Generalized CV conjecture and Krylov complexity in two-mode Hermitian systems via information geometry

We extend the “complexity=volume” (CV) conjecture (Zhang et al., 1990) to quantum states of two-mode Hermitian systems using the framework of information geometry. Specifically, we conjecture that the Krylov complexity of a quantum state equals the volume of the Fubini–Study metric. To test this conjecture, we construct the wave functions for both closed and open two-mode systems. For the closed system, the wave function corresponds to the well-known two-mode squeezed state, while for the open system, we employ the second kind of Meixner polynomials to generate an open two-mode squeezed state. Remarkably, in both cases, the calculated Fubini–Study volume matches the Krylov complexity, providing analytic evidence for the generalized CV relation in this controlled two-mode setting. Our results establish a direct link between operator growth in Krylov space and geometric properties of quantum states, highlighting the potential applications of this framework in quantum information and quantum optics.