Contextuality degree of quadrics in multi-qubit symplectic polar spaces

Abstract

Quantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality proofs, aka. observable-based proofs of the Kochen-Specker Theorem, we formulate the contextuality property as the absence of solutions to a linear system and define for a contextual configuration its degree of contextuality. Then we explain why subgeometries of binary symplectic polar spaces are candidates for contextuality proofs. We report the results of a software that generates these subgeometries, decides their contextuality and computes their contextuality degree for some small symplectic polar spaces. We show that quadrics in the symplectic polar space Wn are contextual for n = 3, 4, 5. The proofs we consider involve more contexts and observables than the smallest known proofs. This intermediate size property of those proofs is interesting for experimental tests, but could also be interesting in quantum game theory.

Exploiting finite geometries for better quantum advantages in Mermin-like games

“…The degree to which one cannot satisfy all line constraints by classical values is known as the degree of contextuality of an arrangement [16]. The degree of contextuality of the Mermin-Peres grid is 1, meaning there is always one context constraint that is not satisfied by NCHV.…”

“…Here we exclude the trivial operator II and ignore overall phase factors. One can construct a general geometry from all 15 operators, which contains a total of 3 negative lines, and 3 unsatisfiable constraints that can be mapped to the negative lines [16]. Such a geometry is known as the doily, and is shown in figure 4.…”

“…The doily configuration has been of special interest in multi-qubit arrangements [16,[19][20][21][22], in particular its relevance in black hole-qubit correspondence (see for example [23]).…”

“…The degree of contextuality for the doily is 3, meaning there are always 3 lines that cannot be satisfied with any NCHV strategy. These lines can be associated with the negative lines [16], which do not intersect each other. For a failure to occur, Charlie must choose one of the 9 points on these lines, and provide that line to one player.…”

“…This subgeometry contains no generalised triangles, and so is also known as GQ (2,4), the generalised quadrangle of 4 + 1 lines through each point, and 2 + 1 points per line (see section 3). It has 15 + 12 = 27 points and 15 + 30 = 45 lines, formed from a doily and 'double-six' arrangement, and a degree of contextuality of 9 [16]. This can be seen as it can also be composed of 3 disjoint doilies, each of which have degree of contextuality 3.…”