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# Blog - quantum network theory

Dear all,

Just wanted to annouce that Tomi Johnson recently finished a draft of a potential blog article on the new topic of, "quantum complex networks". It's stored here

I say, "potential" since we have to get past John's quality check! If this is possible, with a bit of feedback here, I think there could even be a fun little mini-series arising out of this and certainly this stuff has some of its roots in the network theory series!

What does everyone think?

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1.

Thanks Jacob. It would sure be great to hear what people think about it.

Comment Source:Thanks Jacob. It would sure be great to hear what people think about it.
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2.
edited July 2013

Hey, this looks really great! A few preliminary comments should give you guys something to do while the rest of us read the article:

• David Tweed will probably question the claim that degrees obey a power law - just warning you.

• There are a distractingly large number of parentheses in this paragraph:

And this is where stochastic walks come in. Google, who are in the business of ranking the importance of nodes (web pages) in a network (the web), use (up to a small modification) the idealized model of a stochastic walker (web surfer) who randomly hops to connected nodes (follows one of the links on a page). This is called the uniform escape model, since the total rate of leaving any node (web page) is set to be the same for all nodes (pages). Leaving the walker (surfer) to wander (surf) for a long while, Google then takes the probability of the walker (surfer) being on a node (page) to rank the importance of that node (page). In the case that the network is undirected (all links are reciprocated) this long-time probability, and therefore the rank of the node (page), is proportional to the degree of the node (page).

You can probably get rid of most of them.

1. "In a quantum walk, this doesn’t happen, so the walk can’t be characterized by its long-time properties." You haven't proved it's impossible. So, how about "In a quantum walk, this doesn’t happen, so the walk can’t be characterized so easily by its long-time properties."

2. "with a finite number n of nodes." You're not defining 'finite' here, so please don't boldface it.

In particular:

1. I won't be able to create those box in pink on the blog, since you're using Markdown commands like +-- {: .standout}. Maybe there's some way to do something similar using HTML?

2. Section headers should be done using html: <h3> </h3>

3. Boldface should be done using <b> </b> and italics using <i> </i>.

use this:

Anyway, it looks really nice apart from these irksome details!

Comment Source:Hey, this looks really great! A few preliminary comments should give you guys something to do while the rest of us read the article: * David Tweed will probably question the claim that degrees obey a power law - just warning you. * There are a distractingly large number of parentheses in this paragraph: > And this is where stochastic walks come in. Google, who are in the business of ranking the importance of nodes (web pages) in a network (the web), use (up to a small modification) the idealized model of a stochastic walker (web surfer) who randomly hops to connected nodes (follows one of the links on a page). This is called the uniform escape model, since the total rate of leaving any node (web page) is set to be the same for all nodes (pages). Leaving the walker (surfer) to wander (surf) for a long while, Google then takes the probability of the walker (surfer) being on a node (page) to rank the importance of that node (page). In the case that the network is undirected (all links are reciprocated) this long-time probability, and therefore the rank of the node (page), is proportional to the degree of the node (page). You can probably get rid of most of them. 1. "In a quantum walk, this doesn’t happen, so the walk can’t be characterized by its long-time properties." You haven't proved it's impossible. So, how about "In a quantum walk, this doesn’t happen, so the walk can’t be characterized so easily by its long-time properties." 1. "with a **finite** number n of nodes." You're not defining 'finite' here, so please don't boldface it. Also: **please read the instructions on [how to write a blog entry](http://www.azimuthproject.org/azimuth/show/How+to#blog), and follow them!** In particular: 1. I won't be able to create those box in pink on the blog, since you're using Markdown commands like +-- {: .standout} . Maybe there's some way to do something similar using HTML? 1. Section headers should be done using html: <h3> </h3> 1. Boldface should be done using <b> </b> and italics using <i> </i>. 1. Similarly, please change all links to html, so instead of this: [small modification](http://en.wikipedia.org/wiki/Google_matrix) use this: <a href = "http://en.wikipedia.org/wiki/Google_matrix">small modification</a> Anyway, it looks really nice apart from these irksome details!
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3.

Thanks John. Will make those useful changes ASAP. I feel a bit silly for not having noticed the "How to write a blog entry" instructions before.

Comment Source:Thanks John. Will make those useful changes ASAP. I feel a bit silly for not having noticed the "How to write a blog entry" instructions before.
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4.

Blame Jacob, not yourself.

Comment Source:Blame Jacob, not yourself. <img src = "http://math.ucr.edu/home/baez/emoticons/devil.gif" alt = ""/>
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5.

Woops! I'll keep that page in mind in the future. Somehow I don't recall seeing that before. Great!

By the way, I have some results on abstracting the diagram in Tomi's post that I plan to post after this article is all done. I could use some feedback from everyone on making sure we're abstracting it enough, and in just the right way. There are a few things I think are nice that can be said about it from the strucural mathematics side.

Comment Source:Woops! I'll keep that page in mind in the future. Somehow I don't recall seeing that before. Great! By the way, I have some results on abstracting the diagram in Tomi's post that I plan to post after this article is all done. I could use some feedback from everyone on making sure we're abstracting it enough, and in just the right way. There are a few things I think are nice that can be said about it from the _strucural mathematics side_.
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6.

I have made the changes recommended by John, thanks.

Further, following the "How to write a blog entry" guide I have created Blog - quantum complex networks, which we should edit instead of the experiment. Therefore:

Now that I have an article that follows the style guide, I look forward to hearing what else everyone has to say!

P.S. While Jacob's knowledge of where to find information on formatting and procedure might have been lacking, I thought I should point out that a large amount of credit is owed to him for any parts of the article that read well...I thought I would say this before people get too mad with him :)

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7.

Hi!

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8.

Hi - sorry to have been distracted by another article, but I'll look over this article, make comments and post it soon!

Comment Source:Hi - sorry to have been distracted by <a href = "http://johncarlosbaez.wordpress.com/2013/07/10/coherence-for-solutions-of-the-master-equation/">another article</a>, but I'll look over this article, make comments and post it soon!
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9.
edited July 2013

I really like this article, because I've been learning about different versions of the graph Laplacian, and I've been starting to think about their relationships, but this puts it into a nice big picture - and it's a big picture I love, the relation between probability theory and quantum theory!

Also, this article seems very well written to me. I found it a lot easier to get into than your official paper on the subject. That's the great thing about blogging - it makes ideas more inviting.

The only mistake I see is that the articles covers more material than anyone can absorb in one sitting - people who write blog articles for the first time tend to get excited and do this. People have a shorter attention span for blog articles than for scientific papers or even printed magazine articles, because they're sitting at a computer and - typically - surfing the web, looking for thrills. Even I was unable to read the whole thing in one go without getting distracted, and I'm more interested in this subject than most people.

So, if you have the energy, you could chop this article in two, for example starting the second part with the section Stochastic Walk. This would give a easy first part and a somewhat more technical second part. That would make it more likely that some people would read the whole thing. And it would also mean that people who read only the first part would not feel guilty.

But if you do chop it, you need to ‛say goodbye' at the end of the first part and let us know what's coming next time (you almost do this already); then in the second part you need a brand-new introduction that quickly reminds people of the main ideas from the first part (showing them your main chart and reminding people of the definitions of G, A, D, L, S, Q would be great).

1. I would greatly prefer some other title, like ‛Quantum network theory'. The reason is that the adjective in the phrase ‛complex networks' seems more like advertising than a truly informative qualification. When people are studying networks, the tools they develop should apply to simple networks just as well as complex ones. Indeed, many examples people use to test these tools, like the ones in this blog article, are examples of simple networks! Of course quantum mechanics may finally give ‛complex network' a real meaning, since now the complex numbers are getting involved. After complex networks we can study quaternionic networks and then octonionic ones! But still...

2. I'm replacing all uses of * to make a bullet symbol by &bull; - blog articles use HTML instead of Markdown. I'll mention this on the 'how to' page.

3. I'll see if the HTML commands you suggest are able to create a colored box on the Wordpress blog.

Comment Source:I really like this article, because I've been learning about different versions of the graph Laplacian, and I've been starting to think about their relationships, but this puts it into a nice big picture - and it's a big picture I love, the relation between probability theory and quantum theory! Also, this article seems very well written to me. I found it a lot easier to get into than your official paper on the subject. That's the great thing about blogging - it makes ideas more inviting. The only mistake I see is that the articles covers more material than anyone can absorb in one sitting - people who write blog articles for the first time tend to get excited and do this. People have a shorter attention span for blog articles than for scientific papers or even printed magazine articles, because they're sitting at a computer and - typically - surfing the web, looking for thrills. Even I was unable to read the whole thing in one go without getting distracted, and I'm more interested in this subject than most people. So, if you have the energy, you could chop this article in two, for example starting the second part with the section **Stochastic Walk**. This would give a easy first part and a somewhat more technical second part. That would make it more likely that some people would read the whole thing. And it would also mean that people who read only the first part would not feel guilty. But if you do chop it, you need to ‛say goodbye' at the end of the first part and let us know what's coming next time (you almost do this already); then in the second part you need a brand-new introduction that quickly reminds people of the main ideas from the first part (showing them your main chart and reminding people of the definitions of G, A, D, L, S, Q would be great). Some smaller comments: 1. I would greatly prefer some other title, like ‛Quantum network theory'. The reason is that the adjective in the phrase ‛complex networks' seems more like advertising than a truly informative qualification. When people are studying networks, the tools they develop should apply to simple networks just as well as complex ones. Indeed, many examples people use to test these tools, like the ones in this blog article, are examples of _simple_ networks! Of course quantum mechanics may finally give ‛complex network' a real meaning, since now the complex numbers are getting involved. After complex networks we can study quaternionic networks and then octonionic ones! <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif" alt = ""/> But still... 2. I'm replacing all uses of * to make a bullet symbol by &bull; - blog articles use HTML instead of Markdown. I'll mention this on the 'how to' page. 3. I'll see if the HTML commands you suggest are able to create a colored box on the Wordpress blog.
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10.

I added first names of authors to all the references. One nice thing about blogging is that you don't have to obey the silly rule that says you can't show someone's full name. It's nice to get to know the names of people who are writing about a subject, so I have the opposite rule.

Comment Source:I added first names of authors to all the references. One nice thing about blogging is that you don't have to obey the silly rule that says you can't show someone's full name. It's nice to get to know the names of people who are writing about a subject, so I have the opposite rule.
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11.

Okay, the HTML code

<div style = "background: #fff1f1; border: solid black; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; overflow: auto;">

does succeed in creating a pink text box on the Wordpress blog. Thanks for teaching me that!

Comment Source:Okay, the HTML code <div style = "background: #fff1f1; border: solid black; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; overflow: auto;"> _does_ succeed in creating a pink text box on the Wordpress blog. Thanks for teaching me that!
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12.

Thanks for the comments, very much appreciated.

The split. I agree about the split into two parts, and where to do it. I knew the post was too long but was in a bind as I (i) didn't feel I had a right to demand two posts, (ii) couldn't miss out the first bit as that would be all that most people understood, and (iii) couldn't miss out the last bit as it was the topic of the post!

Title. Agreed. In the arXiv article, we do actually study examples of Barabasi-Albert networks, which would fall under most people's interpretation of the term 'complex networks'. Sadly this has not made it into the blog. The other reason for the title is that, for better or worse (probably worse), a lot of those newly entering the intersection of quantum and classical networks seem to be uniting under the name 'quantum complex networks'. For example, it was in the title of the recent workshop at IQC. But perhaps we can lead the way, and use a more appropriate title. The name of the workshop could be included in the post so that Google still finds it if the 'quantum complex network' search term is used. I like 'Quantum network theory', it continues on nicely from 'Network theory'.

Updating the 'How to'. Adding something about &bull; to the 'How to' page would be great. I tried <ul> and <li> but this ruined the equations in the list items. Then when I looked at other blogs in progress, I saw that most had used * anyway. Another thing to add might be an explanation of \displaystyle. I see you have used it in your blogs in progress, and I tried to use it for all complicated latex in the displayed equations, where formatting might be different for inline and displayed equations. But this was just me guessing at its use.

Pink boxes. Glad it worked. I just used Chrome's 'inspect element' function to see what the div element used look like, and copied and pasted the style.

Will let you know again when I have split it into two posts. But first I have the small task of passing my viva voce/thesis defence and becoming a doctor!

Comment Source:Thanks for the comments, very much appreciated. **The split**. I agree about the split into two parts, and where to do it. I knew the post was too long but was in a bind as I (i) didn't feel I had a right to demand two posts, (ii) couldn't miss out the first bit as that would be all that most people understood, and (iii) couldn't miss out the last bit as it was the topic of the post! **Title**. Agreed. In the arXiv article, we do actually study examples of Barabasi-Albert networks, which would fall under most people's interpretation of the term 'complex networks'. Sadly this has not made it into the blog. The other reason for the title is that, for better or worse (probably worse), a lot of those newly entering the intersection of quantum and classical networks seem to be uniting under the name 'quantum complex networks'. For example, it was in the title of the recent workshop at IQC. But perhaps we can lead the way, and use a more appropriate title. The name of the workshop could be included in the post so that Google still finds it if the 'quantum complex network' search term is used. I like 'Quantum network theory', it continues on nicely from 'Network theory'. **Updating the 'How to'**. Adding something about &bull; to the 'How to' page would be great. I tried <ul> and <li> but this ruined the equations in the list items. Then when I looked at other blogs in progress, I saw that most had used * anyway. Another thing to add might be an explanation of \displaystyle. I see you have used it in your blogs in progress, and I tried to use it for all complicated latex in the displayed equations, where formatting might be different for inline and displayed equations. But this was just me guessing at its use. **Pink boxes**. Glad it worked. I just used Chrome's 'inspect element' function to see what the div element used look like, and copied and pasted the style. Will let you know again when I have split it into two posts. But first I have the small task of passing my viva voce/thesis defence and becoming a doctor!
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13.
edited July 2013

Tomi wrote:

(i) didn’t feel I had a right to demand two posts,

I can see why you might worry about that. But in fact you are explaining things very well and you're right in line with Azimuth's current emphasis on network theory (which I'll admit is mainly just driven by my own personal interest in that subject, though I really do believe that done right, it could help us save the planet). So I'd be happy to have you write more on this topic.

(ii) couldn’t miss out the first bit as that would be all that most people understood, and (iii) couldn’t miss out the last bit as it was the topic of the post!

It's fine to have a general introductory post (or posts) that many people will want to read followed by more technical ones that only, say, people who understand operators on a finite-dimensional Hilbert spaces will want to read. Done right, it makes more people happy.

I like ‛Quantum network theory’, it continues on nicely from ‛Network theory’.

Okay, great.

I tried <ul> and <li> but this ruined the equations in the list items.

Unfortunately the things that work on the Wordpress blog are somewhat different from those that work on the wiki. We could deal with this by having Andrew Stacey run the blog as well as the wiki, but for other reasons I don't want to.

As it happens, <ul> and <li> don't work well on the blog, so I use simply &amp;bull; for bulleted lists.

I updated the wiki to suggest using &amp;bull; and now I'll update it again to warn people against using <ul> and <li>.

I just posted a great new blog article by Chris Lee, about the selected papers network, so I want to let that sit there on top for a few days... but it sounds like you'll be busy with your viva voce for a bit. Good luck! Just let me know when the new version is ready, and I'll post the first part.

Comment Source:Tomi wrote: > (i) didn’t feel I had a right to demand two posts, I can see why you might worry about that. But in fact you are explaining things very well and you're right in line with Azimuth's current emphasis on network theory (which I'll admit is mainly just driven by my own personal interest in that subject, though I really do believe that _done right_, it could help us save the planet). So I'd be happy to have you write more on this topic. > (ii) couldn’t miss out the first bit as that would be all that most people understood, and (iii) couldn’t miss out the last bit as it was the topic of the post! It's fine to have a general introductory post (or posts) that many people will want to read followed by more technical ones that only, say, people who understand operators on a finite-dimensional Hilbert spaces will want to read. Done right, it makes more people happy. > I like ‛Quantum network theory’, it continues on nicely from ‛Network theory’. Okay, great. > I tried <ul> and <li> but this ruined the equations in the list items. Unfortunately the things that work on the Wordpress blog are somewhat different from those that work on the wiki. We could deal with this by having Andrew Stacey run the blog as well as the wiki, but for other reasons I don't want to. As it happens, <ul> and <li> don't work well on the blog, so I use simply &amp;bull; for bulleted lists. I updated the wiki to suggest using &amp;bull; and now I'll update it again to warn people against using <ul> and <li>. I just posted a great new blog article by Chris Lee, about the [selected papers network](http://johncarlosbaez.wordpress.com/2013/07/12/the-selected-papers-network-part-3/), so I want to let that sit there on top for a few days... but it sounds like you'll be busy with your viva voce for a bit. Good luck! Just let me know when the new version is ready, and I'll post the first part.
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14.

Tomi - I've got a question for you here.

Comment Source:Tomi - I've got a question for you [here](http://forum.azimuthproject.org/discussion/1227/experiments-with-acyclic-stochastic-petri-nets/?Focus=9294#Comment_9294).
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15.
edited July 2013

Tomi: I'm eagerly awaiting your blog article.

I'll be in the wilds of western China August 6-20, so I may have very limited internet access then. Everything involving me should happen before or after that.

Comment Source:Tomi: I'm eagerly awaiting your blog article. I'll be in the wilds of western China August 6-20, so I may have very limited internet access then. Everything involving me should happen before or after that.
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16.

Hi John! Tomi has his PhD viva I think today. In fact, I assume it just ended or is still going on right now since normally they start them at 5 pm to make sure everyone can be present. I think what's left on the blog is pretty straightforward and he's also back to Torino on Saturday.

Comment Source:Hi John! Tomi has his PhD viva I think today. In fact, I assume it just ended or is still going on right now since normally they start them at 5 pm to make sure everyone can be present. I think what's left on the blog is pretty straightforward and he's also back to Torino on Saturday.
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17.

Okay, great. I had to write a blog article on negative probabilities to keep the blog from getting lonely. By the way, check out the relation between Feynman's remarks on negative probabilities for the heat equation and your thoughts on 'forbidden eigenstates' for Dirichlet operators! I mentioned this in a comment.

Comment Source:Okay, great. I had to write a blog article on [negative probabilities](http://johncarlosbaez.wordpress.com/2013/07/19/negative-probabilities/) to keep the blog from getting lonely. By the way, check out the relation between Feynman's remarks on negative probabilities for the heat equation and your thoughts on 'forbidden eigenstates' for Dirichlet operators! I mentioned this [in a comment](http://johncarlosbaez.wordpress.com/2013/07/19/negative-probabilities/#comment-31860).
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18.

Sorry about the delay, have been travelling and preparing for the viva (which turned out to be unnecessary, all was fine). I've split the article into two. Here are the new links:

Part 1 and part 2

Let me know what changes you want and I'll make them ASAP.

Comment Source:Sorry about the delay, have been travelling and preparing for the viva (which turned out to be unnecessary, all was fine). I've split the article into two. Here are the new links: [[Blog - quantum network theory (part 1)|Part 1]] and [[Blog - quantum network theory (part 2)|part 2]] Let me know what changes you want and I'll make them ASAP.
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19.

I wanted to suggest that you might add the following table... though I can't seem to get the latex to work here!

 Quantum Mechanics Stochastic Mechanics states $\psi: X\rightarrow C$ inner product symmetries equation of motion generator properties

Does anyone know how to get the latex to work here?

Comment Source:I wanted to suggest that you might add the following table... though I can't seem to get the latex to work here! <table> <tbody> <tr> <td></td> <td>Quantum Mechanics</td> <td>Stochastic Mechanics</td> </tr> <tr> <td>states</td> <td>$\psi: X\rightarrow C$</td> <td></td> </tr> <tr> <td>inner product</td> <td></td> <td></td> </tr> <tr> <td>symmetries</td> <td></td> <td></td> </tr> <tr> <td>equation of motion</td> <td></td> <td></td> </tr> <tr> <td>generator properties</td> <td></td> <td></td> </tr> </tbody> </table> Does anyone know how to get the latex to work here?
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20.

Here are the equations, for configuration $x$ in the set of configurations $X$.

• QM states: $\psi: X \rightarrow {\mathbb{C}}$, $L^2(X) = \{\psi:X\rightarrow {\mathbb{C}}: \int_X |\psi(x)|^2 \text{dx}\langle \infty \}$
• QM inner product: $\langle -, -\rangle:L^2(X) \times L^2(X) \rightarrow {\mathbb{C}}$
• QM symmetries: $U:L^2(X) \rightarrow L^2(X)$ and $\langle U\psi, U\phi\rangle=\langle \psi, \phi\rangle$ (reversible, unitary)
• QM eqaution of motion: $i\frac{d}{d t}\psi(t) = H \psi(t)$, $\psi(t) = e^{-i t H} \psi(0)$
• QM generator properties: $H = H^\dagger$

• SM states: $\psi: X \rightarrow {\mathbb{R}}_+$, $L^1(X) = \{\psi:X\rightarrow {\mathbb{R}}: \int_X \psi(x) \text{dx}\langle \infty \}$
• SM inner product: $\int:L^1(X) \rightarrow {\mathbb{R}}$
• SM symmetries: $U:L^1(X) \rightarrow L^1(X)$ and $\int U \psi = \int \psi$ and $\psi \geq 0 \implies U\psi \geq 0$ (semi group)
• SM eqaution of motion: $\frac{d}{d t}\psi(t) = H \psi(t)$, $\psi(t) = e^{t H} \psi(0)$
• SM generator properties: $H_{i\neq j}\geq 0$ , $\sum_i H_{ij} = 0$
Comment Source:Here are the equations, for configuration $x$ in the set of configurations $X$. * QM states: $\psi: X \rightarrow {\mathbb{C}}$, $L^2(X) = \{\psi:X\rightarrow {\mathbb{C}}: \int_X |\psi(x)|^2 \text{dx}\langle \infty \}$ * QM inner product: $\langle -, -\rangle:L^2(X) \times L^2(X) \rightarrow {\mathbb{C}}$ * QM symmetries: $U:L^2(X) \rightarrow L^2(X)$ and $\langle U\psi, U\phi\rangle=\langle \psi, \phi\rangle$ (reversible, unitary) * QM eqaution of motion: $i\frac{d}{d t}\psi(t) = H \psi(t)$, $\psi(t) = e^{-i t H} \psi(0)$ * QM generator properties: $H = H^\dagger$ *** * SM states: $\psi: X \rightarrow {\mathbb{R}}_+$, $L^1(X) = \{\psi:X\rightarrow {\mathbb{R}}: \int_X \psi(x) \text{dx}\langle \infty \}$ * SM inner product: $\int:L^1(X) \rightarrow {\mathbb{R}}$ * SM symmetries: $U:L^1(X) \rightarrow L^1(X)$ and $\int U \psi = \int \psi$ and $\psi \geq 0 \implies U\psi \geq 0$ (semi group) * SM eqaution of motion: $\frac{d}{d t}\psi(t) = H \psi(t)$, $\psi(t) = e^{t H} \psi(0)$ * SM generator properties: $H_{i\neq j}\geq 0$ , $\sum_i H_{ij} = 0$
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21.

Tables are mentioned in the how-to. Look:

cell $U:L^1(X) \rightarrow L^1(X)$ and $\int U \psi = \int \psi$ and $\psi \geq 0 \implies U\psi \geq 0$ (semi group) right-align

it works!

Comment Source:Tables are mentioned in the [how-to](http://www.azimuthproject.org/azimuth/show/How+to#math_typesetting_commands_44). Look: Col1 | Very very long head | Very very long head| -----|:-------------------:|-------------------:| cell | $U:L^1(X) \rightarrow L^1(X)$ and $\int U \psi = \int \psi$ and $\psi \geq 0 \implies U\psi \geq 0$ (semi group) | right-align | it works!
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22.

Thank you! If you guys think the table is a nice addition, say the word and I'll patch it together!

Comment Source:Thank you! If you guys think the table is a nice addition, say the word and I'll patch it together!
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23.

By the way, check out the relation between Feynman’s remarks on negative probabilities for the heat equation and your thoughts on ’forbidden eigenstates’ for Dirichlet operators! I mentioned this in a comment.

I found a similar thing to this using quantum algorithms for evolving probability vectors (the algorithms were originally for state vectors). The ansatz for the probability vector would be a sum of many terms. I could restrict each term to be real, but without modifying the quantum algorithm substantially a term could be negative. So it was a nightmare to interpret each term but the sum had a nice interpretation.

Comment Source:> By the way, check out the relation between Feynman’s remarks on negative probabilities for the heat equation and your thoughts on ’forbidden eigenstates’ for Dirichlet operators! I mentioned this in a comment. I found a similar thing to this using quantum algorithms for evolving probability vectors (the algorithms were originally for state vectors). The ansatz for the probability vector would be a sum of many terms. I could restrict each term to be real, but without modifying the quantum algorithm substantially a term could be negative. So it was a nightmare to interpret each term but the sum had a nice interpretation.
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24.

This is off topic but...

using quantum algorithms

Do you mean, tensor contraction algorithms performed on a classical computer? Quantum algorithms, like the stuff James and I used to work on, are refered to as algoriths run on a quantum computer, not a numerical method.

Comment Source:This is off topic but... > using quantum algorithms Do you mean, tensor contraction algorithms performed on a classical computer? Quantum algorithms, like the stuff James and I used to work on, are refered to as algoriths run on a quantum computer, not a numerical method.
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25.

Yes, numerical algorithms for simulating quantum systems.

Comment Source:Yes, numerical algorithms for simulating quantum systems.
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26.

What do you think about adding a summary table detailing the quantum stochastic relationship?

Comment Source:What do you think about adding a summary table detailing the quantum stochastic relationship?
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27.

While I like the table itself, I would choose not to add it because it's not fully aligned with the style/feel of the post, and the delay to make it so would probably not be worth it. It's best if John is able to take the current stable version and post it as soon as he wants.

Comment Source:While I like the table itself, I would choose not to add it because it's not fully aligned with the style/feel of the post, and the delay to make it so would probably not be worth it. It's best if John is able to take the current stable version and post it as soon as he wants.
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28.

In the future posts, I think a good table could serve as a quick sort of review.

Comment Source:In the future posts, I think a good table could serve as a quick sort of review.
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29.

The problem I wanted to solve was having to review stochastic and quantum mechanics in each post. A good table might help avoid that. Also, I want to ask Mauro Faccin to help with explaining some of this stuff.

Comment Source:The problem I wanted to solve was having to review stochastic and quantum mechanics in each post. A good table might help avoid that. Also, I want to ask Mauro Faccin to help with explaining some of this stuff.
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30.

I can see it's a good review table, but I can't easily see where to put it and how to work it into the flow of the article. The most relevant bits, states, generator properties, symmetries and equations of motion are stated and done in such an identical way for stochastic/quantum that it is quite easy to compare and contrast as if they were a table. The stochastic inner product is not used and might be confusing to mention.

Do you mean to ask Mauro to help with the current article?

Comment Source:I can see it's a good review table, but I can't easily see where to put it and how to work it into the flow of the article. The most relevant bits, states, generator properties, symmetries and equations of motion are stated and done in such an identical way for stochastic/quantum that it is quite easy to compare and contrast as if they were a table. The stochastic inner product is not used and might be confusing to mention. Do you mean to ask Mauro to help with the current article?
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31.

Hi! I think the current blog is good without the table. I'm just saying that tons of posts keep explaining this stuff over again and a good table might be worth looking into. Likewise, Petri nets have been introduced a lot and a table might provide a quicker summary!

Good point about Mauro! He did a ton of work on this stuff and I'm not even sure he's read the post! Let me ask him what he thinks! I was going to try to talk John into letting him explain complex networks and their computer simulation/generation.

It would be good if we started working towards some environmental or biological examples to step towards helping save the planet a bit! We need to think how to help out and Mauro certainly understands several biological applications so it might be good.

Comment Source:Hi! I think the current blog is good without the table. I'm just saying that tons of posts keep explaining this stuff over again and a good table might be worth looking into. Likewise, Petri nets have been introduced a lot and a table might provide a quicker summary! Good point about Mauro! He did a ton of work on this stuff and I'm not even sure he's read the post! Let me ask him what he thinks! I was going to try to talk John into letting him explain complex networks and their computer simulation/generation. It would be good if we started working towards some environmental or biological examples to step towards helping save the planet a bit! We need to think how to help out and Mauro certainly understands several biological applications so it might be good.
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32.
edited July 2013

Why is the SM inner product not <,> : and L(1)->L(1)->R in the table?

Comment Source:Why is the SM inner product not <_,_> : and L(1)->L(1)->R in the table?
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33.

That's correct Jim. Of course you can take inner products of probability vectors. I will change it to describe measurements...

Comment Source:That's correct Jim. Of course you can take inner products of probability vectors. I will change it to describe measurements...
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34.

When I read the table, I replaced the words "inner product" with "norm" in my head. In both cases the norm of a state has to be 1 (due to their interpretation in terms of probabilities). But what you use as the norm in each case is different and corresponds to what Jacob wrote. So I guessed that was what he was thinking.

Comment Source:When I read the table, I replaced the words "inner product" with "norm" in my head. In both cases the norm of a state has to be 1 (due to their interpretation in terms of probabilities). But what you use as the norm in each case is different and corresponds to what Jacob wrote. So I guessed that was what he was thinking.
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35.

What I was thinking, is that these are definitions from $L^p$ spaces. In stochastic mechanics, there is a map assigning

$\int:L^1(X) \rightarrow {\mathbb{R}}$

In the stochastic case, we can think of all possible subsets of the set of configurations $X$ and use the map to define a measure. In this case, the measure would return the probability of there being, say a walker found in the subset of $X$.

For a stochastic state, you of course have the normalization

$\int \psi = 1$.

The way I understood most of this stuff anyway, was from hanging out with John and reading part 12.

Comment Source:What I was thinking, is that these are definitions from $L^p$ spaces. In stochastic mechanics, there is a map assigning $\int:L^1(X) \rightarrow {\mathbb{R}}$ In the stochastic case, we can think of all possible subsets of the set of configurations $X$ and use the map to define a measure. In this case, the measure would return the probability of there being, say a walker found in the subset of $X$. For a stochastic state, you of course have the normalization $\int \psi = 1$. The way I understood most of this stuff anyway, was from hanging out with John and reading [part 12](http://math.ucr.edu/home/baez/networks/networks_12.html).
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36.

Hi John. I imagine it is not clear from the above, but we're happy with the current versions of the blog articles (part 1 and part 2). Let us know if you feel anything should be changed.

Comment Source:Hi John. I imagine it is not clear from the above, but we're happy with the current versions of the blog articles ([[Blog - quantum network theory (part 1)|part 1]] and [[Blog - quantum network theory (part 2)|part 2]]). Let us know if you feel anything should be changed.
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37.

We're in the process of having our paper reviewed and just last week received the first round of reviews. They were pretty good. As everyone knows, there are currently some issues regarding the publication process that have been discussed on Azimuth these days. We can't change the system overnight, and we certainly also don't want to add to the problems surrounding it. We therefore are trying to publish open access as much as possible. Open access is not ideal, since it costs about 1K EUR per paper, but we think it's better than nothing. Currently the most prestigious physics journals with good open access policies include, Nature Communications and Physical Review X. There are other journals with open access policies, but those two are the most prestigious physics journals that we know of that support open access. The paper is in review in PRX and we were very surprised that the editor responded to our first submission, having read in detail our manuscript, she provided feedback for us before she sent it to review. It was helpful, and makes us feel as if we're at least paying for something.

Most of the reviewers comments were easy to respond to. I'll post the technical ones here that I think are interesting and try to disscuss how we can respond to them.

v) With respect to the fact that regular networks have epsilon = 0, i.e. both the ground state and the classical stationary vectors match, it is clear from the relation of both vectors through D$^{1/2}$. Right? The more heterogeneous, the more quantumness is also clear from the same reason. On the other hand a physical, not mathematical, explanation is missing.

vi) epsilon as a quantumness is not a distance measure between the two states, namely the classical and the quantum. What about choosing the fidelity or trace distance (for example) as the distance between the two states.

Comment Source:We're in the process of having our paper reviewed and just last week received the first round of reviews. They were pretty good. As everyone knows, there are currently some issues regarding the publication process that have been discussed on Azimuth these days. We can't change the system overnight, and we certainly also don't want to add to the problems surrounding it. We therefore are trying to publish open access as much as possible. Open access is not ideal, since it costs about 1K EUR per paper, but we think it's better than nothing. Currently the most prestigious physics journals with good open access policies include, _Nature Communications_ and _Physical Review X_. There are other journals with open access policies, but those two are the most prestigious physics journals that we know of that support open access. The paper is in review in PRX and we were very surprised that the editor responded to our first submission, having read in detail our manuscript, she provided feedback for us before she sent it to review. It was helpful, and makes us feel as if we're at least paying for something. Most of the reviewers comments were easy to respond to. I'll post the technical ones here that I think are interesting and try to disscuss how we can respond to them. > v) With respect to the fact that regular networks have epsilon = 0, i.e. both the ground state and the classical stationary vectors match, it is clear from the relation of both vectors through D$^{1/2}$. Right? The more heterogeneous, the more quantumness is also clear from the same reason. On the other hand a physical, not mathematical, explanation is missing. > vi) epsilon as a quantumness is not a distance measure between the two states, namely the classical and the quantum. What about choosing the fidelity or trace distance (for example) as the distance between the two states.
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38.

v) With respect to the fact that regular networks have epsilon = 0, i.e. both the ground state and the classical stationary vectors match, it is clear from the relation of both vectors through D$^{1/2}$. Right? The more heterogeneous, the more quantumness is also clear from the same reason. On the other hand a physical, not mathematical, explanation is missing.

I think that the "physical argument" the reviewer is looking for could go along these lines.

On the complete graph, with homogeneous couplings throughout, there is a physical symmetry that can help us understand exactly why the ground states in the quantum and stochastic walks match.

Let $L$ be the corresponding graph Lapacian and let $P_{ij}$ be the permutation operator that , through the adjoint action on $L$, exchanges node labels. We have in particular that

$[ L, P_{ij}] = 0$

for all $i,j$. This in turn implies that $L$ carries the trivial representation of the adjoint action of $P_{ij}$ on $L$.

Let us label the $k$ permutation operators that exchange two nodes $i$ and $j$ as $P_k$---these generate the permutation group---and then consider the projector

$\hat P := \sum_k P_k$

We have that

$[ L, \hat P] = 0$

We can pick a complete basis of states, and let the projector $\hat P$ act on this basis to find a mutual eigenstate. The projection recovers the all ones vector which is an eigenstate of $\hat P$. From the Perron-Frobenius Theorem, we know that there is only a single, unique vector that has all positive entries. The all ones vector has this property, and since $[ L, \hat P] = 0$, we have indirectly found this vector for $L$ using this symmetry argument. As the all ones vector has uniform entries, the corresponding probability distributions in the quantum and stochastic cases are exactly the same.

As the reviewer points out in his summary of the results:

Finally this quantumness is related to the entropy of the network. The most quantum is the one with more heterogeneity in the connectivity i.e. the BA network, while in regular networks both the classical stationary state and quantum ground state match.

Which is clearly exhibited in this particular case. In alternative to the main idea in the paper which relates to considering finding this eigenvector by considering the node degrees, for the complete graph we can also go about things using this symmetry argument, based on an alternative physical property of the graph (e.g. not its degree distribution though it could have been found using that argument instead). I think we can edit this a bit, make the idea simpler and maybe quickly mention it in the paper to respond to the reviewer here. If done properly, it could add a little bit more depth when we mention the complete graph properties in the paper.

What do you think Tomi?

Comment Source:> v) With respect to the fact that regular networks have epsilon = 0, i.e. both the ground state and the classical stationary vectors match, it is clear from the relation of both vectors through D$^{1/2}$. Right? The more heterogeneous, the more quantumness is also clear from the same reason. On the other hand a physical, not mathematical, explanation is missing. I think that the "physical argument" the reviewer is looking for could go along these lines. On the complete graph, with homogeneous couplings throughout, there is a physical symmetry that can help us understand exactly why the ground states in the quantum and stochastic walks match. Let $L$ be the corresponding graph Lapacian and let $P_{ij}$ be the permutation operator that , through the adjoint action on $L$, exchanges node labels. We have in particular that $[ L, P_{ij}] = 0$ for all $i,j$. This in turn implies that $L$ carries the trivial representation of the adjoint action of $P_{ij}$ on $L$. Let us label the $k$ permutation operators that exchange two nodes $i$ and $j$ as $P_k$---these generate the permutation group---and then consider the projector $\hat P := \sum_k P_k$ We have that $[ L, \hat P] = 0$ We can pick a complete basis of states, and let the projector $\hat P$ act on this basis to find a mutual eigenstate. The projection recovers the all ones vector which is an eigenstate of $\hat P$. From the [Perron-Frobenius Theorem](http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem), we know that there is only a single, unique vector that has all positive entries. The all ones vector has this property, and since $[ L, \hat P] = 0$, we have indirectly found this vector for $L$ using this symmetry argument. As the all ones vector has uniform entries, the corresponding probability distributions in the quantum and stochastic cases are exactly the same. As the reviewer points out in his summary of the results: > Finally this quantumness is related to the entropy of the network. The most quantum is the one with more heterogeneity in the connectivity i.e. the BA network, while in regular networks both the classical stationary state and quantum ground state match. Which is clearly exhibited in this particular case. In alternative to the main idea in the paper which relates to considering finding this eigenvector by considering the node degrees, for the complete graph we can also go about things using this symmetry argument, based on an alternative physical property of the graph (e.g. not its degree distribution though it could have been found using that argument instead). I think we can edit this a bit, make the idea simpler and maybe quickly mention it in the paper to respond to the reviewer here. If done properly, it could add a little bit more depth when we mention the complete graph properties in the paper. What do you think Tomi?
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39.
edited August 2013

Tomi wrote:

Sorry about the delay, have been travelling and preparing for the viva (which turned out to be unnecessary, all was fine).

Congrats on the viva! Sorry about my delay; I was desperately trying to finish some things before my trip to China. I'm leaving in a couple of hours, and I will post your first part now. With luck I'll have time to post part 2 during my two-week trip; otherwise it'll be the first thing I post on my return.

I'm only making a few tiny changes... it looks great.

Comment Source:Tomi wrote: > Sorry about the delay, have been travelling and preparing for the viva (which turned out to be unnecessary, all was fine). Congrats on the viva! Sorry about _my_ delay; I was desperately trying to finish some things before my trip to China. I'm leaving in a couple of hours, and I will post your first part now. With luck I'll have time to post part 2 during my two-week trip; otherwise it'll be the first thing I post on my return. I'm only making a few tiny changes... it looks great.
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40.

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41.

I just arrived in Japan. Comments on the post look great!

Comment Source:I just arrived in Japan. Comments on the post look great!
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42.

• What elementary aspect of complex network theory is worth mentioning and do we have time to write it?

Tomi is on vacation currently and part II is currently ready to go so it's not a requirement.

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43.

By the way, if we went down th WS track, we would simply need to explain six degrees of separation, the history of the idea, and the first experiments that found such a small world property using letters.

Comment Source:By the way, if we went down th WS track, we would simply need to explain [six degrees of separation](http://en.wikipedia.org/wiki/Six_degrees_of_separation), the history of the idea, and the first experiments that found such a small world property using letters.
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44.
edited August 2013

I made a bunch of small stylistic changes. If any look bad to you, Tomi, I can fix them. But I don't have much time on the internet until August 20 and I wanted to use today---sitting in a hotel in Lanzhou waiting for my conference to start---to quickly put this post onto the blog. So, instead of making changes on the wiki and asking you to look them over, I just went ahead and posted the article now.

• I got rid of a pink box surrounding some explanation of new material. I think these boxes are nice to mark review of old stuff, but this wasn't old stuff.

• I find the use of $\phi^i_k$ for a list of vectors confusing; these are actually the components of the vectors $\phi_k$, and to keep people from getting mixed up between superscripts and subscripts---and in general to keep them from drowning in a sea of tiny letters---I removed the superscripts. There are times when we need to work with components, but often we don't.

• I reminded people of a few more basic facts near the beginning; you can't expect people to remember very much.

Comment Source:I went ahead and posted &bull; <a href = "http://johncarlosbaez.wordpress.com/2013/08/13/quantum-network-theory-part-2/">Quantum network theory (part 2)</a> I made a bunch of small stylistic changes. If any look bad to you, Tomi, I can fix them. But I don't have much time on the internet until August 20 and I wanted to use today---sitting in a hotel in Lanzhou waiting for my conference to start---to quickly put this post onto the blog. So, instead of making changes on the wiki and asking you to look them over, I just went ahead and posted the article now. A few comments: &bull; I got rid of a pink box surrounding some explanation of new material. I think these boxes are nice to mark review of old stuff, but this wasn't old stuff. &bull; I find the use of $\phi^i_k$ for a list of vectors confusing; these are actually the components of the vectors $\phi_k$, and to keep people from getting mixed up between superscripts and subscripts---and in general to keep them from drowning in a sea of tiny letters---I removed the superscripts. There are times when we need to work with components, but often we don't. &bull; I reminded people of a few more basic facts near the beginning; you can't expect people to remember very much.
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45.

Part II looks nice!

Comment Source:Part II looks nice!
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46.

Hi John,

Thanks for putting it up. Great changes. I like it a lot. Spotted a couple of things

• There should be a space between "(simple, connected)graph" in the first sentence of Flashback'.
• At some point you use "large-time" where it has previously been "long-time"
• The extra index in $\phi_k^i$ allows for some eigenvectors sharing eigenvalues (you probably knew this, but I couldn't tell for sure from what you wrote). From a quick skim, I don't see any problems with missing out the index $i$. Either people won't notice and won't care, or people will notice but only because they understand the subtlety and therefore are ok.

P.S. Thanks for correcting my comment. I've understood how to format them now.

Comment Source:Hi John, Thanks for putting it up. Great changes. I like it a lot. Spotted a couple of things * There should be a space between "(simple, connected)graph" in the first sentence of Flashback'. * At some point you use "large-time" where it has previously been "long-time" * The extra index in $\phi_k^i$ allows for some eigenvectors sharing eigenvalues (you probably knew this, but I couldn't tell for sure from what you wrote). From a quick skim, I don't see any problems with missing out the index $i$. Either people won't notice and won't care, or people will notice but only because they understand the subtlety and therefore are ok. P.S. Thanks for correcting my comment. I've understood how to format them now.
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47.

Second round of reviews for the paper now in.

• "On the basis of the comments, we are definitely interested in considering the manuscript further with a view toward its eventual publication in Physical Review X."

## Second Report of the First Referee -- XE10064/Faccin

First of all, I want to acknowledge the authors for their response. They have tackled with some detail all my criticisms and questions. It was also a pleasure to read the new manuscript colored for a better checking. Thanks a lot.

As the authors know, PRX is now a high impact journal. Thus, the acceptance or not is not restricted to a physical basis but also to the "originality and significance" criteria. I must say that I like the paper. I have not doubt about this. The analytical analysis in Sect. II is really nice. However, as said before, publication in PRX requires papers to satisfy extra conditions. For me the weakest point is the motivation (a broad motivation including the relevance to researchers in several areas). In my first report I asked the "nasty" question of why it was interesting to study quantum dynamics in complex networks. The authors have answered that "There is ample motivation for studying quantum walks on complex networks, from condensed matter and biological systems, to quantum information processing." This sentence is empty. I do not see the role in biology. Are the authors thinking of the photosynthesis? In the latter, the network has seven nodes and the role of dissipation (not considered in the work) is more than crucial. Are there other examples in biology of quantum walks (defined as in the manuscript)? I do not see the role in condensed matter neither. No examples comes to my mind: Sorry. Finally, I do not know much about random walks in quantum information. Empty sentences leave me frustrated. If "quantum walks on complex networks" are important in condensed matter and biology, there should be some concrete examples. Right?

Finally, I have a last question. The authors argue that the gap is never zero for connected networks by referring to the work of Keizer in J. Stat. Phys (1972). I must say that I can not see in the work of Keizer the strong statement as written by the authors, namely that a connected network has always a unique stationary state. In fact, this confuses me. Consider the so-called Google matrix: It fixes problems with the non-uniqueness of stationary states in classical random walks. So, sorry, but here I am totally confused.

I do not see, as the authors do, compelling reasons for studying quantum processes in complex networks. However, I have been "infected" by the enthusiasm of the second referee and if,

• the authors kill the "empty sentence" providing specific examples and

• reply to my confusion on the preceding paragraph,

I would recommend the paper for publication in PRX.

## Second Report of the Second Referee -- XE10064/Faccin

I am very happy with the new presentation of the manuscript and the responses given by the authors.