Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products?   For example, I have a matrix 16x16 and want it to be decomposed into something like $$A =\sum_{i,j,k,l}h_{ijkl}\sigma_i\otimes\sigma_j\otimes\sigma_k\otimes\sigma_l$$ I can't find any clear explanation how it can be done. I found only this for two-dimentional case:
The Pauli group for $n$-qubits is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the group containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli matrices form a basis for the $2\times 2$ complex matrix vector spaces, that is $\mathbb{C}^{2\times 2}$. Apart from it, from the definition of the tensor product, it is known that the $n$-qubit Pauli group will form a basis for the tensor product space $(\mathbb{C}^{2\times 2})^{\otimes n}$.  I am wondering if the Pauli group in $n$-qubits forms a basis for the complex vector space where the elements of this tensor product space act, that is $\mathbb{C}^{2^n\times 2^n}$. Summarizing, the question would be, is $(\mathbb{C}^{2\times 2})^{\otimes n}=\mathbb{C}^{2^n\times 2^n}$ true?  I have been trying to prove it using arguments about the dimensions of both spaces, but I have not been able to get anything yet.
Julian day of the year in Java Does $a\mid bc$ imply $a\mid b$ or $a\mid c$? I am not sure I understand this I just to an explanation to this question:  If $a, b, c \in \Bbb{Z}^+$ and $a\mid bc$, does it follow that $a\mid b$ or $a\mid c$? Prepositions used with "inquire" How do I enable the "Universe" repository? I have a typical scientific manuscript in a LaTeX .tex file, and I need to convert it to MS Word .doc file.  The reason for having to convert to MS Word is I'm submitting the manuscript to an academic journal and they only accept MS Word (I know...)  The manuscript includes title page, figures, tables, equations (inline and in their own align environment), footnotes, bibliography, and an annex.  The tables are in their own separate tables.tex file, which I include using the \include{tables} command.  Most tables take up a whole landscape page, and were generated sing the package pdflscape.  I am using Windows 7 Professional.    My plan is to use pandoc to go from .tex to .odt, open the latter in Libre Office, and convert to .doc.  I have read a related  but it is too general.  Similarly the examples in the Pandoc  are too simple.  I have played around but I am unable to accomplish what I want.  This is surprising since converting a scientific manuscript is probably the most common use case for Pandoc. Here are some sample failures:  Example 1  I open a command line in the project folder, and execute the following:  pandoc -s document.tex -o document.odt   I get this error message:  pandoc: figure1: openFile: does not exist &lt;no such file or directory&gt;   where figure1 is the name of a figure file (e.g. figure1.png) in the project folder referenced in a line as \includegraphics[width=5.8in]{figure1}.  I suspect pandoc expects a .png extension but not sure how to provide it.  Example 2  Next I try .html, and excute the following:  pandoc -s document.tex -o document.html   The program executes fine.  I open HTML file.  Footnotes are there but figures are missing, tables are displayed as LaTeX, bibliography is missing, in-line math displays well, but math in align environment does not, section labels are displayed, and some other minor issues.  So given that mine is probably a typical use case scenario, my question is this: What commands should I use to get the .odt file I want?  I could not find a fully worked out example on the web.    Here is a specific list of errors.  I'll update how I corrected them based on community suggestions:   Figures not rendering.  Solved by adding .png extension to .tex file in \includegraphics command.  Now figures are included but they are huge, with half of each figure outside the page. No bibliography.  Solved. First, I have one huge consolidated Latex .bib file where I keep all my citations.  I manage it using .  This was giving me problems as I do not keep the cleanest .bib file in town.  So I reduced the problem by using a neat trick in JabRef that allows you to subset your master .bib file using the .aux file generated by Latex when compiling your manuscript.  In JabRef click on Tools > New Subdatabase based on AUX file.  This way I generated a much smaller biblio.bib file with only the articles referenced in my manuscript.  Running pandoc -s document.tex -o document.odt --bibliography=biblio.bib did the trick. Display math. Math in \begin{align} environment displayed in verbatim \latex;  (A partial solution is to use the  Libre Office extension.  Copy and paste the latex math code in the .odt file created by Pandoc into the equation editor, and so on.  Surely this could be built into a macro that can post-process all remaining math.)  UPDATE: Display math works very well using --mathjax extension. Inline math. Inline equation do not always render properly.  Bold math is a problem.  E.g. $\Sigma=\sigma^2\bm{I}$ displays as $\Sigma=\sigma^2\bm{I}$; Labels are displayed (e.g. section labels show as [sec:empirical] blah blah]; All tables display as raw latex. How to get a list of applications installed in Ubuntu dash? In Gnome, we have applications grouped up based on their type (like - games, accessories, et al). But in Unity, we need to manually type in the Unity to get the App.  Is there any keyword/utility to display the applications in GUI?  Ubuntu Version : 14.04 Hardware : Dell Inspiron 3437 Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$ So far: I know that$\mod(p)$ is a field thus one of the factors in the expression has to equal $0$ $\mod(p)$ for some $x$. However I just don't know where to continue from this point? How do I approach a congruency like this? Are there any general tips for problems like these?  Thanks for all the help. How to find relevant and new open research topics? How can one find relevant and new open research topics?  When working on my PhD thesis, I find it hard to find open questions in my very specific research area. Which approach do you take to find new and relevant topics? I was loading an image resource with the following code and it works fine from my dev environment (netbeans) but threw an exception as soon as it ran from the desktop.  new ImageIcon(Images.class.getResource("images/highlighter24.gif"));   I traced it down to my file having different case than in the code.  Fine, my bad, but why the inconsistent behavior?  Gremlins?  Note: I'm running on windows before you ask. If someone is my proxy, I am their _____? Bonus question: If I am a proxy for multiple people, what's the collective noun for that group? Generic List - moving an item within the list So I have a generic list, and an oldIndex and a newIndex value.  I want to move the item at oldIndex, to newIndex...as simply as possible.  Any suggestions?  Note  The item should be end up between the items at (newIndex - 1) and newIndex before it was removed. Why is ATP the most prevalent form of chemical energy storage and utilization in most cells? A locally constant sheaf on a locally connected space is a covering space; Proof? As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here is the question repeated:     A sheaf on a locally connected space $X$ is locally constant if each point $x \in X$ has a basis of open neighborhoods  $\mathcal{N}_x$ such that whenever $U,V \in \mathcal{N}_x$ with $U \subset V$, the restriction $\mathcal{F}V \rightarrow \mathcal{F}U$ is a bijection. Prove that $\mathcal{F}$ is locally constant iff the associated etale space over $X$ is a covering.   I don't have a problem (i think) showing the "covering" $\rightarrow$ "locally constant" direction of the implication. My problem is in the reverse direction. In fact, in the process of trying to solve this problem i think i have come up with a very simple counterexample.  Here's my attempt at a counterexample:  Let $X = \{a,b,c\}$ be  finite space with open subsets $\{\{\},\{a\},\{a,b\},\{a,b,c\}\}$. I claim this is hyper-connected and locally connected.  Let $\mathcal{F}$ be the sheaf with $\mathcal{F}(\{a,b,c\}) = \{q\}$, $\mathcal{F}(\{a,b\}) = \{r,s\}$, and $\mathcal{F}(\{a\}) = \{t\}$, where $r = q|_{\{a,b\}}$, and $t = q|_{\{a\}} = r|_{\{a\}} = s|_{\{a\}}$. I claim this defines a locally constant sheaf. Indeed we can have $\mathcal{N}_{a} = \{\{a\}\}$, $\mathcal{N}_{b} = \{\{a,b\}\}$, and $\mathcal{N}_c = \{\{a,b,c\}\}$.  Now we can calculate the corresponding etale space $p : E \rightarrow X$ as follows: $E = \{\dot{q}(a), \dot{q}(b), \dot{q}(c), \dot{s}(b)\}$, and has these four distinct elements since $\dot{q}(a) = \dot{r}(a) = \dot{s}(a) = \dot{t}(a)$, and $\dot{q}(b) = \dot{r}(b)$. The action of $p$ follows by construction.  However, $p$ does not appear to me to be a covering space. In particular, the only neighborhoods of $b \in X$ are $\{a,b\}$and $\{a,b,c\}$ and we have $p^{-1}(\{a,b\}) = \{\dot{q}(a),\dot{q}(b),\dot{s}(b)\}$ and this cannot be partitioned into homeomorphic images of $\{a,b\}$ because $2 \nmid 3$. Similarly, $p^{-1}(\{a,b,c\}) = \{\dot{q}(a), \dot{q}(b), \dot{q}(c), \dot{s}(b)\}$ and this cannot be partitioned into homeomorphic images of $\{a,b,c\}$ since $3 \nmid 4$.  So i think i have a locally constant sheaf on a locally connected space whose corresponding etale space is not a covering. What have i done wrong? I have seen the claim mentioned elsewhere so i'm convinced i'm wrong somewhere. Perhaps someone knows where a correct proof is published otherwise? One adjective for two or more nouns Could we use one adjective for two nouns? Is the following sentence correct?     It has low speed and voltage. In my class design, I use abstract classes and virtual functions extensively. I had a  feeling that virtual functions affects the performance. Is this true? But I think this performance difference is not noticeable and looks like I am doing premature optimization. Right? Can anyone please tell me whether the following function $d:\mathbb{R}^2 \times \mathbb{R}^2 \to  \mathbb{R}$ is really a metric or not on $\mathbb{R}^2$.  $$ d((x_1,x_2),(y_1,y_2))= \begin{cases} |x_1-y_1|, &amp;\text{ if } x_2=y_2\\ |x_1|+|y_1|+|x_2-y_2|, &amp;\text{ if } x_2 \neq y_2. \end{cases} $$  If it really is indeed a metric, please give the proof of triangle inequality.(I'm OK with the other two conditions). How to upload string as file with jQuery or other js framework What is the best way to share/distribute Excel macro? Animation Blending Basics Say I have a framework with characters that can run and walk and of course, stand still.  Am I right in thinking that animation blending is the process of smoothly transitioning between these by generating dynamic animations at runtime?  If so, what is the basic theory of how this works, and how would I implement it? I have a 1-year residence permit from one Schengen country that is now expiring. Upon its expiration, I would like to get my 90/180 days validity as a tourist to stay, or return here as soon as possible.  I understand that the tourist short-stay scheme is based on the 90/180 days "counter". But what happens if my previous stay was not tourism but residence (as searching year). Wouldn't that mean that I can quickly exit and re-enter the Schengen area to activate my tourist stay?   Also, is there a way to change this purpose of stay without exiting the Schengen area?