Quantum Contextual Hypergraphs, Operators, Inequalities, and Applications in Higher Dimensions

TL;DR

This work presents a dimension-agnostic framework for generating and analyzing Kochen–Specker and non-Kochen–Specker contextual sets using McKay–Megill–Pavičić hypergraphs (MMPHs). It introduces eight generation methods that produce NB MMPHs and demonstrates dimension-upscaling up to 32 dimensions, along with novel inequalities (v- and e-inequalities) and a postprocessed quantum fractional independence number that reliably discriminates contextuality. The paper catalogs KS and non-KS MMPHs across 3–32 dimensions, provides coordinatizations for selected masters, and reveals structural properties and graphical representations that illuminate contextuality. Four applications—larger-alphabet QKD, oblivious communication, generalized Hadamard/S-H matrices, and stabilizer operations—illustrate practical avenues where hypergraph contextuality can impact quantum communication and computation, underscoring the significance of scalable, hypergraph-based approaches for high-dimensional quantum information science.

Abstract

Quantum contextuality plays a significant role in supporting quantum computation and quantum information theory. The key tools for this are the Kochen--Specker and non-Kochen--Specker contextual sets. Traditionally, their representation has been predominantly operator-based, mainly focusing on specific constructs in dimensions ranging from three to eight. However, nearly all of these constructs can be represented as low-dimensional hypergraphs. This study demonstrates how to generate contextual hypergraphs in any dimension using various methods, particularly those that do not scale in complexity with increasing dimensions. Furthermore, we introduce innovative examples of hypergraphs extending to dimension 32. Our methodology reveals the intricate structural properties of hypergraphs, enabling precise quantifications of contextuality of implemented sets. Additionally, we investigate several promising applications of hypergraphs in quantum communication and quantum computation, paving the way for future breakthroughs in the field.

Figures (14)

• Figure 1: (a) The 8-7 NBMMPH (pavicic-pra-22, Supplementary Materials, Figure 3) or the bug; cf. (koch-speck, $\Gamma_1$); (b) filled bug---13-7 BMMPH; grey dots represent vertices with $m=1$.
• Figure 2: Obtaining non-KS MMPHs via three different methods (colored lines represent hyperedges): (a) 8-dim KS MMPH obtained via M1 or M6; (b) 8-dim non-KS MMPH $\overline{\rm subhypergraph}$ of (a) obtained by M5 (deleting vertices inside the rectangle loop); (c) 4-dim KS MMPH obtained via M1 or M6; (d) 4-dim non-KS MMPH $\overline{\rm subhypergraph}$ obtained from (c) by M2 and M3 (successive deletions of $m=1$ (grey) vertices and hyperedges so that, at each step, the resulting MMPH is a non-KS MMPH---until we reach the smallest critical non-KS without $m=1$ vertices); (e) 3-dim (noncontextual) BMMPH subhypergraph obtained by M1 and M2; (f) 3-dim non-KS MMPH $\overline{\rm subhypergraph}$ obtained from (e) by M2 and M3 (successive deletions of vertices and hyperedges so that, at each step, the resulting MMPH is a non-KS MMPH---until we reach a critical non-KS without $m=1$ vertices); strings and coordinatizations of (a,c,e) are given in the Appendix \ref{['app:0']} since they were not given elsewhere; strings and coordinatizations of (b,d,f) can be derived from those of (a,c,e).
• Figure 3: TIF diagrams according to figures from cabello-svozil-18 (except (b)) (colored lines represent hyperedges): (a) (cabello-svozil-18, Figure 1a), cf. Figure \ref{['fig:bug']}; (b) 4-dim NBMMPH; (c) (cabello-svozil-18, Figure 4a); (d) (cabello-svozil-18, Figure 5a); (e) (cabello-svozil-18, Figure 7a); (a,c,d,e) are non-KS MMPHs; (b) would be a KS MMPH if it had a coordinatization, but it does not, so it is simply a NBMMPH.
• Figure 4: BMMPH 9-3.
• Figure 5: 4-dim non-KS NBMMPH 25-15, a $\overline{\rm subhypergraph}$ of 26-15 (pavicic-quantum-23, Figure 6b); $\alpha=7>\alpha^*=\frac{25}{4}=6.25$; vertices contributing to $\alpha$ are red-squared; string and coordinatization are given in Appendix \ref{['app:2']}.

Theorems & Definitions (36)

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