^{1,a)}, Brendan D. McKay

^{2,b)}, Norman D. Megill

^{3,c)}and Krešimir Fresl

^{4,d)}

### Abstract

Using a graph approach to quantum systems, we show that descriptions of 3-dim Kochen–Specker (KS) setups as well as descriptions of 3-dim spin systems by means of Greechie diagrams (a kind of lattice) that we find in the literature are wrong. Correct lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable future exhaustive generation of 3-dim KS setups by means of our recently found *stripping technique*, bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements)—up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular, on properties such as superposition and orthoraguesian equations.

One of us (M.P.) would like to thank his host Hossein Sadeghpour for a support during his stay at ITAMP.

This work is supported by the U.S. National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics (ITAMP) at Harvard University and Smithsonian Astrophysical Observatory and Ministry of Science, Education, and Sport of Croatia through the Project No. 082-0982562-3160.

Computational support was provided by the cluster Isabella of the Zagreb University Computing Centre and by the Croatian National Grid Infrastructure.

I. INTRODUCTION

II. PRELIMINARY DEFINITIONS AND THEOREMS AND THE SEMIQUANTUM LATTICES

A. Hilbert lattices

B. Overview of equations holding in Hilbert lattices

C. States

D. Vector-valued states

E. Superposition

F. Orthoarguesian equations

G. Greechie diagrams

H. Semiquantum lattices

III. WHY 3D KOCHEN–SPECKER SETUPS CANNOT BE DESCRIBED WITH GREECHIE DIAGRAMS, AND HOW THEY CAN BE

IV. LATTICES THAT ADMIT ALMOST NO HILBERT LATTICE EQUATIONS

V. MMP HYPERGRAPHS WITH EQUAL NUMBER OF VERTICES AND EDGES GENERATED FROM CUBIC BIPARTITE GRAPHS

VI. PROPERTIES OF LATTICES WITH EQUAL NUMBERS OF ATOMS AND BLOCKS

VII. SEPARATE LEVEL REPRESENTATION OF THE MMP HYPERGRAPHS

VIII. CONCLUSIONS

### Key Topics

- Hilbert space
- 78.0
- Lattice theory
- 66.0
- Subspaces
- 53.0
- Plasma diagnostics
- 42.0
- Field theory
- 19.0

## Figures

3- and 4-dim Greechie diagrams and their corresponding Hasse diagrams shown above them (Ref. 56) (Fig. 18, p. 84).

3- and 4-dim Greechie diagrams and their corresponding Hasse diagrams shown above them (Ref. 56) (Fig. 18, p. 84).

(a) OML L42 (Ref. 17) in which Eq. (16) fails; (b) (Ref. 19) in which Eq. (16) fails; (c) (Ref. 19) in which Eq. (17) fails.

(a) OML L42 (Ref. 17) in which Eq. (16) fails; (b) (Ref. 19) in which Eq. (16) fails; (c) (Ref. 19) in which Eq. (17) fails.

KS original setup (Ref. 23) *How to* turn their figure into a MMP is in Ref. 58.

KS original setup (Ref. 23) *How to* turn their figure into a MMP is in Ref. 58.

Peres’ KS MMP hypergraph.

Peres’ KS MMP hypergraph.

Bub’s MMP with 49 atoms and 36 blocks. Notice that 12 bigger dots with a pattern (red online) represent just four atoms: 4, 5, 6, and .

Bub’s MMP with 49 atoms and 36 blocks. Notice that 12 bigger dots with a pattern (red online) represent just four atoms: 4, 5, 6, and .

Conway–Kochen’s MMP. Notice that we cannot drop blocks containing , , 7, and G because atoms 4, 5, 6, and , from them also share two other blocks each. Why we cannot drop atoms 7, G, H, J, etc., is explained in the text.

Conway–Kochen’s MMP. Notice that we cannot drop blocks containing , , 7, and G because atoms 4, 5, 6, and , from them also share two other blocks each. Why we cannot drop atoms 7, G, H, J, etc., is explained in the text.

(a) Greechie diagrams that correspond to modular lattices with 1one to four blocks (over each other); (b) their vectors triples; (c) the smallest nonmodular lattice; (d) its vector triples.

(a) Greechie diagrams that correspond to modular lattices with 1one to four blocks (over each other); (b) their vectors triples; (c) the smallest nonmodular lattice; (d) its vector triples.

A new kind of lattice (MMPL) in which Bub’s setup passes 3OA. The inequality relation in Eq. (36) is represented by the thick red line.

A new kind of lattice (MMPL) in which Bub’s setup passes 3OA. The inequality relation in Eq. (36) is represented by the thick red line.

(a) A 13–7 OML that admits a strong set of states but violates Hilbert-space orthoarguesian equations; (b) a 16–9 OML that does not admit a strong set of states but satisfies orthoarguesian equations; (c) a 16–10 OML that admits neither a strong nor a full set of states and violates all orthoarguesian equations.

(a) A 13–7 OML that admits a strong set of states but violates Hilbert-space orthoarguesian equations; (b) a 16–9 OML that does not admit a strong set of states but satisfies orthoarguesian equations; (c) a 16–10 OML that admits neither a strong nor a full set of states and violates all orthoarguesian equations.

Starting configuration for generation of 41–41 MMPs.

Starting configuration for generation of 41–41 MMPs.

36–36 OML that admits exactly one state and is dual to itself. It is given in the standard compact representation in the first figure and in our separate cycle representation in the other three figures. The figures are explained in detail in Sec. VII.

36–36 OML that admits exactly one state and is dual to itself. It is given in the standard compact representation in the first figure and in our separate cycle representation in the other three figures. The figures are explained in detail in Sec. VII.

First figure shows a 35–35 lattice presented by means of its biggest loop, hexadecagon; it exhibits a left-right symmetry with respect to an axis through vertices V and Y. Three other figures show the same OML in the separate cycle representation. They are explained in detail in Sec. VII.

First figure shows a 35–35 lattice presented by means of its biggest loop, hexadecagon; it exhibits a left-right symmetry with respect to an axis through vertices V and Y. Three other figures show the same OML in the separate cycle representation. They are explained in detail in Sec. VII.

39–39–06 OML dual to itself.

39–39–06 OML dual to itself.

40–40–34 OML dual to itself.

40–40–34 OML dual to itself.

40–40–38 OML dual to itself.

40–40–38 OML dual to itself.

## Tables

Summary of known equations holding in (quantum) Hilbert lattices

Summary of known equations holding in (quantum) Hilbert lattices

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