Hypegraph-Based Contextuality Mladen Paviˇ ci´ c Center of Excellence for Advanced Materials and Sensors (CEMS) , Research Unit Photonics and Quantum Optics , Institute Ruder Boˇ skovi´ c (IRB) , Zagreb, Croatia. ees Informatique Quantique 2019 , Journ´ Nov 28 & 29, 2019 - Besan¸ con, France. Besan¸ con2019 Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 1 / 26
Introduction MMPHs vs. vectors McKay-Megill-Paviˇ ci´ c hypergraph (MMPH) strings system of nonlinear equations vectors c d a b a b x + a b a b z = 0 y + = x y z coordinatization a c a c a c a c z = 0 = x + y + x y z b b c b c b c b c z = 0 x + y + = x y z a d a d a d a d z = 0 x + y + = x y z a e e a a e a e a e z = 0 = x + y + x y z orthogonalities d e d e d e d e z = 0 = x + y + x y z exponential complexity MMPH c e edge b d statistically polynomial complexity a vertex MMPH string: cba,ade. Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 2 / 26
Introduction Formalism vs. MMPH’s linearity Vertex notation; Edge; Hypergraph hypergraph = pair v - e ; v = a set of elements called vertices; e = a set of non-empty subsets of e called edges; edge = a set of vertices related to each other — e.g., orthogonal to each other. Each vertex is denoted by one of the following characters: " # $ % & ’ ( ) 1 2 ... 9 A B ... Z a b ... z ! * - / : ; < = > ? @ [ \ ] ˆ ‘ { | } ˜, +1, +2, . . . +Z, +a, +b, . . . +˜, ++1, ++2, . . . ++Z, ++a, ++b, . . . Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 3 / 26
Introduction Definition of MMPH McKay-Megill-Paviˇ ci´ c hypergraph (MMPH) An MMPH is an n -dim hypergraph in which (i) Every vertex belongs to at least one edge; (ii) Every edge contains at least 2 vertices; (iii) Edges that intersect each other in m − 2 vertices contain at least m vertices, 2 ≤ m ≤ n . Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 4 / 26
History-1 MMPH isomorphism-free generation Isomorphism-free MMPH generation:M Paviˇ ci´ c,J-P Merlet, B D McKay & N D Megill, J. Phys. A , 38 , 1577 (2005) 1 2 3 3 5 2 3−dim 4 1 1 2 3 7 5 5 7 6 6 4 4 1 2 2 1 3 3 9 8 7 5 7 9 5 5 7 5 7 9 9 6 6 4 6 8 4 4 6 4 8 8 2 1 2 1 1 2 2 3 3 3 1 3 MMPH generation tree: 10 vertices; 7 8 6 9 5 3 vertices per edge; loop of size 5 A 4 1 2 3 Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 5 / 26
Non-contextuality vs. contextuality MMPH contextuality Enter MMPH non-contextuality (N-C) & contextuality n -dim MMPH non-binary contextual set , n ≥ 3, is a hypergraph whose each edge contains at least two and at most n vertices to which it is impossible to assign 1s and 0s in such a way that No two vertices within any of its edges are both assigned the value 1; In any of its edges, not all of the vertices are assigned the value 0. ******************************************************** An MMPH set to which it is possible to assign 1s and 0s so as to satisfy the above two conditions is a N-C MMPH binary set . An MMPH non-binary set with edges of mixed sizes to which vertices are added so as to make all edges of equal size each containing n vertices is called filled MMPH set. Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 6 / 26
Non-contextuality vs. contextuality MMPH non-binary visualization MMPH non-binary set conditions visualised MMPH non−binary set violates the following conditions: ( ) ii ( ) i 0 1 0 either or 0 0 1 Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 7 / 26
Non-contextuality vs. contextuality Measure of MMPH contextuality Measuring MMPH non-contextuality vs. contextuality quantum hypergraph index ( HI q ) = sum of probabilities of getting detector clicks for all considered vertices classical hypergraph index ( HI c ) = maximal number of 1s assigned to vertices so as to satisfy the two conditions from the previous slide. Non-contextual inequality - contextual distinguisher Contextual, non-binary sets: HI q > HI c Non-contextual, binary sets: HI q ≤ HI c Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 8 / 26
Non-contextuality vs. contextuality Coordinatization MMPH coordinatization and contextuality Kochen−Specker no coordinatization original non−binary contextual non−binary 192−118 set contextual pentagon binary non−contextual pentagon Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 9 / 26
MMPH Kochen-Specker masters MMPH KS masters 1 MMPH KS masters: Coordinatization inherited Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 10 / 26
MMPH Kochen-Specker masters MMPH KS masters 2 M. Paviˇ ci´ c and N.D. Megill, Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces, Entropy 20 (12), 928 (2018) Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 11 / 26
MMPH Kochen-Specker masters MMPH KS masters 3 MMPH KS masters together with their coordinatization created from simple vector components Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 12 / 26
MMPH Kochen-Specker masters MMPH examples Examples: M. Paviˇ ci´ c, M. Waegel, N. Megill, and P.K. Aravind, Automated generation of Kochen-Specker sets, Scientific Reports , 9 , 6765 (2019). Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 13 / 26
3-dim MMPHs and their contextuality 3-dim MMPHs M. Paviˇ ci´ c, Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets, Physical Review A , 95 , 062121–1-25 (2017). Gray vertices are usually dropped in the literature g Bub L O H G Conway− i k P −Kochen c f Peres Zap1T i 3 f l j e g ch F p H K klm h 1 C m P q k d 6 f 51−37 o e s R PV l D t U V 7 h U J m Q E n U IJ V j 9 d b n r B W I E S c b N W a N j S D H 57−40 A 7 u e K O 49−36 T 6 8 C W C n L D X i IM 9 S g 8 B J K QR v E o T R A d L 3 4 6 7 8 92 b M N O 1 3 2 4 5 G F Y X Z 4 2 5 A BQ Y MFG Y 20−gon Z a 18−gon X 22−gon 5 33−36 31−37 33−40 Literature names Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 14 / 26
3-dim MMPHs and their contextuality Literature In the literature all vertices that appear in only one edge are dropped, but ... Peres wrote “It can be shown that if a single ray is deleted from the set of 33, the contradiction disappears.” [A. Peres, J. Phys. A , 24 , L175 (1991)] “In the original proof of Kochen and Specker the number of elements is 117. The present record, due to Kochen and Conway, is 31 vectors.” [I. Pitowsky, J. Math. Phys. , 39 , 218 (1998) Similar statements throughout the literature. But none of them: Bub’s 33-36, Conway-Kochen’s 31-37, Peres’ 33-40, and Kochen-Specker’s 117-118, is actually critical, i.e., if a single vertex/ray/vector or edge were deleted, the “contradiction” would not disappear. Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 15 / 26
3-dim MMPHs and their contextuality Yu-Oh S. Yu and C.H. Oh, State-Independent Proof of Kochen-Specker Theorem with 13 Rays, Phys. Rev. Lett. 108 , 030402 (2012). (a) Peres’ 57-40 crit- (b) Removal of gray 1 1 (a) (b) 2 2 I I ical MMPH non- bi- vertices from 25-16 H 3 H 3 J J 4 4 G G nary KS set contain- yields MMPH non- K K F 5 F 5 L L ing Yu-Oh filled 25-16 P M P M binary Yu-Oh 13-16 6 6 E E j N O i k MMPH binary non- e Q r set. See (b,c) below. D R D q 7 7 u dC S 8 p t T o C 8 c h g l KS set. B s A 9 b m v B A 9 f U a V n Z W O Y X N (a) (b) y 3 (c) z 1 (d) Kochen & Specker 110=y 3 notation 112 112 h 3 h 0 111=h 111=h 0 y 3 3 110=y y 1 y 1 3 121 121 z 3 101=y 001=z 3 isomorphic 101=y h h 3 2 2 y 2 y 2 1 z 2 h 0 010=z 2 equivalent y 2 y 3 112 211 211 112 121 121 h h 2 z 3 111=h 1 or 111=h 2 1 z 2 y 2 y 3 h 2 211 211 z 1 011=y 1 100=z 1 011=y 1 y 1 y 1 MMPH notation Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 16 / 26
3-dim MMPHs and their contextuality Operator-based contextuality Yu & Oh did not prove the Kochen-Specker theorem but they introduced a kind of operator-based contextuality and its non-contextual inequality Yu & Oh picked 13 vertices out of 25 to construct an expression of state/vector defined 3x3 operators that eventually reduces to a multiple of a unit operator. I. Bengtsson, K. Blanchfield, and A. Cabello, A Kochen-Specker Inequality from a SIC, Phys. Lett. A , 376 , 374 (2012) and Z.P. Xu, J.L. Chen, and H.Y. Su, State-independent contextuality sets for a qutrit. Phys. Lett. A , 379 , 1868 (2015) make use of projectors whose expressions also reduce to a multiple of a unit operator. Paviˇ ci´ c (CEMS, Zagreb, Croatia) Hypegraph Contextuality Besan¸ con2019 17 / 26
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