Browsing by Author "Okay, Cihan"
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Item Open Access Commutative d-torsion K-theory and its applications(A I P Publishing LLC, 2021-10-04) Okay, CihanCommutative d-torsion K-theory is a variant of topological K-theory constructed from commuting unitary matrices of order dividing d. Such matrices appear as solutions of linear constraint systems that play a role in the study of quantum contextuality and in applications to operator-theoretic problems motivated by quantum information theory. Using methods from stable homotopy theory, we modify commutative d-torsion K-theory into a cohomology theory that can be used for studying operator solutions of linear constraint systems. This provides an interesting connection between stable homotopy theory and quantum information theory.Item Open Access Dimension functions for spherical fibrations(Mathematical Sciences Publishers, 2018) Okay, Cihan; Yalçın, ErgünGiven a spherical fibration ξ over the classifying space BG of a finite group G we define a dimension function for the m-fold fiber join of ξ, where m is some large positive integer. We show that the dimension functions satisfy the Borel-Smith conditions when m is large enough. As an application we prove that there exists no spherical fibration over the classifying space of (formula presented) with p- effective Euler class, generalizing a result of Ünlü (2004) about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in upcoming work of Alejandro Adem and Jesper Grodal as a corollary of a previously announced program on homotopy group actions due to Grodal.Item Open Access Mermin polytopes in quantum computation and foundations(Rinton Press Inc., 2023-06-27) Okay, Cihan; Chung, Ho Yiu; İpek, SelmanMermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes β obtained from the Mermin scenario, parametrized by a function β on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes 0 and 1 depending on the parity of β. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of 0 turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. 1 can be seen as a nonlocal toy version of A-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the 2-qubit case, we provide a decomposition of the A-polytope using 1, whose vertices are classified, and the nonsignaling polytope of the (2, 3, 2) Bell scenario, whose vertices are well-known.Item Open Access The monomial Burnside functor(Bilkent University, 2009) Okay, CihanGiven a finite group G, we can realize the permutation modules by the linearization map defined from the Burnside ring B(G) to the character ring of G, denoted AK(G). But not all KG-modules are permutation modules. To realize all the KGmodules we need to replace B(G) by the monomial Burnside ring BC(G). We can get information about monomial Burnside ring of G by considering subgroups or quotient groups of G. For this the setting of biset functors is suitable. We can consider the monomial Burnside ring as a biset functor and study the elemental maps: transfer, retriction, inflation, deflation and isogation. Among these maps, deflation is somewhat difficult and requires more consideration. In particular, we examine deflation for p-groups and study the simple composition factors of the monomial Burnside functor for 2-groups with the fibre group {±1}.Item Open Access On the extremal points of the Λ-polytopes and classical simulation of quantum computation with magic states(Rinton Press, Inc., 2021-11) Okay, Cihan; Zurel, M.; Raussendorf, R.We investigate the Λ-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, Λn, for every number n of qubits. We establish two properties of the family {Λn,n∈N}, namely (i) Any extremal point (vertex) Aα∈Λm can be used to construct vertices in Λn, for all n>m. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage Aα. In addition, we describe a new class of vertices in Λ2 which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of Λn, the above results extend efficient classical simulation of quantum computations beyond the presently known range.Item Open Access Simplicial quantum contextuality(Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2023-05-22) Okay, Cihan; Kharoof, Aziz; İpek, SelmanWe introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine’s theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason’s theorem and Kochen–Specker theorem can be expressed naturally within this new language.Item Open Access The role of cohomology in quantum computation with magic states(Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2023-04-13) Raussendorf, R.; Okay, Cihan; Zurel, M.; Feldmann, P.A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality. Here we extend this web to quantum computation with magic states. In this computational scheme, the negativity of certain quasiprobability functions is an indicator for quantumness. However, when constructing quasiprobability functions to which this statement applies, a marked difference arises between the cases of even and odd local Hilbert space dimension. At a technical level, establishing negativity as an indicator of quantumness in quantum computation with magic states relies on two properties of the Wigner function: their covariance with respect to the Clifford group and positive representation of Pauli measurements. In odd dimension, Gross' Wigner function-an adaptation of the original Wigner function to odd-finite-dimensional Hilbert spaces-possesses these properties. In even dimension, Gross' Wigner function doesn't exist. Here we discuss the broader class of Wigner functions that, like Gross', are obtained from operator bases. We find that such Clifford-covariant Wigner functions do not exist in any even dimension, and furthermore, Pauli measurements cannot be positively represented by them in any even dimension whenever the number of qudits is n ≥ 2. We establish that the obstructions to the existence of such Wigner functions are cohomological.Item Open Access Topological methods for studying contextuality: N-Cycle scenarios and beyond(MDPI AG, 2023-07-27) Kharoof, Aziz; İpek, Selman; Okay, CihanSimplicial distributions are combinatorial models describing distributions on spaces of measurements and outcomes that generalize nonsignaling distributions on contextuality scenarios. This paper studies simplicial distributions on two-dimensional measurement spaces by introducing new topological methods. Two key ingredients are a geometric interpretation of Fourier–Motzkin elimination and a technique based on the collapsing of measurement spaces. Using the first one, we provide a new proof of Fine’s theorem characterizing noncontextual distributions in N-cycle scenarios. Our approach goes beyond these scenarios and can describe noncontextual distributions in scenarios obtained by gluing cycle scenarios of various sizes. The second technique is used for detecting contextual vertices and deriving new Bell inequalities. Combined with these methods, we explore a monoid structure on simplicial distributions.