About the use of fidelity in continuous variable systems

We present examples of continuous variable (CV) states having high fidelity to a given target, say $F > 0.9$ or $F > 0.99$, and still showing striking differences in their physical properties, including classical and quantum states within the set, separable and entangled ones, or nearly Gaussian and strongly non-Gaussian ones. We also show that the phenomenon persists also when one imposes additional constraints on the energy or the squeezing fraction of the states, thus generally questioning the use of fidelity to assess properties of CV systems.