Persistence of controlled-rotation leverage asymmetry

Determine whether the controlled-rotation construction in the directed ZZ–CRX quantum generator produces a leverage-like asymmetry that persists in the complete dual-observable generator, rather than only in the isolated idealized controlled-rotation term.

Background

The directed ZZ–CRX architecture uses controlled rotations to represent state-dependent and asymmetric temporal modulation motivated by leverage-like dependence in financial returns. In the idealized single-term analysis, a control qubit in the $|0\rangle$ state activates an $R_X$ rotation on the target, producing relative suppression of the target’s $Z$-expectation for sufficiently short evolution times.

The paper emphasizes that the isolated-term mechanism does not by itself establish the behavior of the full generator. The complete architecture combines the controlled rotations with basis-dual interactions, boundary operations, latent encoding, and noncommuting ordered circuit blocks. Establishing whether the intended asymmetry survives these additional operations is therefore necessary to validate the proposed operator-level interpretation of leverage-like effects.