Generalized CV Conjecture and Krylov Complexity in Two-Mode Hermitian Systems via Information Geometry

TL;DR

This paper extends the CV conjecture to two-mode Hermitian quantum systems using information geometry, demonstrating that Krylov complexity matches the Fubini-Study volume in both closed and open systems.

Contribution

It introduces a generalized CV conjecture linking Krylov complexity to geometric volume in two-mode systems, supported by explicit analytic calculations.

Findings

Krylov complexity equals Fubini-Study volume in two-mode systems

Constructed wave functions for closed and open systems using squeezed states and Meixner polynomials

Analytic evidence supports the generalized CV relation in this setting

Abstract

We extend the CV conjecture 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.