Suppose that there is a system of two photons 1 and 2, each of which is in a mixed state $1/2|R\rangle\langle R| + 1/2 |L \rangle\langle L|$, where $|R \rangle$ and $\langle L|$ are two orthonormal pure polarization states. (So the composite state would be a product of two mixed states.) How is this case different from the case where each of 1 and 2 is in a pure state $1/\sqrt{2}(|R\rangle+|L\rangle)$? Can you distinguish these two cases experimentally? Because it looks like the measurement outcomes should be the same in both cases: either both are in $|R\rangle$, both are in $|L\rangle$, or one is in $|R\rangle$ and the other in $|L\rangle$. Please educate me.
I am in trouble with polarization and entanglement. Let's consider three cases : Case 1) : Statistical mixture of $|H\rangle$ and $|V\rangle$ polarized photons Case 2) : Photons in a superposition state $1/\sqrt{2}(|H\rangle+|V\rangle)$ Case 3) : Photons which are entangled with twin ones in $1/\sqrt{2}(|H,H\rangle+|V,V\rangle)$ state Which experiments can be conducted to differentiate the case  statistical mixture from the case  superposition state ?   Which experiments can be conducted to differentiate the case  statistical mixture from the case  entangled photons in superposition state ?   Using a $45^\circ$ polarizer I think you can differentiate case 1/case 2 but not case 1/case 3 I don't know how to differentiate case 1/case 3 except maybe using quantic tomography and Wigner function. Is it true ? Is there any other simpler way ? Thanks a lot for your answer and sorry for this maybe dummy question...
I've read that it is possible to add a method to an existing object (i.e., not in the class definition) in Python.   I understand that it's not always good to do so. But how might one do this? How do I access a page's HTTP response headers via JavaScript?  Related to , which was modified to ask about accessing two specific HTTP headers.     Related: I came across a combinatorial proof of the Fermat's Little Theorem which states that    If $p$ is a prime number then the number ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$.  Let me write down the proof.    PROOF  We have pearls of $a$ colours . From these we make necklaces of exactly $p$ pearls .   First,we make a string of pearls . There are $a$$p$ different strings.If we throw away  the $a$ one one -coloured pearls  ($a$$p$-$a$) strings will remain.We connect the ends of each string to get necklaces.We find that two strings that differ only by a cyclic permutation of its pearls result in indistinguishable necklaces.But there are $p$ cyclic permutations of $p$ pearls on a string . Hence the number of distinct necklaces is    [($a$$p$-$a$)/$p$].  Because of this interpretation this is an integer.Thus ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$. [HENCE PROVED]     MY CONFUSIONS    What is the use of the fact that $p$ is a prime in this proof?  If this proof is true then why is this not true for every positve integer $p$? Unexpected voltage with switch turned off I am doing some work on a ceiling light and I notice that, even though the light-switch is off, my voltage tester shows voltage present.  Answers to other questions (below) explain about phantom voltages but how exactly do I test for these and know what voltages are real and what are phantom so that I can work safely?    Phantom voltage questions How to reduce the spacing between responses in the alterqcm package? This question has a continuation here   The alterqcm package is no longer maintained by Alain Matthes. Two other developers maintain it. I wrote to them 15 days ago by email to report the error reported here:  I have not received a reply. I am trying to reduce the spacing between the lines of the QCM as indicated in  by @Werner and it does not work. In addition, by using the David Carlisle's code given in , I get an X instead of a square.  Full code of the screenshot: \documentclass[12pt]{article} %\usepackage[utf8]{inputenc} %\usepackage[upright]{fourier} \usepackage{etoolbox}% http://ctan.org/pkg/etoolbox \usepackage[english]{alterqcm} \def\square{X} % &lt;---------- David Carlisle comment \def\nogreekalph{??} % &lt;---- David Carlisle comment \parindent0pt \begin{document} % Original display \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \bigskip  % Smashed display !!!!!! Dont work now  {\begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {\smash{Proposition 1}},  {\smash{Proposition 2}},  {\smash{Proposition 3}}} \end{alterqcm}}  \bigskip  % Global smashed display !!!! Dont work now  \makeatletter \patchcmd{\aq@prop}{\vbox{#1}}{\vbox{\smash{#1}}}{}{}% patch \aq@prop \makeatother \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \end{document}  Is there a solution? How to inject Spring Bean for factory method requiring MyClass.class parameter How can I send mail from an iPhone application I want to send an email from my iPhone application. I have heard that the iOS SDK doesn't have an email API. I don't want to use the following code because it will exit my application:  NSString *url = [NSString stringWithString: @"mailto:foo@example.com?cc=bar@example.com&amp;subject=Greetings%20from%20Cupertino!&amp;body=Wish%20you%20were%20here!"]; [[UIApplication sharedApplication] openURL: [NSURL URLWithString: url]];   So how can I send an email from my app? $(this) inside of AJAX success not working I need a binary search algorithm that is compatible with the C++ STL containers, something like std::binary_search in the standard library's &lt;algorithm&gt; header, but I need it to return the iterator that points at the result, not a simple boolean telling me if the element exists.  (On a side note, what the hell was the standard committee thinking when they defined the API for binary_search?!)  My main concern here is that I need the speed of a binary search, so although I can find the data with other algorithms, as mentioned below, I want to take advantage of the fact that my data is sorted to get the benefits of a binary search, not a linear search.  so far lower_bound and upper_bound fail if the datum is missing:  //lousy pseudo code vector(1,2,3,4,6,7,8,9,0) //notice no 5 iter = lower_bound_or_upper_bound(start,end,5) iter != 5 &amp;&amp; iter !=end //not returning end as usual, instead it'll return 4 or 6   Note: I'm also fine using an algorithm that doesn't belong to the std namespace as long as its compatible with containers. Like, say, boost::binary_search. For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous. I was just trying to solve an exercise in Munkres' Analysis on Manifolds to get better at continuous functions and topology in $\mathbb{R}^{n}$, and I got stuck. Here it is:      Let $X = A \cup B$, $A$ and $B$ are subspaces of $X$. Let $f:X \rightarrow Y.$ Suppose that the restricted functions $f_{|A}:A \rightarrow Y$ and $f_{|B}:B \rightarrow Y$ are continuous. Show that if both $A$ and $B$ are closed in $X$, then $f$ is continuous.    Our definition of continuity is:  $f$ is continuous at $x \in X$ if for each open set $V$ of $Y$ containing $f(x)$, there is an open set $U$ of $X$ containing $x$ such that $f(U) \subset V$. $f$ is continuous if it is continuous at each point of its domain.  What I've tried so far:  Suppose $A$ and $B$ are closed in $X$. Suppose $A \cap B = \emptyset$. Then $X -A$ is open in $X$. But $X - A = B$ which is closed in $X$, which is a contradiction. Hence, $A \cap B \ne  \emptyset$. I'm not sure if this will be useful in the problem or not. If $x \in X$, then it's either in $A$ or in $B$. Suppose without loss of generality that it's in $A$. Then for all open sets $V$ of $Y$ s.t. $f(x) \in V$, $\exists$ an open set $U_{1}$ of $A$ s.t. $U_{1} \subset A, a \in U_{1}$ and $f(U_{1}) \subset V$. This is almost what we want, except that we want to find an open set $U$ of $X$ s.t. $U \subset A, a \in U$ and $f(U) \subset V$. This is because $U_{1}$ being open in $A$ does not necessarily mean that it's open in $X$ because $A$ is closed in $X$. (i know how to expand this argument out if needed) My idea now is to try to find a subset $U$ of $U_{1}$ such that $U$ is open in $X$. We have two cases:  (1) $U_{1}$ does not intersect the boundary of $A$. (2) $U_{1}$ intersects the boundary of $A$. Now I'm not sure what to do. I'm wondering if it would help to investigate the relationship between Bd(A) and Bd(B).. In any case (no pun intended), any help/advice would be greatly appreciated. :)  Sincerely,  Vien How can I effectively use the Coppersmith? I've been playing Dominion with the base + Intrigue sets for a while now, and one of the cards that I have trouble wrapping my mind around is the Coppersmith.  In theory, it seems like a great card - it allows you to make all of your Coppers into Silvers, and in some cases - even Gold. However, I have yet to build a deck during a game that manages to use it effectively. Usually, in order to pull one or two of those with a nice amount of coppers, I need to create a deck that relies heavily on +cards/+actions chains, and the Coppersmith only begins to affect the game too late for it to be of any use.  So... how can this card be effectively used? This question is regarding the minecraft Beta 1.8 prerelease -- I have a beautiful house in single player survival which I'd like to record while flying.  I want to be able to change the game mode of my survival map to creative to be able to fly through it without installing a mod. I know it is possible if you run the map on a multiplayer server via the command /gamemode &lt;user&gt; 0|1 but I want to achieve this in single player :)  I think this would be possible through editing your level.dat file as I think that's where the game mode for the map is stored. combining results of two select statements Get content from a url using php Where should I start learning? There are various topics to learn like Objects, Classes, Apex, Visualforce, SOSL, SOQL, Triggers, etc. There are so many options out there to start learning. I couldn't decide what topic to learn first, and then step by step get proficiency in all aspects of Saleforce. What is a "callback" in C and how are they implemented? From the reading that I have done, Core Audio relies heavily on callbacks (and C++, but that's another story).    I understand the concept (sort of) of setting up a function that is called by another function repeatedly to accomplish a task.  I just don't understand how they get set up and how they actually work.  Any examples would be appreciated. Excel destroying special character when saved as CSV One of the most common security best practices these days seems to be . I've never really found a good explanation for that, but I'm assuming it's to minimize the risk of a  malicious or infected script within the webroot from reading the database password.  But, you still have to let WordPress access it, so you need to expand open_basedir to include the directory above the document root. Doesn't that just defeat the entire purpose, and also potentially expose server logs, backups, etc to attackers?  Or is the technique only trying to prevent a situation where wp-config.php would be shown as plain-text to anyone requesting http://example.com/wp-config.php, instead of being parsed by the PHP engine? That seems like a very rare occurance, and it wouldn't outweigh the downsides of exposing logs/backups/etc to HTTP requests.  Maybe it's possible to move it outside the document root in some hosting setups without exposing other files, but not in other setups?    Conclusion:  After a lot of back-and-forth on this issue, two answers have emerged that I think should be considered the authoritative ones. Aaron Adams makes a good case in  wp-config, and chrisguitarguymakes a good . Those are the two answers you should read if you're new to the thread and don't want to read the entire thing. The other answers are either redundant or inaccurate. I am running Ubuntu 14.04 and Notepad++ 6.6.4 (through WINE)  Under the 'Run' tab, I clicked on "Launch in Firefox" , but nothing happened   I uninstalled Notepad++ and WINE, then shutdown. I re-installed Notepad++ and WINE and still have the same problem.   Does anyone know what the problem is? Let $f: M \to N$ be a mono in $\mathsf{Mon}$. I want to prove that the underlying function $U(f): U(M) \to U(N)$ is injective.   How can this be done ? Thanks in advance.