Fleming's bound for the decay of mixed states

TL;DR

This paper generalizes Fleming's inequality to the decay of mixed quantum states, establishing bounds on the decay function and characterizing states that saturate these bounds in finite-dimensional quantum systems.

Contribution

It extends Fleming's bound to mixed states, providing a new inequality for decay functions and identifying all states that achieve equality for all times.

Findings

Derived a generalized decay bound for mixed states.

Characterized mixed states that saturate the bound at all times.

Proved conditions under which equality holds or does not hold.

Abstract

Fleming's inequality is generalized to the decay function of mixed states. We show that for any symmetric hamiltonian $h$ and for any density operator $\rho$ on a finite dimensional Hilbert space with the orthogonal projection $\Pi$ onto the range of $\rho$ there holds the estimate $\Tr(\Pi \rme^{-\rmi ht}\rho \rme^{\rmi ht}) \geq\cos^{2}((\Delta h)_{\rho}t) $ for all real $t$ with $(\Delta h)_{\rho}| t| \leq\pi/2.$ We show that equality either holds for all $t\in\mathbb{R}$ or it does not hold for a single $t$ with $0<(\Delta h)_{\rho}| t| \leq\pi/2.$ All the density operators saturating the bound for all $t\in\mathbb{R},$ i.e. the mixed intelligent states, are determined.