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

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 - 2014 Jordi Tura

Summary Motivation Structure of the Local Polytope A parametrized class of 2-body symmetric BI Detection of Physical States Possible Generalizations 15th - May - 2014 Jordi Tura

Summary Motivation Structure of the Local Polytope A parametrized class of 2-body symmetric BI Detection of Physical States Possible Generalizations 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

Preliminaries Bell Experiment Bell Inequality Local Polytope A B C NS LHV Q NS Alice A Bob B Charlie C Bell Inequality Local Polytope 15th - May - 2014 Jordi Tura

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, 377409 (1993)] 15th - May - 2014 Jordi Tura

Practical problems Most restrictive parameter: Take only 2-body correlators Problem still too difficult! 15th - May - 2014 Jordi Tura

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:1312.0265 Submitted to J. Phys. A. 50 years of Bell's Theorem 15th - May - 2014 Jordi Tura

Symmetry groups to consider Full Cycle Translational Invariance + Nearest Neighbors Spin chains with periodic boundary conditions Nearest Neighbors. D = 6 A B D C 15th - May - 2014 Jordi Tura

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:1306.6860 Being reviewed by referees in Science 15th - May - 2014 Jordi Tura

Summary Structure of the Local Polytope Motivation A parametrized class of 2-body symmetric BI Detection of Physical States Possible Generalizations 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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. 15th - May - 2014 Jordi Tura

Fully Symmetric Case More precisely, such expressions are: Structure of the Local Polytope More precisely, such expressions are: 15th - May - 2014 Jordi Tura

Fully Symmetric Case Structure of the Local Polytope The polytope can be generated just from 3 parameters! Actually, 2 parameters are enough 15th - May - 2014 Jordi Tura

Fully Symmetric Case Only points in generate vertices of Structure of the Local Polytope Only points in generate vertices of Polytope Dimension Vertices 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Symmetries in the BI 8 symmetries only 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI General form The Bell Inequalities constraining A particular choice of coefficients: 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Tightness tight non-tight When is (1) a tight Bell Inequality? Amount of vertices saturated by (1): 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Density of BI How many of the total are of this form? 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Quantum Violation – A closer look Bell Operator is pentadiagonal in the Dicke basis 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Quantum Violation – A closer look The state maximally violating (1) can be written as 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Quantum Violation – A closer look The spectrum of depends only on 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Quantum Violation – A closer look The spectrum of depends only on 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Maximal Quantum Violation Example 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Maximal Quantum Violation Example 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Reduced state is local! Bipartite reduced density matrix does not violate CHSH But multipartite nonlocality is detected 15th - May - 2014 Jordi Tura

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% 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

Physical States The Lipkin-Meshkov-Glick Hamiltonian: Lipkin-Meshkov-Glick Model The Lipkin-Meshkov-Glick Hamiltonian: Solvable analytically → Half-filled Dicke state 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

Physical States Implementation Symmetrized correlators can be obtained directly from collective spin components 15th - May - 2014 Jordi Tura

Summary Possible Generalizations Motivation Structure of the Local Polytope A parametrized class of 2-body symmetric BI Detection of Physical States Possible Generalizations 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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] 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

Thanks for your attention! 15th - May - 2014 Jordi Tura

15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Tight classical bound Considering can we get its classical bound? If Tight classical bound is 15th - May - 2014 Jordi Tura

Translationally Invariant Case Class of BI Just one class of BI Has QV for odd N=2k+1 Constant QV -0.592875 15th - May - 2014 Jordi Tura

Translationally Invariant Case Violation by entangled state 15th - May - 2014 Jordi Tura

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) 15th - May - 2014 Jordi Tura

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. 15th - May - 2014 Jordi Tura

Fully Symmetric Case Structure of the Local Polytope We can think of a Local Strategy as a coloring of a graph: 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Idea What happens if, instead of having We consider ? Smooth manifold interpolating the Local Polytope 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

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 15th - May - 2014 Jordi Tura

A class of 2-body symmetric BI Conditions for meaningful BI Imposing gives the condition Conditions for a good Bell Inequality: 15th - May - 2014 Jordi Tura