# Temporal correlation beyond quantum bounds in non-hermitian dynamics

## TL;DR

This paper demonstrates that non-Hermitian Hamiltonians can violate the quantum bounds of the Leggett-Garg inequality, reaching the algebraic maximum and revealing new dynamics beyond standard quantum mechanics.

## Contribution

It shows that non-Hermitian two-level systems can surpass quantum bounds on Leggett-Garg inequality violations, achieving the algebraic maximum of 3, unlike Hermitian systems.

## Key findings

- Violates Leggett-Garg bound of 3/2 with non-Hermitian Hamiltonians.
- Achieves algebraic maximum violation of 3 in two-level systems.
- Exceeds quantum speed limits through non-linear Bloch dynamics.

## Abstract

We study the dynamics of two level systems described by non-hermitian Hamiltonians with real eigenvalues. Within the framework of hermitian quantum mechanics, it is known that maximal violation of Leggett-Garg inequality is bounded by $3/2$ (Luder's bound). We show that this absolute bound can be evaded when dynamics is governed by non-hermitian Hamiltonians. Moreover, the extent of violation can be optimized to reach its algebraic maximum of $3$ which is otherwise only feasible when the Hilbert space is infinite dimensional in the hermitian case. The extreme violation of Leggett-Garg inequality is shown to be directly related to the two basic ingredients: (i) The Bloch equation for the two level system has non-linear terms which allow for accelerated dynamics of states on the Bloch sphere exceeding all known quantum speed limits of state evolution; and (ii) We need to ensure that the quantum trajectory of states always lies on a single great circle (geodesic path) on the Bloch sphere at all times.