Split Cayley hexagons of order two are distinguished finite geometries living in the three-qubit symplectic polar space in two different forms, called classical and skew. Although neither of the two yields observable-based contextual configurations of their own, classically -embedded copies are found to fully encode contextuality properties of the most prominent three-qubit contextual configurations in the following sense: for each set of unsatisfiable contexts of such a contextual configuration there exists some classically-embedded hexagon sharing with the configuration exactly this set of contexts and nothing else. We demonstrate this fascinating property first on the configuration comprising all 315 contexts of the space and then on doilies, both types of quadrics as well as on complements of skew-embedded hexagons. In connection with the last-mentioned case and elliptic quadrics we also conducted some experimental tests on a Noisy Intermediate Scale Quantum (NISQ) computer to substantiate our theoretical findings.
Abstract We introduce and describe a new heuristic method for finding an upper bound on the degree of contextuality and the corresponding unsatisfied (i.e. non-reproducible by any NCHV model) part of a quantum contextual configuration with three-element contexts (i.e. lines) located in a multi-qubit symplectic polar space of order two. While the previously used method based on a SAT solver was limited to three qubits, this new method is much faster and more versatile, enabling us to also handle four- to six-qubit cases. The method is illustrated by the structure of unsatisfied parts of the associated contextual configurations represented by quadrics and full spaces; these entail well-known finite geometries like the split Cayley hexagon of order two, the dual polar space of rank three and order two as well as certain incidence graphs of binary projective spaces of small dimensions.
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 W n 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.
Recently Waegell and Aravind [J.Phys.A: Math.Theor.45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen-Specker theorem.Here it is demonstrated that two of these sets/configurations, namely the 18 2 -12 3 and 2 4 14 2 -4 3 6 4 ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types V 22 (37; 0, 12, 15, 10) and V 4 (49; 0, 0, 21, 28) in the classification of Frohardt and Johnson [Comm.Algebra 22 (1994), 773-797].Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.
Contextuality is one of the fundamental deviations of quantum mechanics from classical physics.The Kochen-Specker theorem shows that non-contextual classical physics with hidden variables is inconsistent with the predictions of quantum mechanics.Parity proof is one of the many different approaches applied to prove Kochen-Specker theorem for quantum system with different dimensionality.This method of proof requires Kochen-Specker sets that were composed of an odd number of bases and an even number of projectors.In most of the cases, the number of Kochen-Specker sets produced is large and its production is generally aided by computer calculation.However, manual generation of the Kochen-Specker sets are also reported previously for qutrit and two-qubit systems.We put forward the first and surprisingly simple method to generate manually Kochen-Specker sets, with respect to the Mermin pentagram, that can be used for the parity proof of the Kochen-Specker theorem in a state space of eight-dimensional three-qubit system.
The
n-qubit Pauli group is an essential ingredient to most quantum applications, from computing and error correction to benchmarking and simulation. Despite comprising merely a discrete set of operators, it exhibits many quintessential features of quantum theory, including contextuality, which has been identified as a key resource to quantum advantage in a variety of different flavours. Here, we extend this analysis, introducing the notion of a `noncontextual property' whose nonexistence proves the Kochen-Specker theorem, similarly to and generalising common arguments based on the nonexistence of valuations. We relate this notion formally to the existence of Boolean-valued frame functions, and characterise all such frame functions in the case of the
n-qubit Pauli group.
For two qubits, we show that the Pauli group admits noncontextual properties, despite admitting no valuations. We then establish this as the only nontrivial such case with
n≥2, by proving that any Boolean-valued frame function on stabiliser states is constant for more than two qubits. We also perform a similar analysis for the symplectic theory underlying the
n-qubit Pauli group, for which nonconstant Boolean-valued frame functions exist for all
n, yet only in restricted form. By comparison, this shows that contextuality in the
n-qubit Pauli group is not only a consequence of the projective nature of the Pauli group as a representation of its underlying symplectic vector space, but of the geometry of symplectic polar spaces itself. In geometric terms, our result determines all Cameron-Liebler sets of maximal totally isotropic flats in the binary affine-symplectic space.