The Finite Geometry of Breaking Quantum Secrets

TL;DR

This work develops a finite-geometric framework for the pentagon and heptagon quantum codes, revealing how stabilizer observables organize into incidence structures in symplectic polar spaces and how contextuality emerges from line/plane embeddings. By exploiting $2{+}3$ and $3{+}4$ splits, it links entanglement and contextuality to explicit secret-breaking protocols for $(3,5)$ and $(4,7)$ threshold schemes, using negative lines and planes to derive bisep decompositions and teleportation-like recovery steps. The authors also connect these finite-geometric constructions to discrete space-time concepts via Klein and Plücker correspondences, suggesting a geometric underpinning for holographic codes and emergent spacetime, and propose generalizations to other CSS codes. Overall, the paper shows that finite geometry provides a unifying language for encoding, entanglement, and contextuality in quantum secret sharing, with potential implications for discrete models of space-time and quantum information processing.

Abstract

Using a finite geometric framework for studying the pentagon and heptagon codes we show that the concepts of quantum secret sharing and contextuality can be studied in a nice and unified manner. The basic idea is a careful study of the respective $2+3$ and $3+4$ tensorial factorizations of the elements of the stabilizer groups of these codes. It is demonstrated in detail how finite geometric structures entailing a specific three-qubit (resp. four-qubit) embedding of binary symplectic polar spaces of rank two (resp. three), corresponding to these factorizations, govern issues of contextuality and entanglement needed for a geometric understanding of quantum secret sharing. Using these results for the $(3,5)$ and $(4,7)$ threshold schemes explicit secret breaking protocols are derived. Our results hint at a novel geometric way of looking at contextual configurations.

Figures (4)

• Figure 1: The doily with the split labeling. This means that the nontrivial five qubit observables of ${\mathcal{S}}$ of (\ref{['presentation1']})-(\ref{['presentation3']}) are split into two. One label is a two qubit one and the other is a three qubit one. Here the two qubit labels are corresponding to the red "digits" and the three qubits ones to the remaining ones of our list of the $15$ nontrivial elements of the group ${\mathcal{S}}$. Observables connected by a "line" are mutually commuting both for the two and three qubit labelings. The bold faced lines are the negative lines. All other lines are positive ones. Notice that the doily structure is emerging only after invoking the $2+3$ split. Without this split all of the $5$ qubit observables are mutually commuting.
• Figure 2: The four qubit part of the seven qubit heptacode (CSS-code). In this figure $63$ nontrivial four qubit observables are shown. These are the ones of the heptacode that are showing up in qubits $3567$. Compare these observables with the relevant (black) part of the full list of $63$ seven qubit ones (after neglecting signs) as given in Appendix \ref{['sec:app2']}. In this arrangement the red circles correspond to the $18=9+9+9$ observables not containing the identity. The blue, green and yellow circles contain $33=9+9+9+6$, $9$, $3$ observables. They are containing the identity in one, two and three slots respectively. These data identify our embedded configuration as type $23$ in the classification as given in Table 5. of Ref.SanigaTax.
• Figure 3: The $15$ observables from the central part of Figure \ref{['fig:heptaly']} form a doily i.e. a $W(3,2)$ embedded into $W(7,2)$ shown in black. The red three-qubit labels corresponding to the remaining digits of the $15$ stabilizer group elements form an embedding of a $W(3,2)$ into $W(5,2)$. The bold faced lines are negative lines in both cases. The remaining ones are positive. Compare this figure with Figure \ref{['fig:doily']}.
• Figure 4: The $15$ observables from the four qubit part of the $3+4$ split of the heptacode containing the identity in the first slot of the four qubit part. This is qubit No:3 in the seven qubit picture. See Appendix \ref{['sec:app2']}. These observables can be arranged into a triple of negative Fano planes intesecting in a line. The Fano planes are negative because they contain negative lines. In particular the black line of intersection is a negative one. The red plane is the one of Eq.(\ref{['negplane']}) featuring our detailed example of a secret breaking protocol effected by joint manipulations of parties $3567$ of Section \ref{['sec:break']}. There the secret is finally showing up in qubit No:$3$. As explained in the text there are $8$ different ways for making this triple to contain only positive planes, i.e. eligible for a stabilizer subgroup interpretation for all members of the triple. These $8$ different ways correspond to the $8$ terms in the decomposition of Eq.(\ref{['decomphepta']}). Our figure should be imagined as a one connected to the corresponding black negative line of the doily of Figure \ref{['fig:troily']}. Due to cyclic symmetry in qubits $356$ actually there are three such triples of Fano planes. The new triples are obtained by cyclic permutation of the observables in the first three slots i.e. in slots $356$ of our figure. The new triples then also should be connected to the relevant black negative doily lines of Figure \ref{['fig:troily']}. See also Appendix \ref{['sec:heptaplanes']}