**2. Connectivity spaces**

**3. Two general quantum formalisms**

**1. Introduction**

**3.2 State vector**

formalism

formalism

**3.3 Density operator**

formalism

formalism

**3.1 Common aspects of the two formalisms**

**Recalls & notations**

**Connectivity structures of quantum (entangled) states**

**4. States**

**5. Multi-local devices**

**non-probabilistic approachs**

**Connectivity structure of a familly of random variables**

**6. Random variables**

**8. Conclusion**

**References**

Workshop on Diffeology etc.

Aix en Provence

June 25-26-27, 2014

**Connectivity Structures of Quantum Entanglement**

**by Stéphane Dugowson (**

Supméca

)

Supméca

)

**Connectivity structures of multi-local devices**

2.1 Definition

2.2 Examples

2.4. Generation

2.3 Brunn theorem

2.5 Connectivity Order

Diffeological spaces

June 2013,

in Aix-en-provence

The connected parts of a diffeological space

constitute a connectivity structure...

Any topological space gives rise (functorally) to some connectivity spaces (topological connectedness, arcwise connectedness, etc.)

**Thank you !**

The connectivity structure generated by

A

1

2

4

Here, only finite dimensional spaces

Quantum systems

?

Measurement

Composite systems

isolated

States

Projective measurement

Composite systems

and intrication

States

Projective measurement

Composite systems

and intrication

(and we use unitary vectors to represent it)

For a (sub)system, it is possible to have no state

What could be a (projective) measurement on a subsystem for which it is possible to have no state ???

Partial measurement

Rmk : The Schmidt decomposition theorem is usefull only when there are

2

subsystems

trace

Lien avec le formalisme FV : états purs

Rappel : produit tensoriel d'opérateurs

Correlation and separability in terms of density states

State of a subsystem

partial trace

?

Brunnian conjectures

Relations between all those structures

Naturality, functoriality, physicality, etc...

Definition

Quantum experiments

Separability (or locality)

Definition of tensorial connectivity structures

Examples

?

Disentanglement connectivity structures

Density connectivity structures

Sugita idea

Correlation and entanglement of a density state on J

Correlation connectivity structure

Entanglement connectivity structure

To see the connectivity structure of a link...

you can

cut

or

forget

some components

what about quantum entanglement ?

to cut = to measure

to forget = to consider states of subsystems

(disantanglement)

Aravind, 1997

Borromean Entanglement Revisited

Ayumu Sugita 2007

An interesting analogy between quantum entangled states and topological links was suggested by Aravind. In particular, he emphasized a connection between the Greenberger-Horne-Zeilinger (GHZ) state and the Borromean rings. However, he made the connection in a way that depends on the choice of measurement basis. We reconsider it in a basis-independent way by using the reduced density matrix.

of the GHZ state

Generalization : connectivity structures (better than links) associated with any (finite) entangled quantum system

J

-states and their partial separability

Entanglement types on

J

Disentanglement connectivity structure on

I

EPR

GHZ

eventail

of connectivity structures

of a global state

Decisive experiment on L = I-J

(experiment

on

L, decisive

for

J)

Entanglement types on

J

of a global state

States on

J

resulting from a decisive experiment

Aravind's idea

globally entangled

jumbled

globally separable

partially

well-entangled

totally jumbled

partially

well-separable

clearly separable

totally separable

finest

coarsest

GI

BIP

MT

IP

ML

NCS

Globally entangled

Partially Well-Entangled

(bonne intrication partielle)

Totally Jumbled

Partially entangled

Jumbled

Not clearly separable

globally entangled

jumbled

globally separable

partially

well-entangled

totally jumbled

partially

well-separable

clearly separable

totally separable

globally entangled

jumbled

globally separable

partially

well-entangled

totally jumbled

partially

well-separable

clearly separable

totally separable

globally entangled

jumbled

globally separable

partially

well-entangled

totally jumbled

partially

well-separable

clearly separable

totally separable

globally entangled

jumbled

globally separable

partially

well-entangled

totally jumbled

partially

well-separable

clearly separable

totally separable

globally entangled

jumbled

globally separable

partially

well-entangled

totally jumbled

partially

well-separable

clearly separable

totally separable

globally entangled

jumbled

globally separable

partially

well-entangled

totally jumbled

partially

well-separable

clearly separable

totally separable

Sub-devices

Tensor product

Some classes of devices

Thinness

Tensorial structures

Multilocal Devices

Domanial structures

Ludic Structures

A notion due to C. Chalons, in a thesis supervised by A. Khelif.

A ludic degree is a class of some relations (multilocal classes), for some equivalence relation with local properties.

A. Khelif defined the connectivity structure of every ludic degree.

Ludic degrees seem to provide a good frame to study local and non-local properties of quantum mechanics

Brunnian conjectures

Relations between all those structures

Naturality, functoriality, physicality, etc...

Relations between state structures and device structures ?

A classical correlation :

the Bertlmann socks

as multilocal devices

**7. Recapitulative table**

Séminaire CLE

(Catégories, Logique, Etc...)

Université Paris Diderot

June 18, 2014

Séminaire QUARTZ-LISMMA

Supméca Paris

July 2, 2014

https://sites.google.com/site/logiquecategorique/Contenus/CSQE

"Structures connectives de l'intrication quantique" (July 9, 2014)

Text on the open archive HAL :

http://hal.archives-ouvertes.fr/hal-01025949

http://hal.archives-ouvertes.fr/docs/01/02/59/49/PDF/20140718%20SCdIQ.pdf

Padmanabhan K. Aravind

"Borromean entanglement of the GHZ state" (1997)

http://users.wpi.edu/~paravind/Publications/borrom.pdf

Ayumu Sugita

"Borromean Entanglement Revisited" (2007)

http://arxiv.org/abs/0704.1712

https://sites.google.com/site/logiquecategorique/Contenus/CSQE

Lecture videos

Stéphane Dugowson