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Graph approach to quantum systems
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Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

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

Image of FIG. 4.
FIG. 4.

Peres’ KS MMP hypergraph.

Image of FIG. 5.
FIG. 5.

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 .

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

(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.

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

(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.

Image of FIG. 10.
FIG. 10.

Starting configuration for generation of 41–41 MMPs.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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.

Image of FIG. 13.
FIG. 13.

39–39–06 OML dual to itself.

Image of FIG. 14.
FIG. 14.

40–40–34 OML dual to itself.

Image of FIG. 15.
FIG. 15.

40–40–38 OML dual to itself.


Generic image for table
Table I.

Summary of known equations holding in (quantum) Hilbert lattices


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Graph approach to quantum systems