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

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!

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

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

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

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

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

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

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.

First attempts M Żukowski, Phys. Lett. A 177, 290 (1993) D. Kaszlikowski, M. Żukowski. PRA 61, 022114 (2000) Systtem: two-qubit Werner state Observers can freely choose any local projection.

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, 230403 (2004) Violation for GHZ states grows faster than for WWWŻB inequality.

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, 062315 Bell inequalities for qudit GHZ states JPA 49, 035302 Bell inequalities with discrete number of observables (PRA 74,062109) Multiqubit entanglement criteria PRL 100, 140403

Quadratic Entanglement Criteria S 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

Bell inequality avoiding the KS contradicton.

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

Geometric inequalities as a test field Wieśniak, Dutta, Ryu, JPA 49, 035302 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

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

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

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