1. Detecting nonlocality in many-body quantum states
Jordi Tura
DIQIP & QAlgo Joint meeting
15th – May – 2014

2. Detecting nonlocality in many-body quantum states
Jordi Tura1
Remigiusz Augusiak1
Ana Belén Sainz1 1) 2)
Tamás Vertesi2
Maciej Lewenstein1, 3
Antonio Acín1, 3 3)
15th - May
Jordi Tura

3. Summary Motivation Structure of the Local Polytope
A parametrized class of 2-body symmetric BI
Detection of Physical States
Possible Generalizations
15th - May
Jordi Tura

4. Summary Motivation Structure of the Local Polytope
A parametrized class of 2-body symmetric BI
Detection of Physical States
Possible Generalizations
15th - May
Jordi Tura

5. Motivation Least effort to detect nonlocality
Tests measuring only 2 parties?
For entanglement works!
Experimentally friendlier
Precise measurements of macroscopic observables
Drawbacks
Tightness (weaker Bell Inequalities)
Extra assumptions when performing a Bell Test
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6. Preliminaries Bell Experiment Bell Inequality Local Polytope A B C NS
LHV Q NS
Alice A
Bob B
Charlie C
Bell Inequality
Local Polytope
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7. Preliminaries All Bell Inequalities Convex Hull problem
(n,m,d) scenario
Dimension of Local Polytope
Number of vertices
Complexity
[B. Chazelle, An optimal convex hull algorithm in any fixed dimension, Discrete Comput. Geom. 10, (1993)]
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8. Practical problems Most restrictive parameter:
Take only 2-body correlators
Problem still too difficult!
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9. Idea Further reduce the dimension of Action of a Symmetry Group: A B D
J. T., A. B. Sainz, T. Vértesi, A. Acín, M. Lewenstein, R. Augusiak
Translationally invariant Bell inequalities with two-body correlators, arXiv:
Submitted to J. Phys. A. 50 years of Bell's Theorem
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10. Symmetry groups to consider
Full Cycle
Translational Invariance + Nearest Neighbors
Spin chains with periodic boundary conditions
Nearest Neighbors. D = 6 A B D C
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11. Symmetry groups to consider
All permutations
Full Symmetry
Dicke states
D = 5 A B D C
J. T., R. Augusiak, A. B. Sainz, T. Vértesi, M. Lewenstein, A. Acín
Detecting nonlocality in many-body quantum states, arXiv:
Being reviewed by referees in Science
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12. Summary Structure of the Local Polytope Motivation
A parametrized class of 2-body symmetric BI
Detection of Physical States
Possible Generalizations
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13. Fully Symmetric Case
Structure of the Local Polytope: (N,2,2) scenario
We want to construct Bell Inequalities of the form
Deterministic values assigned to
Depend on strategy? A B D C
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14. Fully Symmetric Case
Structure of the Local Polytope
Fully symmetric correlators don't take into account order of parties.
Only #local strategies matters
We can express
in terms of a,b,c,d.
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15. Fully Symmetric Case More precisely, such expressions are:
Structure of the Local Polytope
More precisely, such expressions are:
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16. Fully Symmetric Case
Structure of the Local Polytope
The polytope can be generated just from 3 parameters!
Actually, 2 parameters are enough
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17. Fully Symmetric Case Only points in generate vertices of
Structure of the Local Polytope
Only points in generate vertices of
Polytope
Dimension
Vertices
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18. A class of 2-body symmetric BI
Symmetries in the BI
8 symmetries only
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19. Summary A parametrized class of 2-body symmetric BI Motivation
Structure of the Local Polytope
A parametrized class of 2-body symmetric BI
Detection of Physical States
Possible Generalizations
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Jordi Tura

20. A class of 2-body symmetric BI
General form
The Bell Inequalities constraining
A particular choice of coefficients:
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21. A class of 2-body symmetric BI
Tightness
tight
non-tight
When is (1) a tight Bell Inequality?
Amount of vertices saturated by (1):
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22. A class of 2-body symmetric BI
Density of BI
How many of the total are of this form?
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23. A class of 2-body symmetric BI
Quantum Violation
Numerical evidence
Highest violation when
Permutational invariance implies a block-decomposition of the Bell Operator
(n,2,2) scenario
Qubits suffice
Real Observables
Highest quantum violation found in the Symmetric Space
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24. A class of 2-body symmetric BI
Quantum Violation – A closer look
Bell Operator is pentadiagonal in the Dicke basis
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25. A class of 2-body symmetric BI
Quantum Violation – A closer look
The state maximally violating (1) can be written as
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26. A class of 2-body symmetric BI
Quantum Violation – A closer look
The spectrum of depends only on
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27. A class of 2-body symmetric BI
Quantum Violation – A closer look
The spectrum of depends only on
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28. A class of 2-body symmetric BI
Maximal Quantum Violation
Example
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29. A class of 2-body symmetric BI
Maximal Quantum Violation
Example
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30. A class of 2-body symmetric BI
Reduced state is local!
Bipartite reduced density matrix
does not violate CHSH
But multipartite nonlocality is detected
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31. A class of 2-body symmetric BI
Robustness
Misalignments
Against white noise → 20%
Particle losses
Known #lost → 33%
Unknown #lost → 10%
Depolarization → p ~ 0.1
Detection efficiency → 92%
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32. Summary Detection of Physical States Motivation
Structure of the Local Polytope
A parametrized class of 2-body symmetric BI
Detection of Physical States
Possible Generalizations
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Jordi Tura

33. Physical States The Lipkin-Meshkov-Glick Hamiltonian:
Lipkin-Meshkov-Glick Model
The Lipkin-Meshkov-Glick Hamiltonian:
Solvable analytically → Half-filled Dicke state
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34. Physical States Class of 2-body symmetric BI n = 2 → We recover CHSH
Lipkin-Meshkov-Glick Model
Class of 2-body symmetric BI
n = 2 → We recover CHSH
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35. Physical States We can prove for these measurements
Lipkin-Meshkov-Glick Model
We can prove for these measurements
that QV exists for any n, tending to -1
Numerically, measurements can be refined to achieve a better scaling
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36. Physical States All Dicke States can be detected
Lipkin-Meshkov-Glick Model
All Dicke States can be detected
High/Low number of excitations
Medium number of excitations
Even n
Odd n
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37. Physical States
Implementation
Symmetrized correlators can be obtained directly from collective spin components
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38. Summary Possible Generalizations Motivation
Structure of the Local Polytope
A parametrized class of 2-body symmetric BI
Detection of Physical States
Possible Generalizations
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39. Possible Generalizations
General (n,m,d) scenario with k-body correlators
Need to extend the definition of correlator (d>2)
Number of basic variables:
Candidates to vertex:
Vertices described by polynomials of degree k
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40. Possible Generalizations
General (n,m,d) scenario with k-body correlators
Given a vector of correlators, is there an efficient way to determine whether it is local?
can be determined with LP
constraints,
Polynomial in n, but inefficient for practical purposes
parametrize the algebraic set interpolating
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41. Possible Generalizations
General (n,m,d) scenario with k-body correlators
Given an algebraic variety , it is NP-Hard to check if
Closely related to deciding if a polynomial is positive
But one can relax this problem to a SDP Hierarchy of Theta-bodies (↔ a polynomial is s.o.s.)
[G. Blekherman, P. A. Parrilo, R. Thomas, Semidefinite Optimization and Convex Algebraic Geometry, MPS-SIAM Series on Optimization]
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42. Conclusions Minimal amount of knowledge to certify nonlocality
Two-body statistics
Practical parametrization of local polytope
3-parameter “dense” class of BI
Weaker BI by construction
Still powerful enough to detect Physical States
Robust to many sources of errors
Method developed can be applied to more complex scenarios
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43. Thanks for your attention!
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44. 15th - May
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45. A class of 2-body symmetric BI
Quantum Violation
Numerical evidence
Lowest eigenstate has the property
We can restrict our analysis to
Bell Operator is Pentadiagonal
Every shifts 1 position in the Dicke States representation
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46. A class of 2-body symmetric BI
Tight classical bound
Considering
can we get its classical bound? If
Tight classical bound is
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47. Translationally Invariant Case
Class of BI
Just one class of BI
Has QV for odd N=2k+1
Constant QV
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48. Translationally Invariant Case
Violation by entangled state
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49. Translationally Invariant Case
Violation by entangled state
Can we get a state of this form?
Correlators depend on arrangement of colors in the graph
Necklaces (combinatorics)
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50. Translationally Invariant Case
Violation by entangled state
Finding classical bound is not so easy either
Equivalent to finding a cycle of length N=2k+1 minimizing its cost.
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51. Fully Symmetric Case
Structure of the Local Polytope
We can think of a Local Strategy as a coloring of a graph:
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52. Fully Symmetric Case Then it is easy to obtain the expressions for
Structure of the Local Polytope
Then it is easy to obtain the expressions for
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Jordi Tura

53. Fully Symmetric Case Then it is easy to obtain the expressions for
Structure of the Local Polytope
Then it is easy to obtain the expressions for
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Jordi Tura

54. Fully Symmetric Case Then it is easy to obtain the expressions for
Structure of the Local Polytope
Then it is easy to obtain the expressions for
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Jordi Tura

55. Fully Symmetric Case Proving existence of QV Idea
Quantum Violation
Proving existence of QV
Idea
All diagonally dominant matrices are positive semidefinite
Pentadiagonal matrix is almost split in boxes
Approximate negative eigenvalue from not diagonally dominant box
Numerically good approximation
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56. A class of 2-body symmetric BI
Idea
What happens if, instead of having
We consider ?
Smooth manifold interpolating the Local Polytope
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57. A class of 2-body symmetric BI
Finding Classical Bound
Taking
We may increase the #vertices, however BI are still valid
We want to minimize
subject to
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58. A class of 2-body symmetric BI
Finding Classical Bound
For 2-body correlators, it turns out that
is linear in
Condition for extrema
We are not interested in the case
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59. A class of 2-body symmetric BI
Conditions for meaningful BI
Imposing gives the condition
Conditions for a good Bell Inequality:
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