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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, a relative energy difference less than 10% for quite arbitrary (unknown) coherent states, the fidelity must exceed the level 0995. For other kinds of states, the restrictions can be much stronger.

Comparing energy difference and fidelity of quantum states