arXiv subjects

Berke Ünal

Publications and source records attributed to Berke Ünal.

3 recordsLinked to original sources

arXiv2026

Norm properties of quantum uniformity measures

We prove that for $k=4$ the quantum uniformity measures are norms, whereas for every $k\geq5$ the triangle inequality fails. This gives a complete answer to an open question of Bu, Gu, and Jaffe. The $k=4$ case is proved using the Cauchy--Schwarz inequality for the quantum uniformity pairing $\langle\cdot,\cdot\rangle_{Q^4}$ introduced here, together with the triangle inequality in a suitable finite-dimensional Bochner $L^8$ space. For $k\geq5$, we construct systems of linear equations over a finite field and derive a contradiction from the structure theory of norming linear systems.

math-ph↗

arXiv2026

The Quad-$C_5$ Graph: Maximum Contextuality Gap on Eight Vertices

Quantum measurements can exhibit contextuality: their outcomes cannot always be explained by assigning pre-existing values that are independent of which compatible measurements are performed together. The Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality provides the canonical minimal test of this effect for a single three-level quantum system, or qutrit, using five measurement events arranged as a pentagon. Here we ask whether a larger but still compact set of measurement events can produce a stronger separation between quantum predictions and the corresponding noncontextual limit. We perform an exhaustive search over all 11,117 connected non-isomorphic graphs with eight vertices, where each vertex represents a measurement event and edges connect pairs of events that cannot occur together. We identify a sparse ten-edge graph, which we call Quad-$C_5$, as the unique maximizer of this separation at the reported numerical precision. The graph can be understood as four overlapping KCBS pentagons, with every edge shared by two pentagons. Quad-$C_5$ already demonstrates contextuality in a qutrit, for which we obtain an exact analytical result and find the same violation above the noncontextual bound as in the original KCBS test. The larger quantum-noncontextual separation allowed by the graph, however, becomes accessible in a four-level quantum system, where numerical optimization reaches the full graph-theoretic quantum bound and yields a larger contextuality gap than the standard eight-vertex Wagner-graph benchmark while requiring fewer pairwise constraints. Quad-$C_5$ therefore provides a compact connection between minimal qutrit contextuality and stronger contextuality tests available in higher-dimensional quantum systems.

quant-ph↗

arXiv2026

Quantum uniformity norms are pullbacks of matrix-valued uniformity norms

We show that the quantum uniformity norms introduced by Bu, Gu, and Jaffe are the pullbacks of the matrix-valued uniformity norms of Gowers and Hatami under the map $S\mapsto(a\mapsto W_aSW_a^*)$, which we call the Weyl-orbit embedding. This identification yields the Gowers--Cauchy--Schwarz inequality and the triangle inequality in all orders, answering a question of Bu, Gu, and Jaffe, and characterizes the Clifford hierarchy in terms of unitary-valued Leibman polynomial Weyl orbit maps. We then study the structure of the Weyl-orbit embedding. Its range is the fixed-point algebra of a natural isometric action, and the associated norm-one projection is a trace-preserving conditional expectation. We also establish the corresponding dual pullback identity. We further prove Hanner's inequality for both uniformity norms and derive uniform convexity with explicit lower bounds for their moduli of convexity. This yields a variational characterization of the quantum uniformity norms, with the Weyl-orbit map as the unique norm-minimizing representative of each fiber, and an isometric quotient realization of the quantum uniformity spaces. Finally, we decompose the ambient matrix-valued uniformity space canonically into isometric $1$-complemented twisted copies of the quantum uniformity space.

math.GR↗