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Connectivity Structures of Quantum Entanglement

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Stéphane Dugowson

on 19 July 2014

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Transcript of Connectivity Structures of Quantum Entanglement

2. Connectivity spaces
3. Two general quantum formalisms
1. Introduction
3.2 State vector
formalism

3.3 Density operator
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
)

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
Full transcript