Upper bounds on the relative energy difference of pure and mixed Gaussian states with a fixed fidelity

TL;DR

This paper derives formulas for the maximum relative energy difference between Gaussian states given a fixed fidelity, revealing that pure states have higher bounds and establishing a fidelity threshold for small energy differences.

Contribution

It introduces exact and approximate formulas for upper bounds on the relative energy difference of Gaussian states at fixed fidelity, including reciprocal formulas and specific bounds for pure and mixed states.

Findings

Bounds are higher for pure states than mixed states.

Maximal bounds are achieved by squeezed vacuum states.

A fidelity of 0.998866 ensures less than 10% energy difference.

Abstract

Exact and approximate formulas for the upper bound of the relative energy difference of two Gaussian states with the fixed fidelity between them are derived. The reciprocal formulas for the upper bound of the fidelity for the fixed value of the relative energy difference are obtained as well. The bounds appear higher for pure states than for mixed ones, and their maximal values correspond to squeezed vacuum states. In particular, to guarantee the relative energy difference less than 10%, for quite arbitrary Gaussian states, the fidelity between them must exceed the level 0.998866.