Percolation Distribution in Small-World Quantum Networks

Round 1

Reviewer 1 Report

From the beginning on the impression could emerge that small-world networks are identical with quantum networks which is not actually the case, because the small-world principles are also present in classical networks. Several formulations in this paper are thus subject to misunderstandings. Probably, this is also a question of the English employed which should be re-formulated from time to time (while misprints should be corrected).

On the other hand, the relationship between the percolation threshold and entanglement behaviour is well explained. An important connection is implied in case of the hexagonal network described in this paper, due to possible consequence for spin networks in the sense of Penrose.

It would be very helpful by the way, if the sparsely given references could be connected to former standard references in the field for improving the reader's orientation.

Author Response

Point 1: From the beginning on the impression could emerge that small-world networks are identical with quantum networks which is not actually the case, because the small-world principles are also present in classical networks. Several formulations in this paper are thus subject to misunderstandings.

Response: We don't think small-world networks are identical to quantum networks.  But we didn't express this point in our manuscript properly. We have fixed it by properly using "small-world quantum network" when refering to our researh subject. 

Point 2: It would be very helpful if the sparsely given references could be connected to former standard references in the field for improving the reader's orientation.

Response: 2 more references are added. References are cited more often in the text fo better understanding, especially when explaining sponge-crossing and concurrence in part 4. 

Point 3: English languages

Response: Misspellings are found and corrected. Some expressions are polished for easier reading. 

Reviewer 2 Report

Authors describes a problem of percolation distribution in quantum networks, especially Authors checks how entanglement distribution is performed in quantum networks. The level of percolation is important to obtain a better network in such sense that networks require lower amount of resources to perform a communications. In general, discussed problem is important and worth of wider discussion. How ever in current form paper requires some additional effort to clear presented results e.g.:

• meaning of acronym/abbreviation WS is not known, “WS” appears at line 81 without any explanation,
• construction of discussed quantum networks is not know!!! after reading of paper, Authors should introduce necessary quantum (and unitrary?) operation which can be used to construct examined network,
• especially operations entitled as q-swap, 4-swap, 4-pswap are unknown, and it action is only described verbally without precise mathematical notation,
• notation used on figures 4, 5, 6, 7, 9,10,11 as P_sc, C_sc are unknown, symbols are used without explanation in text, we have only note that SC means sponge-crosing without citation and other discussion, concurrency is defined as c=sin 2\theta but this suggest special case (related to singlet state), but text do not discussed how simple equation for concurrence is obtained,
• in case of Figure 5 “Y” axis is denoted as number of GCC which suggest that nodes are given as an integer number, but we scale of axis is given as real value from zero to point one,
• mentioned figures are also given as bitmap, so they are barely readable after print, and hard to read even in PDF viewer,
• Figure 6 axes is denoted as <C> it is expectation value of C, in text Authors describes quantity as clustering coefficient line 148, more precise definition of C is needed,
• Typographic error at line 161 for square root of two,
• Role of Table 1 is unknown. Presented rules are the same, even if the use letter p and c.

Therefore, after necessary major correction and substantial rewriting of paper’s text, this contribution is worth of publication, because discussion and results shown in paper seems to be valuable for other interested readers in this area of science.

Author Response

Point 1: meaning of acronym/abbreviation WS is not known, “WS” appears at line 81 without any explanation; Typographic error at line 161 for square root of two.

Response: Words missing and mistakes are fixed: full name for WS is added, the square root and some subscripted mistakes are corrected. The parallel rule of ConPT in Table 1 was wrong, and has been corrected. The connectivity rules in Table 1 are used to calculate the concurrency. 

Point 2: construction of discussed quantum networks is not know!!! after reading of paper, Authors should introduce necessary quantum (and unitrary?) operation which can be used to construct examined network, especially operations entitled as q-swap, 4-swap, 4-pswap are unknown, and it action is only described verbally without precise mathematical notation,

Response: The construction of the quantum network is added in Figure 1. The samll-world quantum network we study is not special except that each edge represents a pure patially entagled state. We mainly focus on the effects of different operations on the same network. Operations of q-swap and n-pswap are introduced by examples in Figure 3 and Figure 5, also in Line 91 and 112. These operations change the structure of the underlying networks no matter what states the edges represent, so equation may not be necessary to describe these operations.

Point 3: notation used on figures 4, 5, 6, 7, 9,10,11 as P_sc, C_sc are unknown, symbols are used without explanation in text, we have only note that SC means sponge-crosing without citation and other discussion, concurrency is defined as c=sin 2\theta but this suggest special case (related to singlet state), but text do not discussed how simple equation for concurrence is obtained,

Response: Explanations are added in the figures. P_sc and C_sc were introduced in the first paragraoh of Part 4. Detailed descriptions and references are added for better understanding in the new version. A general equation for concurrency c is added, so is the realated reference. 

Point 4: in case of Figure 5 “Y” axis is denoted as number of GCC which suggest that nodes are given as an integer number, but we scale of axis is given as real value from zero to point one; mentioned figures are also given as bitmap, so they are barely readable after print, and hard to read even in PDF viewer,

Response: The figures now have larger fonts for easier reading, at least in a PDF viewer. "The number of GCC" was wrongly expressed: GCC is a ratio, so its value ranges from 0 to 1. 

Point 5: Figure 6 axes is denoted as <C> it is expectation value of C, in text Authors describes quantity as clustering coefficient line 148, more precise definition of C is needed,

Response: Detailed explanation and equation of clustering coefficient <C> is added.

Point 6: Role of Table 1 is unknown. Presented rules are the same, even if the use letter p and c. 

Response: The parallel rule of ConPT in Table 1 was wrong in the last manuscript and has been corrected. The connectivity rules in Table 1 are used to calculate the concurrency which can be seen in Line 156. 

Round 2

Reviewer 2 Report

In paper Authors introduced some changes into paper text which answered for raised concerns in previous review. Some other technical issues have been corrected but paper still has minor typographic errors e.g. line 81 space before left bracket is missed. Caption for Figure 1 also has the same typographic error before (BSM).

Other methodological changes improve overall scientific score of paper. Therefore, in my opinion paper could be published in current form after some typographic improvements.