Comparing Energy Difference and Fidelity of Quantum States

TL;DR

This paper derives bounds relating energy difference and fidelity for various quantum state families, providing practical thresholds for state similarity in quantum information tasks.

Contribution

It presents exact analytical bounds for energy difference and fidelity for several quantum state families, enhancing understanding of their relationship.

Findings

Fidelity must exceed 0.995 to keep energy difference below 10% for coherent states.

Bounds vary significantly across different state families.

Results inform practical quantum state comparison and verification.

Abstract

We look for upper bounds of the relative energy difference of two pure quantum states with a fixed fidelity between them or upper bounds of the fidelity for a fixed relative energy difference. The results depend on the concrete families of states chosen for the comparison. Exact analytical expressions are found for several popular sets of states: coherent, squeezed vacuum, binomial, negative binomial, and coherent phase states. Their consequence is that to guarantee, for example, the relative energy difference less than 10% for quite arbitrary (unknown) coherent states, the fidelity must exceed the level 0.995. For other kinds of states, the restrictions can be much stronger.