cox

Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Classical systems

• ✠

❅ ❅ ❅ ❅ ❅ ❘

LCM = LCM1 × LCM2 LCM1 LCM2 π1 π2

Figure : Going down with set-theoretical projections π1 and π2

.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Improper mixtures

|ψ = 1 2(|+ ⊗ |− + |− ⊗ |+) (1) ρ = |ψψ| (2) ρ1 = tr2(ρ) = 1 2(|++| + |−−|) (3) Improper mixtures But (3) is not a pure state!!! And thus, it is not an element of the Piron lattice.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Improper mixtures

|ψ = 1 2(|+ ⊗ |− + |− ⊗ |+) (1) ρ = |ψψ| (2) ρ1 = tr2(ρ) = 1 2(|++| + |−−|) (3) Improper mixtures But (3) is not a pure state!!! And thus, it is not an element of the Piron lattice.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Quantum systems (Piron lattices)

• ✠

❅ ❅ ❅ ❅ ❅ ❘

LvN LvN1 LvN2 ξ1? ξ2?

Figure : We cannot apply partial traces in order to go down from LvN to LvN1, and LvN2.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

But if we extend the lattices...

C = {ρ : ρ = ρ†; ρ ≥ 0; tr(ρ) = 1}. (4) Improper mixtures The faces of C form a lattice (in a canonical way) which is isomorphic to the von Neumann lattice. The set o convex subsets of C can be endowed with a lattice structure also.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

But if we extend the lattices...

C = {ρ : ρ = ρ†; ρ ≥ 0; tr(ρ) = 1}. (4) Improper mixtures The faces of C form a lattice (in a canonical way) which is isomorphic to the von Neumann lattice. The set o convex subsets of C can be endowed with a lattice structure also.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

But if we extend the lattices...

• ✠

❅ ❅ ❅ ❅ ❅ ❘ ✻ ❄ ✲ ✛

LC LC1 LC2 LC1 × LC2 Λ τ τ1 τ2 π1 π2

Figure : The different maps between LC1, LC2, LC1 × LC2, and LC

F . Holik, C. Massri and N. Giancaglini, “Convex Quantum Logic”, International Journal of Theoretical Physics May 2012, Volume 51, Issue 5, pp 1600-1620.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Outline

Abstract General Framework Boolean algebra Quantum probabilities Kolmogorov

• R. T. Cox

Quantum Probabilities The von Neumann program General Scheme General Method Compound quantum systems Conclusions

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Summarizing, the path that we have followed is:

◮ We have presented a formal framework in which probability

theory is developed out of the algebraic properties of the underlying logic of the theory.

◮ According to von Neumann, a framework in which logic and

probability theory are derived simultaneously is in order.

◮ We start with the logico-geometrical organization L of

events in a theory.

◮ This structuration determines the form of the probability

measures (in a certain sense, in the direction pointed by von Neumann). But this leads to the representation of states as a convex set (in some examples, like QM, the lattice of faces of the convex set is isomorphic to the event structure).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Summarizing, the path that we have followed is:

◮ We have presented a formal framework in which probability

theory is developed out of the algebraic properties of the underlying logic of the theory.

◮ According to von Neumann, a framework in which logic and

probability theory are derived simultaneously is in order.

◮ We start with the logico-geometrical organization L of

events in a theory.

◮ This structuration determines the form of the probability

measures (in a certain sense, in the direction pointed by von Neumann). But this leads to the representation of states as a convex set (in some examples, like QM, the lattice of faces of the convex set is isomorphic to the event structure).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Summarizing, the path that we have followed is:

◮ We have presented a formal framework in which probability

theory is developed out of the algebraic properties of the underlying logic of the theory.

◮ According to von Neumann, a framework in which logic and

probability theory are derived simultaneously is in order.

◮ We start with the logico-geometrical organization L of

events in a theory.

◮ This structuration determines the form of the probability

measures (in a certain sense, in the direction pointed by von Neumann). But this leads to the representation of states as a convex set (in some examples, like QM, the lattice of faces of the convex set is isomorphic to the event structure).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

Summarizing, the path that we have followed is:

◮ We have presented a formal framework in which probability

theory is developed out of the algebraic properties of the underlying logic of the theory.

◮ According to von Neumann, a framework in which logic and

probability theory are derived simultaneously is in order.

◮ We start with the logico-geometrical organization L of

events in a theory.

◮ This structuration determines the form of the probability

measures (in a certain sense, in the direction pointed by von Neumann). But this leads to the representation of states as a convex set (in some examples, like QM, the lattice of faces of the convex set is isomorphic to the event structure).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

◮ The system L and its subsystems L1 and L2 have

associated convex sets.

◮ In this way, the logic, the geometry and the probability

theory are presented as an organized corpus, ending in the convex set of density matrixes (the most general states).

◮ These convex sets have their own lattices. Using these, we

can link states of the compound system to states of the

• subsystems. And thus, describe entanglement inside the

QL approach.

◮ Probably, this could be a step forward in the treatment of

compound quantum systems n the QL approach.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

◮ The system L and its subsystems L1 and L2 have

associated convex sets.

◮ In this way, the logic, the geometry and the probability

theory are presented as an organized corpus, ending in the convex set of density matrixes (the most general states).

◮ These convex sets have their own lattices. Using these, we

can link states of the compound system to states of the

• subsystems. And thus, describe entanglement inside the

QL approach.

◮ Probably, this could be a step forward in the treatment of

compound quantum systems n the QL approach.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

◮ The system L and its subsystems L1 and L2 have

associated convex sets.

◮ In this way, the logic, the geometry and the probability

theory are presented as an organized corpus, ending in the convex set of density matrixes (the most general states).

◮ These convex sets have their own lattices. Using these, we

can link states of the compound system to states of the

• subsystems. And thus, describe entanglement inside the

QL approach.

◮ Probably, this could be a step forward in the treatment of

compound quantum systems n the QL approach.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

◮ The system L and its subsystems L1 and L2 have

associated convex sets.

◮ In this way, the logic, the geometry and the probability

theory are presented as an organized corpus, ending in the convex set of density matrixes (the most general states).

◮ These convex sets have their own lattices. Using these, we

can link states of the compound system to states of the

• subsystems. And thus, describe entanglement inside the

QL approach.

◮ Probably, this could be a step forward in the treatment of

compound quantum systems n the QL approach.

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com

Abstract General Framework Quantum probabilities General Scheme Compound quantum systems Conclusions

References

◮ [R. T. Cox] Cox, R.T. The Algebra of Probable Inference;

The Johns Hopkins Press: Baltimore, MD, USA, (1961).

◮ [QL] G. Birkhoff and J. von Neumann, Annals Math.37

(1936) 823-843.

◮ [Probability and QL] M. R´

edei, The Mathematical Intelligencer, 21, (1999) 7-12.

◮ [Application of Cox 1] P

. Goyal and K. Knuth, Symmetry, 3, 171-206 (2011) doi:10.3390/sym3020171.

◮ [Application of Cox 2] F

. Holik, A. Plastino and M. S´ aenz, Annals Of Physics, 340, 1, 293-310, January (2014).

Logic, geometry and probability theory Federico Holik: olentiev2@gmail.com