Designing generalized elegant Bell inequalities in higher dimensions from a Tsirelson bound

Elegant Bell inequality is well known for its distinctive property, being maximally violated by maximal entanglement, mutually unbiased bases, and symmetric informationally complete positive operator-valued measure elements. Despite its significance in quantum information theory demonstrated based on its unique violation feature, it remains the only known one with the characteristic. We present a method to construct Bell inequalities with violation feature analogous to elegant Bell inequality in higher local dimensions from a simple, analytic quantum bound. A Bell inequality with the generalized violation feature is derived in three dimension for the first time. It exhibits larger violation than existing Bell inequalities of similar classes, including the original elegant Bell inequality, while requiring arguably small number of measurements.

💡 Research Summary

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The paper addresses a long‑standing gap in the theory of Bell non‑locality: while Gisin’s “Elegant Bell Inequality” (EBI) is celebrated for its simultaneous optimality with three distinct quantum‑information structures—maximally entangled states, mutually unbiased bases (MUBs), and symmetric informationally complete positive‑operator‑valued measures (SIC‑POVMs)—this remarkable property has been confined to two‑dimensional (qubit) systems. The authors propose a systematic method to lift this “elegant” feature to arbitrary prime dimensions, thereby constructing a new family of Bell inequalities that retain the same three‑fold optimality.

The core of the construction relies on the Weyl‑Heisenberg (W‑H) group. In a prime dimension (d), the shift operator (X) and phase operator (Z) generate the group ({X^{p}Z^{q}\mid p,q\in