Review of: "Fidelity of quantum blobs"

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Potential competing interests: No potential competing interests to declare.In the manuscript "Quantum distinguishability and symplectic topology," the author investigates the fidelity as a means to distinguish quantum states.Instead of the standard approach in terms of quantum mechanics, an approach based on "symplectic topology" is used.What it comes down to is a geometry-inspired approach that computes the overlap between spherical regions representing states in a kind of phase space.The work is unfortunately rather questionable.There are several issues that need are discussed in the following points.a) The author provides a justification of this work based on a comparison between quantum mechanics and classical mechanics.However, such a comparison ignores the fact that there are other classical theories apart from classical mechanics.The comparison therefore gives the erroneous impression that quantum mechanics possesses certain features not present in classical theories, whereas some of the other classical theories do in fact posses the same features.For example, the superposition of complex amplitude fields is a well-known aspect of (classical) Fourier optics.b) The author defines a phase space with a complex $p$ by multiplying $p$ by a phase factor that contains the Planck constant with an undefined variable/parameter $\alpha$.The latter is not discussed.Since it is a non-physical quantity, there is nothing preventing it from taking on any value, in which case the Planck constant has no effect.In other words, one can simply replace h\alpha -> \phi, where \phi is an arbitrary phase value.Instead of a spherical region of uncertainty as depicted in Fig 2, the region of uncertainty would be toroidal.Here is a serious problem with the manuscript.Apart from the completely unphysical nature of this construction, the mathematical picture presented by the author is also not correct.c) In the section "Quantum blobs" the author introduces a state consisting of N particles known with "maximum precision," which is then referred to as a "saturated state."It is not clear what is meant by the term "maximum precision," nor what it means for a state to be "saturated."When a state is represented as a Wigner function in phase space, the smallest region it can occupy is called the minimum uncertainty area.Such a state would necessarily have to be a pure state.Does the "maximum precision" imply that the state only occupies the minimum uncertainty area?Does the term "saturated" mean the same thing as "pure"?In fact, it is not clear what the actual quantum states are that are represented by the blobs.Or stated differently, how does a blob represent a particular quantum state such as a squeezed state, for example?The author needs to clarify the meaning of terms in the context of existing known concepts of phase space or, better yet, conform to the use of established terms and concepts about phase space.Although the author wishes to introduce a different phase space formalism, such a formalism would need to be presented in terms of much clearer definitions that can be understood in terms of (or by comparison with) existing formulations.d) At some point in the section "Quantum blobs" the author refers to the N-particle state as a coherent state.This

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