This course introduces to the foundations, theory, and applications of quantum logic.

It is presented by the distinguished professor Karl Svozil from the Institute of Theoretical Physics of the Wien University.

The course consists of two days of classes (6hrs of classes approx.), and preceeds the conference Worlds of Entanglement 2019.

**Syllabus of Introduction to Quantum Logic: Philosophical, Mathematical,
Physical, and Computational aspects, foundations of quantum (in) determinism
**

**Day 1 (March 5th, 2019): Foundations of quantum mechanics**: Introduction to Dirac-Von Neumann type formalization of quantum mechanics based on Hilbert space:

*) Hilbert space as (complete) linear vector space endowed with a scalar

product mapping two vectors into scalars (aka numbers)

*) dual spaces, linear functionals and the Riesz representation theorem (identifying linear functionals with scalar products of a unique vector).

*) linear transformations; in particular, adjoint, self-adjoint (symmetric), unitary (orthogonal, permutation) ones

*) orthogonal (versus oblique) projections and their construction from vectors

*) Spectral theorem for normal (self-adjoint, unitary) operators: decomposition in terms of mutually orthogonal projections whose sum is a resolution of unity

+) quantum use of vectors/one-dimensional orthogonal projections identified as pure states

+) mixed states from partial traces; conversely (non-unique) construction of pure states from mixed ones

+) quantum use of self-adjoint operators as observables, and of unitary operators (isometries aka linear transformations which preserve vector length) as quantum state evolution

+) complementarity formalized by non-commutativity

*) Commutativity: definition and finding the maximal operator from which all mutually commuting operators can be functionally defined, and their respective spectral forms

**Day 2 (March 6th, 2019): Foundations of quantum logics
**

*) quantum propositions defined as orthogonal projections (Birkhoff and Von Neumann)

*) formation of contexts as maximal sets of mutually commuting one-dimensional orthogonal projections

*) formation of quantum clouds as collections of intertwining (interconnected) contexts

*) interesting non-classical properties of quantum clouds with respect to classical two-valued states (aka truth-false valuations):

+) true-implies-false clouds (Kochen-Specker, Belinfante, Stairs, Clifton),

+) true-implies-true clouds (Kochen-Specker, Stairs, Clifton, Johansen, Vermaas, Belinfante, Pitowskys, Hardy, Cabello),

+) clouds featuring classical inseparability of quantum propositions (atoms, one-dimensional orthogonal projections) (Kochen-Specker),

+) quantum clouds disallowing classical two-valued states (Kochen-Specker, Pitowsky indeterminism, Abbott-Calude-Conder-Svozil constructions)

(*) audience & speaker permitting and not totally exhausted: partition logics which allow complementarity but not value indefiniteness; interpretable by quasi-classical models such as finite state automata and generalized urn models

*) Strategies to find non-classical probabilities while maintaining classicality for mutually commuting (aka co-measurable) observables: Gleason-type and Wright-type measures

+) Boole-Choquet-Vorob'ev-Froissa

rt-Pitowsky approach to classical Bell-type inequalities

*) some philosophical rants on implications of the aforementioned findings

+) experimental / historical side, speaking about the Bell tests Philippe Grangier´s team did in Orsay in the 80’s, and some more recent developments.

**Reference Books**:

https://link.springer.com/book/10.1007%2F978-3-319-70815-7

https://www.amazon.com/exec/obidos/ASIN/9814021075

Prof. Karl Svozil

Institut für Theoretische Physik, Technische Universität Wien

https://en.wikipedia.org/wiki/Karl_Svozil

https://en.wikipedia.org/wiki/Karl_Svozil

Prof. Philippe Grangier, director of the quantum optics group at the Charles Fabry. The laboratory is a Joint Research Unit between the Optics Graduate School and the CNRS and in partnership with the University Paris-Sud.

https://fr.wikipedia.org/wiki/Philippe_Grangier

**Classical classroom setting + Discussions.**

Program

**05/03/2019**

9:30-10:00

Registration

10:00-10:10

Welcome

10:10-11:30

Class1

11:30-12:00

Coffee Break

12:00-13:30

Class2

13:30-15:00

Lunch

15:00-16:00

Class3

16:00-16:15

Coffee Break

16:15-17:30

Class4

**06/03/2019**

10:00-11:30

Class1

11:30-12:00

Coffee Break

12:00-13:30

Class2

13:30-15:00

Lunch

15:00-16:00

Class3

16:00-16:15

Coffee Break

16:15-17:30

Class4

17:30

Closing Words

60 EUR (47.500 CLP)

Coffee breaks included