# Secret Loss of Unitarity due to the Classical Background

## TL;DR

This paper demonstrates that quantum subsystems interacting with classical backgrounds can lose unitarity due to entanglement, challenging semi-classical assumptions and revealing fundamental limits of classical approximations in quantum systems.

## Contribution

It introduces a model comparing fully quantum and semi-classical treatments, showing unitarity loss in quantum subsystems due to entanglement with classical backgrounds.

## Key findings

- Unitarity of quantum oscillators can be lost over time due to entanglement.
- A critical time scale for unitarity loss is derived.
- Unitarity can be lost even in adiabatic limits.

## Abstract

We show that a quantum subsystem can become significantly entangled with a classical background through a process with little or none semi-classical back-reactions. We study two quantum harmonic oscillators coupled to each other in a time-independent Hamiltonian. We compare it to its semi-classical approximation in which one of the oscillators is treated as the classical background. In this approximation, the remaining quantum oscillator has an effective Hamiltonian which is time-dependent, and its evolution appears to be unitary. However, in the fully quantum model, the two oscillators can entangle with each other. Thus the unitarity of either individual oscillator is never guaranteed. We derive the critical time scale after which the unitarity of either individual oscillator is irrevocably lost. In particular, we give an example that in the adiabatic limit, unitarity is lost before other relevant questions can be addressed.