An Exact Five-Step Method for Classicalizing N-level Quantum Systems: Application to Quantum Entanglement Dynamics

TL;DR

This paper introduces an exact five-step method to convert quantum dynamics of N-level systems into a classical framework, accurately capturing phenomena like entanglement using geometric and Hamiltonian formalisms.

Contribution

The authors develop a novel, exact five-step algorithm to classicalize N-level quantum systems, including entanglement, using complex projective geometry and Hamiltonian mechanics.

Findings

Successfully classicalized two-qubit entanglement dynamics

Reproduced quantum probabilities and concurrence classically

Captured complex quantum phenomena with classical analogs

Abstract

In this manuscript, we present a general and exact method for classicalizing the dynamics of any $N$-level quantum system, transforming quantum evolution into a classical-like framework using the geometry of complex projective spaces $\mathbb{CP}^{N-1}$. The method can be expressed as five-step algorithmic procedure to derive a classical Hamiltonian and a symplectic structure for the Poisson brackets, yielding $N-1$ Hamilton's equations that precisely replicate the quantum dynamics, including complex phenomena like entanglement. We demonstrate the method's efficacy by classicalizing two interacting qubits in $\mathbb{CP}^3$, exactly reproducing quantum observables such as quantum probabilities, quaternionic population differences and the concurrence, capturing entanglement dynamics via a classical analog.