This paper has been submitted for consideration for an upcoming Special Issue of the International Journal of Quantum Information with the theme 'Distributed Quantum Computing'.
In due course the reviewers will post their reports into this thread, at which point the author can enter into an exchange with them and/or revise the manuscript.
Third parties are also welcome to contribute to this thread, however the eventual decision of the editors (Dan Browne and Simon Benjamin) will normally be based principally on the reviewer and author postings

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

alteration of author email address

At the beginning, we would like to thank the reviewer for the valuable remarks. We supply a revised version of the article and the following list of replies to the comments:

a) In fact, the estimation of quantum states and hence their
properties from limited information goes back further. It
is in Buzek, JMO 44, 2607 (1997) where Jaynes' principle
is discussed. The Horodeckis article picks up on this and
shows that Jaynes principle may sometimes provide "fake
entanglement" and then advocate the minimization of entanglement
as an approach. In a revised version we are now citing their paper.

b) We admit that the result of the twirling operation is not obvious. However, it is a well-known fact that any mixed state can be depolarized into a stabilizer diagonal state. We include therefore Ref. [M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, H.-J. Briegel, "Entanglement in Graph States and its Applications", Proceedings of the International School of Physics "Enrico Fermi" on "Quantum Computers, Algorithms and Chaos", Varenna, Italy, July, 2005].

c) 1) The assumption that all measurement outcomes can be taken to be positve follows from the fact that negative outcomes could be corrected by (local) unitaries acting on the state rho, while leaving the optimization problem of eq. (2) invariant. Of course, it depends on the specific choice of the stabilizers, which unitaries have to be applied. For example, let the stabilizers be given by those of eq. (1), and let m be the number of negative measurement outcomes. We can then relabel the stabilizers in such a way that K_1,…,K_m are the ones which lead to negative outcomes a_1,…,a_m. Now, the application of Z_1 …Z_m rho Z_1…Z_m leads to a state rho’ with the same purity ( tr(rho’^2)=tr((Z_1…Z_m \rho Z_1…Z_m) (Z_1…Z_m \rho Z_1…Z_m)) =tr(rho^2) ), and it is also easy to see that all measurement outcomes are now positive since tr(rho’*K_j)= - tr(rho*K_j) for j=1,…,m. Hence, one can just assume that all measurement outcomes are positive.

2) To specify the condition a_k1+a_k2 >= 1, let us note that this is a necessary condition for a non-zero fidelity, since the minimal fidelity that can be concluded from stabilizer measurements is given by F_min = max[0, (1/2)*(a_1+…+a_N)] as shown in Ref. 10. In the revised version we will say this more explicitly and add the reference. Regarding the maximal mixed state, let us also note that the objective is to find the minimal purity given the stabilizer measurements. Then, as shown in Sec. IIA, the minimization needs only to be carried out over stabilizer diagonal states, which have a nonzero fidelity with the target state only if the above condition is satisfied. In case of the maximally mixed state I/d (with I being the identity matrix of dimension d), measurements of the stabilizer operators belonging to a graph would result in zeros, since the measurement operators are traceless, e.g. X \otimes Z. Therefore, these measurements would not allow to infer a nonzero fidelity.

d) In the revised version, we extend the discussion to the case where GHZ states are prepared and measured. The specific numerical values given in this section are typical dephasing rates and measurement times for ion trap experiments.

e) The reason for the formulation of the error estimate is that the purity estimation can still be treated as a semidefinite program. In this way, an analytical expression can be obtained. It might be that the approach presented by Audenaert and Scheel [New J. Phys. 11 023028 (2009)] could be used in this context. However, this is out of the scope of the paper and it is very unlikely to deliver analytical results.

f) 1) Indeed, in eq. 45 a minus sign is missing, which is now fixed.

2) In this section we address the question “How disordered is the experimentally created state in terms of von Neumann entropy?” The entropy of the target state, the pure stabilizer state, is zero (as it is for all pure states). Therefore the bound on the entropy given by the minimal-purity solution provides a lower bound on the disorder in the system.

3) In the revised version we present an extended, clarified proof still based on the subadditivity of the entropy. The LHS of the inequality is the joint entropy of the system, the RHS is the sum of the entropies of the marginals. Max-Entropy is obtained, when equality holds. Now, we show explicitly that the probability distribution lambda_i provides the exact value for the entropy, not only an estimate (and these values are given in the table).

4) It is indeed correct that alternatives for the derivation of the max-entropy state exist. In the referee’s example one would then still need to determine the Lagrange multipliers. Therefore, we believe that the path presented in our paper may be more direct.

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

Small changes in accordance with referee's comments

Referee alpha has written an email to the editors stating,
On this basis, I am pleased to accept the paper for publication in the IJQI Special Issue on Distributed QIP.

The thread will remain open for any interested researcher to post questions, observations etc.