This article represents an advance for continuous-variable quantum
information theory. It provides several contributions for quantum and
classical communication with bosonic modes:

1) The article conjectures a new quantum entropy power inequality
called the entropy photon-number inequality (EPNI) in analogy with the
classical entropy power inequality. The result holds for thermal
states, but the proof of this conjecture for general states remains
elusive. The authors plan to see if there is any connection with the
proof of the classical entropy power inequality.

2) They use the EPNI to prove two long-standing dual minimum output
entropy conjectures. These conjectures are connected to the capacity
of both a thermal-noise bosonic channel and broadcast bosonic channel
for transmitting classical information. This result demonstrates that
the EPNI plays a fundamental role in continuous-variable quantum
information theory (if it is true).

3) They finally use the EPNI to determine the private classical
capacity of a single-mode noiseless bosonic wiretap channel and
explain how to extend the result to multiple modes. They then use the
result of Smith to show that the private classical capacity is equal
to the single-letter capacity for transmitting quantum information
because the bosonic wiretap channel is a degraded channel.

All proofs in this paper are elegant. The longest proof (the private
capacity) requires only seven lines though it does cite prior work.
The proofs of the minimum output conjectures each require only about
four lines.

The paper is well-written overall, but I have a few minor suggestions:

I would like to see a more detailed version of the argument in
footnote 3. The footnote gives a heuristic, but I would still like to
see a more rigorous mathematical argument. I expect the authors will
present it in a more complete, journal version of this paper.

It would be nice if the authors could consistently label the capacity
of a channel as either the classical capacity or the quantum capacity.
They stick with this consistency in the beginning of the article but
then lose some of it toward the end especially in the conclusion of
the article. This lack of consistency will most likely confuse
beginner readers.

I also prefer the notation for a quantum
channel that gives directionality to the quantum channel. A quantum
operation is nonunitary and I prefer a notation that indicates this.

Suggestions for future work:

It would be interesting to find both the private classical and quantum
capacity of the bosonic wiretap channel when interacting with a
thermal environment. The current theorem is interesting but a
thermal-noise result would have implications for realistic quantum
communication and continuous-variable quantum key distribution. This
result would be great because continuous-variable systems are
currently practical and maybe we could start devising clever schemes
to approach this capacity.

It would also be interesting to find the capacity of a different model
of a bosonic wiretap channel in which the attacker Eve uses a linear
phase-insensitive amplifier to enhance her knowledge of the
information Alice sends (similar to that in arXiv:quant-ph/0506193). Some
researchers have discovered that a combined amplifier-beamsplitter
attack has implications for the security of certain threshold schemes
for continuous-variable quantum key distribution. Such a capacity
result then would apply to this generalized attack strategy.