1. Subadditivity of geometrical Bell inequalities for Qudits
Dr hab. Marcin Wieśniak, prof. UG

2. Einstein-Podolsky-Rosen (1935)
Assumptions
Quantum mechanics holds
No information cannot be passed with superluminal speed
Hypothesis: QM can be supplemented with additional, hidden parameters, which account for (apparent randomness)
State of two particles
When Alice measures the position of her particle, position of Bob’s particle is known immediately. Likewise, if Alice measures the position, Bob’s particle’s position is known. That would mean that they had to predefined. But quantum mechanics forbids that.
Can a qantum state be supplemented by additional parameters?
Bohr: no need
Bohm: yes!

3. Bell’s theorem (1964) Local realistic models
Speed of usable information is limited
Results of all possible measurements exist prior to their revelation
Observers can freely and independently choose between local alternative measurements
Local realistic models
Locality+Realism+Freedom+Quantum Mechanics=Conjecture

4. Greenberger-Horne-Zeilinger Paradox (1989)
Source
All-versus-nothing paradox

5. Werner-Wolf-Weinfurter-Żukowski-Brukner inequalities (2001)
qubits
Two setting per side
Arbitrarily many particles
Superexponential number of variants
Necessary conditions for violation
Different inequalities
The highest quantum-to-classical ratio occurs when the GHZ paradox is recreated.
Almost all pure states violate these inequalities

6. Condition for violation
The condition for violation is necessary, but not sufficient!

7. Werner states-violation vs entanglement
Peres-Horodecki criterion
Unentangled states remain semi-positive under partial tranposition.
Werner state-mixture of a maximally enangled state and the white noise
entanglement
1/Sqrt(2)
P=1
1/3
Best known BI
(Vertesi 0,6595)
Violation

8. Variants Werner-Wolf-Weinfurter-Żukowski-Brukner
Laskowski, Paterek, Żukowki, Brukner
Wieśniak Nawareg Żukowski
Uses sucorrelations
Very weak
Can be used to detect nonclassicality of globally uncorrelated states
More settings per side
Violated by all pure state
Necessary and sufficient condition for violation

9. Geometrical Bell Inequalities-concept
Imagine convex set S of vectors (an individual member of this set will be s). We want to verify if vector v belongs to this set. If this is not the case, there is a single component of v sticking out of S. Thus when we find such a component it is certain that v is not a member of S S v
Our vectors will be values of the correlation function and the local realistic model prediction for various local apparata stettings.

10. First attempts M Żukowski, Phys. Lett. A 177, 290 (1993)
D. Kaszlikowski, M. Żukowski. PRA 61, (2000)
Systtem: two-qubit Werner state
Observers can freely choose any local projection.

11. Rotational Invariance as an additional constraint on local realism K
Rotational Invariance as an additional constraint on local realism K. Nagata, W. Laskowski, M. Wieśniak, M. Żukowski, PRL 93, (2004)
Violation for GHZ states grows faster than for WWWŻB inequality.

12. Rotational Invariance as new direction in research
Nagata
Wieśniak+Maruyama,Dutta
Laskowski
+Paterek …
PRA72,012325
jPA40,13101
JPA41,155308
Chin.PL27,030305
Quadratic entanglement Criteria
PRA 85,
Bell inequalities for qudit GHZ states
JPA 49,
Bell inequalities with discrete number of observables (PRA 74,062109)
Multiqubit entanglement criteria
PRL 100,

13. Quadratic Entanglement CriteriaS
The set of separable states is convex->mixing the correlation tenrs cannot increase the norm.
Products of Pauli matrices either commute or anticommute. If they anticommute,
We then introduce cut-anticommutativity. Two operators are said to anticommute with respect to a cut if they anticommute on one part, or the the other. For states separable in this cut, similar complementarity relations hold yx xx
One choose a set of operators and represents them as vertices, then connects them with edges if they anticommute. Then 0s and 1s are assigned to vertices, but two 1s cannot be connected by an edge. The procedure is repeated for partitions xy yy

14. Bell inequality avoiding the KS contradicton.

15. How to measure higher-dimensional systems
Complex Roots of Unity
Unitary
Traceless
Non-degenerate
Real Eigenvalues
Hermitian
Pauli Matrices:
Hermitian
Unimodular eigenvalues
Form-preserving under rotations
Tracelesss
(Necessarily) non-degenate
All (two) eigenvalues are in the same relation to each other
Uniformly distributed vectors
Hermitian
Unimodular
Traceless
Locally-non-degenerate
Eigenvalues in a similar realtion to each other
Scalar function of sum of local outcomes
Traceless
Hermitian
Global observable
They are not related by a rotation, but we can still use a map between them

16. Geometric inequalities as a test field
Wieśniak, Dutta, Ryu, JPA 49,
Local vectors
Strategy 1: multiplying real local oucomes
(d-1)/d,-1/d,-1/d
Strategy II: summing local outcomes modulo d. Each detector gets a label 0 to d-1. Labels of detectors that have clicked are summed modulo d
Strategy III: Complex outcomes-roots of Unity

18. New inequalities
Conjecture: local realistic models are dictated by the optimal vectors in any slice of the correlation function
For more parties, the characteristic function has more complicated
form

19. Fortunately, the correlation function is self-convoled with the characteric function, which was confirmed by numerical integration D 2 3 4 5 6 7
L(d)
1.571
1,98
2,416
2,764
3,090
3,507
A(d) 1
1.01
1.010
1.0223
0,949

20. Conclusions
Geometric criterion is a versalite test for non-classicality
Various systems and their interpertations can be used
Only information content is relevant, not the specific description
Quadratic form allows for many transformations
For all dimensions, we observe a rapid exponential growth of QCR ratio with the number of subsystems