Earl T. Campbell,
"How to exploit local information when distilling entanglement,"
arXiv:0906.1527v2.

This review has gotten very long; here you will find major comments
section by section, then below a summary of the key requested changes,
a few minor comments, then finally some meta-comments on the open
review process.

Summary:

The author presents Zinf and LoMM as two new protocols for
entanglement purification, making use of local information to optimize
the choice of local filter prior to the first round of purification,
then uses imbalance in the undesired states to guide purification.
Zinf and LoMM are compared to Horodecki filtering, and shown to
produce higher fidelity after one round of purification. The author
also proposes a method for scheduling symmetric purification
operations to minimize the physical resources consumed, presumably at
the expense of temporal ones.

Review:

I cannot yet recommend the paper for acceptance; there are some major
concerns about it as it stands. However, once those issues are
cleared up, it will likely be publishable.

Overall, the protocols themselves seem to be sound, though I have a
few concerns about their physical practicality and how distinct they
are from prior work. Some of the work seems similar to the seminal
work of Deutsch et al. (DEJMPS, ref. [12] in the paper), which is
referenced but not compared in detail, making it difficult to
determine the originality of this paper.

My primary technical concern with the paper is that it claims to
provide more efficient means of purification, but the yield is not
shown. It is certainly intuitive that careful use of local
information should be better than not using the information, and the
plots seem to show that, but the difficulty of accomplishing the
several steps in the process is vague.

The final measure of the value of a purification scheme should be
measured in terms of the number of "base pairs" (unpurified pairs
entangled using the underlying physical entangling mechanism). We
should be given some indication of the tradeoff of temporal and
spatial resources, measured in round trips and qubits used at each
end. We need to be told the success probability of each step, as well
as the output fidelity.

A paper on a purification scheme should allow the reader to determine:

* the minimum required physical qubit resources (check, got that)
* the number of time steps (round-trip times) required to produce a
Bell pair of some target fidelity using the minimum physical
resources and assuming some base entangled state (not
presented in this paper)
* how effectively the given scheme can use additional physical
resources, when available (not presented in this paper)

I will make a few suggestions/comments on the protocols themselves,
but most of my review will concern Sec. IV, repeated distillation,
which is most closely related to my own area of expertise. A lot of
my comments will have to do with making the text clearer and more
accessible. Some of my suggestions will cast the paper more toward an
engineering paper, which may or may not be to the taste of the author
and editor.

Abstract: The sentence about the Horodecki protocol is unnecessary
here. The message in the abstract needs to be sharpened and made more
concrete.

Sec. I:

In your description of twirling in Sec. I.A, are you using a different
definition from the original Bennett et al. one? I believe their
definition used rotations, whereas yours seems to use
rotations. Theirs not only suppresses the off-diagonal terms, it
averages the three triplet states by transforming one to another.
With rotations, your protocol leaves all four Bell states
untouched. Or have a missed some definition somewhere that makes them
match?

More importantly, although you reference DEJMPS here, DEJMPS does
*not* use twirling; it gives an explicity formulation for the growth
of fidelity based on known differences in the probability of the
undesired states (as well as explicitly analyzing the behavior of
purification with pairs of different fidelity, as is done in pumping
or other purification scheduling algorithms).

I think that the description of twirling is the first use of and
as the subsystems in your Bell pair; define them.

The "standard recurrence protocol" may be familiar to you and a good
fraction of readers, but a couple of sentences/equations on it might
improve the accessibility of the paper to those who aren't familiar
with purification, as well as reassure other readers that you are
using the same definition they carry in their heads. In fact, I would
argue that a basic description is a better use of space than a full
paragraph on twirling.

I am not terribly conversant with the matrix, and I suspect some
other readers also will not be. Must ? (I
suspect not; in
your text.) Without going into a lot of detail, a couple of sentences
on properties of the matrix will help.

In your description, "It is only when and ..." shouldn't and be constrained to be , giving a
matrix with zeroes along the top row and left column *except* for
?

After Eq. 5, you comment that , , and are reals, but are
there any other relevant constraints on them, or the matrix?

Although the paper is already long, the eigenvalues of the Wootters
matrix are important, and the matrix itself is not defined here; can
it be defined quickly and clearly?

Sec. II:

A:

doesn't seem to be defined.

A small, simple figure here showing the resources and locations might
help.

The classical communications necessary to support this protocol should
be detailed.

In step 1, what is the success probability of this operation? Is it
performed at *both* ends? Despite the "locally filter" nature of the
quantum operations, I think classical communication of at least the
success or failure of the operations is required, right?

Below Eq. 8, add "with as control and as target" or the
like.

B:

Eq. 14 is wrong; the last term should be , I
think.

As in A, a discussion of the success probability of the filtering is
probably needed.

Sec. III:

Fig. 1 could use an in-figure legend to distinguish the curves.

As noted before, the success probabilities are not really defined in
the paper in a way that would allow a reader to re-derive your
curves.

Sec. IV.B.:

Fig. 2: I would like to see curves in terms of yield: what is the
expectation value for the number of "completed" pairs (passing some
fidelity threshold) per time step?

I do not believe that your description of nested purification is the
same as D\"ur, Briegel, Cirac and Zoller (PRA 59, 1999). That PRA
paper is fantastic, a seminal paper in the study of repeaters and
purification, but the description of the nested purification schemes
actually combines *three* issues that should probably be treated
separately:

a) the design of the physical operations/circuit for purification
(often called the "protocol", though I personally reserve that term
for the mostly classical communications necessary to make repeaters
work);
b) the choice of *which pairs* to purify (what we have called the
"purification scheduling" problem); and
c) the entanglement swapping that spans longer distances (generally,
doubling at each step) at the price of lost fidelity, forcing a loop
back into purification.

Earlier in that paper, DBCZ define schemes A and B (Bennett et
al. (BBPSSW) and Deutsch et al. (DEJMPS), respectively) and C
(original to that paper, I think), which are purely(!) about
purification, and hence more directly related to your scheme than the
discussion of nesting later in the paper. A & B are both
symmetrically scheduled, differing in the choice of physical
operations and assumptions about the state, whereas the scheduling
plan for C is what is now called "pumping", technically independent of
the choice of physical operations.

In your discussion of "nested distillation", you describe combining
symmetric scheduling WITH pumping, pumping a few times at each "level"
before combining the result with another one similarly refined, and
raising the level. Note that DBCZ use "level" to refer also to how
many times entanglement *swapping* has been performed, so that the
distance (measured in number of hops) is , which is different
from your definition. The distinction is made more clear in the more
recent D\"ur & Briegel Rep. Prog. Phys. paper (citation below),
although it still seems that their original intent was to match
"level" with distance and to simply "pump" at each distance.

It is also worth pointing out that scheme B explicitly considers
imbalance in the undesired Bell states, which is one of your goals,
although they do not explicitly use local filtering or the
off-diagonal elements, with their analysis focusing on differences
among the on-diagonal states. You need to be much more explicit about
the difference between your scheme and theirs.

Our "banded purification" scheme (the reviewer says immodestly), in
which only pairs of similar, though not necessarily identical,
fidelity are allowed to purify with each other, can be viewed as a
generalization of the nested distillation scheme, although we were
very careful to treat the purification scheduling and entanglement
swapping as separate problems.

Banded scheduling is useful not only because of the improvement in the
rate at which low-fidelity pairs improve, but also because truly
symmetric scheduling is impossible when you consider memory effects;
Hartmann et al. (HKBD, below) showed the impact on throughput, but
it's also important to realize the divergence between theory and
reality in scheduling purification. (As an engineer, I might list the
HKBD paper as the most important repeater engineering paper of the
last couple of years.)

DBCZ and D&B report that the resources required for their nested
pumping scheme are polynomial. It is important to note that it is the
*physical* resources that are so restricted; for nested pumping
levels (or "bands", in our scheme), you need only qubits per
node (see below). However, the total number of base pairs consumed is
still exponential in ; , if you use rounds of pumping at
each level and pumping succeeds with probability one (many more,
obviously, if ). It can be viewed as an -place, base-
counter, in fact, and when the counter overflows, you are "done". For
truly symmetric purification, would be 2. Fortunately, is
small (no more than six even when initial base-pair fidelities are
very low, ~0.6).

Actually, to be more correct, you need qubits per node for each
distance at which that node operates. For hops, then, the
end nodes in the chain need qubits to avoid deadlock. That's
enough to guarantee that at least one distance can make forward
progress,though , allowing all distances to make progress, is
clearly better.

It is possible, within limits, to trade off temporal and spatial
resources to accommodate that exponential number of base pairs. In
the limit of a large number of qubits per node, many end-to-end (E2E)
pairs are under construction concurrently and many purification
operations are taking place at each time step and over many distances.
The latency for any individual final pair to "graduate" and go to the
next distance would be, for symmetric purification, , but the
resource consumed in that time would be for some constant
(ignoring the improvement in success probability as fidelity
improves).

It is instructive to view purification and entanglement swapping,
then, as classical computer systems problems: scheduling resources,
avoiding deadlock, and minimizing communication waits. (Oh, wait, my
engineer's bent is showing again. That's the way *I* view the
problem, but you are free to focus on more limited aspects in a
different fashion.)

(or any two subscripts) is undefined.

Jiang et al. (Ref. [4]), if I recall, use pumping, but again be
careful about the different use of the term "level". Their mechanism
for entanglement swapping is somewhat different from others'. One key
to the success of their system is the high initial fidelity. Pumping
does not work with a large disparity between the fidelity of
base-level pairs and the target threshold.

Nest Zinf distillation:

With only one subsection, numbering is not necessary.

Your algorithm (p. 8) seems to have either some unnecessary steps or
inaccuracies: as it stands, steps 6 & 8 are the same, and 7 says to
repeat 1-5, giving 1-6,1-5,8. However, that would be the same as
1-6,1-6. This algorithm, in my opinion, would be more readable as
pseudo-code than the current step-by-step approach that includes
looping and branching as steps.

Moreover, I don't understand step 4. Unless I am misreading
something, step 4 is performed without entanglement on the A1-B1 qubit
pair (which were measured in step 3). This step seems unlikely to
actually purify anything, so either there is a bug here, or I don't
understand the intent.

My concern with the unitaries in Eq. 24 is whether or not there is a
straightforward physical means of implementing them. In particular,
the assertion that this approach removes one qubit from the required
number is, in my opinion, unsupportable except in the context of a
particular physical implementation and compilation of the algorithm.

Finally, although it is natural to treat a purification paper as a
paper on distributed quantum computation (DQC), tying it into DQC a
little more explicitly might be valuable. Your goal seems to be
support of small-node quantum multicomputers, such as those of Oi et
al., Jiang et al., and Kim & Kim. Showing that a focus on optimizing
for small-node systems is valuable will at least help the force of the
argument, if not the technical material itself.

Summary of requested major changes:

1. Compare to the DEJMPS mechanism rather than BBPSSW (or do both),
unless you have some strong, valid reason not to. Acceptance of the
paper will hinge on this distinction.
2. Recast performance graphs (or add additional graphs) in terms of
yield, pairs of some output fidelity per base-level pair created.
3. Correct the discussion of prior art in purification scheduling and
choice of physical ops.
4. Recommended, not required: I think Sec. IV.B will be clearer and
more easily verified to be correct if couched in terms of the
spatial/temporal tradeoff of resources.
5. Fix the nested Xinf algorithm.

Minor comments:

There are a number of spelling and grammatical mistakes in the paper:
"distil", "mixturs", "dipicts".

"is delicate property" --> "is a delicate property"
"providing a two qubits" --> "providing two qubits"
"with concurrence measure" --> "with the concurrence measure"
"probability successfully" --> "probability of successfully"
"are do not" --> "do not"?
"mixed state it terms"

"Horodecki" should probably be pluralized as "Horodeckis".

A careful read-through will pick up many more mistakes, I have not
listed all of them.

References:

In addition to more carefully contrasting with prior work as described
above, this paper should be read for the value it brings to the
engineering side:

@article{hartmann06,
author = {L. Hartmann and B. Kraus and H.-J. Briegel and
W. D\"ur},
title = {On the role of memory errors in quantum repeaters},
year = 2007,
journal = pra,
volume = 75,
pages = {032310}
}

This paper is perhaps an easier and more comprehensive read than some
of the seminal ones:

@article{dur2007epa,
title={{Entanglement purification and quantum error correction}},
author={D\"ur, W. and Briegel, H.J.},
journal={Rep. Prog. Phys.},
volume={70},
pages={1381--1424},
year={2007},
doi={doi:10.1088/0034-4885/70/8/R03},
comment={Good review paper on repeaters and purification.}
}

which includes 119 references, not all on purification and repeaters,
but worth a look.

Some small-node quantum multicomputer papers you might like to look
at:

(later published somewhere?)
@MISC{jiang-2007,
author = {L. Jiang and J.~M. Taylor and A.~S. Sorensen and M.~D. Lukin},
title = {Scalable Quantum Networks based on Few-Qubit Registers},
url = {http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0703029},
year = {2007},
howpublished = {quant-ph/0703029}
}

@article{oi06:_dist-ion-trap-qec,
author = {Daniel K. L. Oi and Simon J. Devitt and Lloyd
C. L. Hollenberg},
title = {Scalable error correction in distributed ion trap
computers},
journal = pra,
year = 2006,
volume = 74,
pages = {052313}
}

(published somewhere?)
@article{kim2007ioa,
title={{Integrated Optical Approach to Trapped Ion Quantum Computation}},
author={Kim, J. and Kim, C.},
journal={eprint arXiv: 0711.3866},
year={2007}
}

Edit by Operator: comments on the open review concept moved to dedicated discussion thread, http://quantalk.org/135.

Finally, a disclaimer: when approached by the editor to write this
review, I asked if he considered it a conflict of interest that I have
a paper under review for the same IJQI special issue. The editor
assured me that the size of the issue is not fixed, so it is not a
zero-sum game. My paper will not be more likely to be accepted if I
trash this paper. Having an open review system has the potential to
expose such conflicts, at the risk of "false positives" on perception
of bias (a rumor goes around that so-and-so was biased, when he/she
was not).

Again, I would like to thank Rodney Van Meter for his useful comments. I have made substantial revisions to the manuscript (available as arXiv:0906.1527) and hope that he will now deem it suitable for publication. To begin with I will comment on how I have addressed his main suggested changes:

As discussed in my previous post, the results presented in the original manuscript were for “recurrence with Bell twirling” which has the same performance as DEJMPS. Hence, addressing this suggested change has not involved any change of analysis but a change of presentation. I no longer ambiguously refer to “recurrence”, but always refer to DEJMPS. To clarify I have also provided a description of the DEJMPS protocol, and have cut down the discussion on twirling.

I have performed the requested yield calculations for symmetric scheduling, and these are presented in figure 4. As I summarise in the abstract: “Our protocols converge to higher fidelities in fewer rounds, and when local information is significant one of our protocols consistently improves yields by 10 fold or more.”. Where the one protocol is Zinf, with LoMM suffering pathologically small success probabilities in some regimes.

I have restructured the discussion of purification scheduling and changed terminology to be more consistent with the terminology used in some of the suggested references. However, I am less certain what Van Meter means by "choice of physical ops". I assume, he is referring to his later comments concerning compiling of local unitaries, which I address later in this post.

This section has been broken up into two different sections, which I hope has clarified the readability. I have given some discussion to the spatial/temporal tradeoffs, and have cited some of the suggested references made by Van Meter. I have also shifted terminology to be more engineering friendly in these sections. However, the discussion on this tradeoff is not extensive. As Van Meter has observed the paper is already quite long, and I am keen not to increase the length of the paper. My main aspiration is to convey the importance of exploiting local information in entanglement distillation, and so I wish to keep the scope focused on this topic.

There where no errors in the nested-Zinf (now called pumped-Zinf) protocol. However, their where a lot of complicated aspects to the protocol that were not conveyed well in the original manuscript. Van Meter has picked up on: (i) that the original presentation has a number of steps repeated; and (ii) that there are some steps where there is no entanglement present on qubits A1 and B1 but local operations are being performed. In the revised manuscript, I present the protocol with half as many steps! However, these steps must be performed an even number of times. I also explain that an even number of steps is required because the local information cancels itself out on only an even number of rounds, and this accounts for (i). As for (ii), the operations without entanglement are performing a POVM measurement, and in the revised manuscript I explain this. Since, the POVM measurement occurs after (rather than before) the usual distillation steps, no additional ancilla are required. In addition to these main points, I have also a small figure (no. 5) illustrating two rounds of the protocol, and have also added details through this section.

Many of Van Meter’s comments relate to the above 5 major suggested changes. I now turn to his comments that are not directly covered by the above 5 points:

The abstract is now much shorter and sharper. The yield calculations make it much easier to quantify how "one of our protocols consistently improves yields by 10 fold or more"

I have added an extra half paragraph describing properties of this matrix, including (i) how expectation values can be extracted; (ii) how it maps to the Bloch sphere representation; and (iii) how it maps under local unitaries. I have also corrected the typo you notice, where the constraint should have read entails . The discussion should also clarify which elements of the matrix contain local information

Well clearly the corresponding density matrix is physical! However, I do not know of a neat way to express this in terms of a,b and c.

This can be defined quickly and clearly, and so I have added this requested detail to the paper (see equations 8 and 9 and surrounding text)

Now defined in terms of denominator of expression.

Have added figure 1 that I hope accomplishes this purpose.

Since there are four protocols introduced (including prior protocols), discussing the communication complexity for every protocol would significantly add to the length and I wish to keep the paper as concise as possible. However, I have added a brief remark when filtering is first introduced: “We will not focus on the communication cost of protocols, but note here that every local filtering operation requires one classical bit of information to communicate success.”. I hope that Van Meter finds this satisfactory, as it is my opinion that brevity must trump additional detail in this instance.

added

Yes, your right! Corrected.

All numerical calculations are discussed in later sections. As I outline later on, in general, numerical calculations are inevitable.

As requested, a legend has been added to all graphs.

The filter success probabilities are defined in terms of trace operations over filtered density matrices. Unfortunately, the full expressions are typically very complex and vary greatly for different density matrices and different protocols. Typically, I have found it better to numerically calculate of these quantities, by finding the post filtering density matrix and performing the trace.

As an intermediate step to finding these quantities, one has to find the correct filtering operation. Even finding the filtering operations does not generally admit a simple form. For example, consider the Horodecki filter. To find the Horodecki filter, one must first partially transpose a 4x4 matrix and calculate the eigenvectors. This involves solving a degree 4 polynomial, with constants that are a function of the 16 elements of the matrix. Even at this stage it seems doubtful that there exists an elegant simple expression for the filter. Hence, we are forced to resort to numerically analysis long before we even compute the trace.

In the latter portion of the paper I have changed terminology from nested-Zinf to pumped-Zinf. Hence, clarifying that this section introduces an entanglement-pumping scheme rather than a scheme that explicitly deals with the details of entanglement swapping across nodes. Furthermore, I now cite the DBCZ paper and refer the reader to this resource for more information on repeater architectures. I now hope that the usage of the terms pumping and pumping level are now clear.

Using imbalance in Bell states is not enough to cope with strong local information. Even DEJMPS performs poorly when distilling such states, and hopefully this is clear in the new manuscript.

I agree that the building block protocols and the scheduling can be considered as independent problems. In my manuscript, I am more concerned with the former design problem, and so the revised manuscript does not go into great details concerning the scheduling problem. However, I have made some changes to my terminology (such as using the phrase scheduling). Also, although I have not gone into great detail concerning these issues I have put more care into listing them and cited some of the suggested references.

The Jaing et al scheme is an excellent one, where they use two levels (or tiers): one for bit flips and one for phase flip errors. By removing one error in each tier they can achieve arbitrarily high fidelities (assuming perfect local operations). I do not want to get into the details of how their scheme works, but rather just wished to survey the amount of resources their scheme requires. I hope the manuscript achieves this goal. I am slightly confused about your comment above that “the success of their system is the high initial fidelity”. As I understand their scheme, they can reach unit fidelity with two tiers of pumping for any entangled Bell diagonal state. Their core insight was that in the first level of pumping, it might be advantageous to keep pumping past the point where you are improving fidelity, such that fidelity starts to decrease. Although the fidelity may be decreasing, all the noise is transferred to phase noise, and so it is relatively easy to distil in the next tier of pumping.

Sections restructured in revised manuscript.

I am a bit puzzled about your comment here. It is common in entanglement distillation to assume that all local unitaries are equally easy to implement! To clarify this slightly the revised manuscript now states that: “Our modified protocol uses only local unitaries and computational basis measurements and so requires no greater technological sophistication than prior protocols.”. Maybe you are concerned that my protocol cannot be composed of “component” gates like CNOTS, Hadamards, etc (e.g. the Clifford group). However, local operations are typically achieved by switching Hamiltonians on-off for some duration. Hence, in such local models the duration of the on period determines the exact gate, so if we can perform a Hadamard, we can equally easily perform the family Exp[ i t H] for any time t. Furthermore, although the DEJMPS protocol only uses a discrete set of gates, the Horodecki protocol requires a universal local gate set (just as I do). Indeed, I know of no protocol that can distill states with F<= ½ without using gates outside the Clifford group. If this is still not clear, or Van Meter judges that my additional sentence is not sufficient to clarify this to the reader, then I will gladly add further discussion to the manuscript.

Most of Van Meter’s comments have been consistent with comments made by a second referee. However, in this instance his opinion is in conflict with the second referee. The second referee thought that, whilst it was clear that the protocol could be used in the context of few-qubit quantum computers, it was not apparent how the protocol would be useful in a broader context. As such, I cannot focus even more on small-node systems without risking making the paper too narrow in scope to satisfy the second referee. Note that, the new yield calculations for symmetrically scheduled tasks should show the second referee that Zinf (and to a lesser extent LoMM) are useful in this context. Indeed, I hope that readers will take home that Zinf can be used as a building block for almost any approach, with the revised manuscript concluding that: “Zinf offers increased yields when spatial resources are abundant and can function when spatial resources are very constrained. Since it had much to offer on both extremes of the spatial scale, we also expect that it will have applications in the intermediate regimes.”

Van Meter noticed a number of typos that have been amended.

Where appropriate, I have added citations to many of Van Meter’s suggested references, and have also amended the terminology of the manuscript to be more consistent with these works.

I have also made a number of other changes to the manuscript that I hope will improve its conciseness and clarity. If Van Meter is notices any changes and is unsure why they have been made, I will be glad to discuss them.

This covers all of the referee’s comments, and I hope that the referee finds that the revised manuscript is now suitable for publication. Indeed, I feel that the manuscript has been much improved by both referees comments.

Regards,
Earl.

Overall, the paper has improved substantially in clarity. The new
Figure 4 is both enlightening and somewhat surprising; I admit to
having expected that the results for Zinf would be much more like
DEJMPS. (Hence, doing the work to create it was in fact valuable.)

I do have one remaining technical concern about Fig. 4, below.
Assuming that can be answered, with a few small changes, I expect to
recommend acceptance of the paper. I will review any response to
these points as quickly as possible.

"will outperform all other protocols in this regime" is a strong
statement. Is it proven true, or should it be "outperforms all other
known protocols" for certain conditions and assumptions?

Fig. 4: One feature of the figure puzzles me. In a1, the yield for
Zinf remains 1.0 until about G = 0.002, if I'm reading it right. This
suggests that even with $\epsilon = 0.4$ and $G = 0.002$, that $F \ge
0.99$, which is the target output fidelity. (Or, at least, that the
state contains enough entanglement that local operations can modify
the state to reach the output fidelity.)

The puzzle: if $F \ge 0.99$, why don't *all* of the protocols have a
yield of 1.0 in this region? If the fidelity has already met our
threshold, aren't we done?

Or have I missed something here? Are you excluding the possible
(expected?) final local rotations from the other protocols?

Forgive me for being dumb, but if $\rho_{PL} =
0.6|\Psi^+\rangle\langle\Pi^+| + 0.4|1\rangle\langle1|$,
how do you transform that to $0.99 |\Psi^+\rangle\langle\Pi^+| +
\textrm{junk}$ using only local operations?

It is also not obvious to me why the yield for DEJMPS and LoMM are
both so low with vanishing damping basis angle in b1. (Is this
mentioned in the text? If so, I missed it.)

Caption: is your definition of "yield" correct, or backwards?
Shouldn't it be "the average number of Bell pairs of fidelity F = 0.99
produced per base pair consumed"?

Sec. V.B:

Explicitly mention that A1 and B1 are reused in the protocol after
measurement/reinitialization.

Minor tidbits:

The new Fig. 1 is helpful, but I think it would be even moreso if you
labeled A1, A2, B1, B2 on the two $\rho$s on the left.

Although I appreciate the attempt at livening up your prose,

"However, the local information can never be completely filtered away!
Having already remarked that Horodecki filtering is not optimal, the
astute reader may suspect that local filtering into the appropriate
canonical form will perform better! Indeed, this will prove to be a
good starting point."

is probably a little too droll for an international journal.
Mentioning the reader explicitly is not normal journal style. (At
the very least, remove the exclamation points.)

Likewise, although the grammar and spelling are much better, there are
still a couple of minor things; I don't know if the IJQI staff will
clean them up. For example, "et al." needs a period.

I would assume that initials in bibliographic entries also need
periods.

Here is the updated BibTeX for our paper that you cited:

@article{van-meter07:banded-repeater-ton,
author = {Rodney Van{ }Meter and Thaddeus
D. Ladd and W. J. Munro and Kae Nemoto},
title = {System Design for a Long-Line Quantum Repeater},
journal = {IEEE/ACM Transactions on Networking},
year = 2009,
month = jun,
volume = 17,
number = 3,
pages = {1002--1013},
doi = {10.1109/TNET.2008.927260}
}

You're right, my bad on the scale on Fig. 4 a1 & b1.

Wouldn't it be better to have them the same in all four sub-graphs? They should still clearly show the key points, and it will be easier to visually compare.

I have made a change to the article this thread is about. The reason for the change was:

Revised in light of review comments on quantalk