ÿþ<html> <head> <title>Albert van der Sel : Intro "weak measurements" in Quantum Mechanics</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> </head> <body> <h1>A few simple notes on some special results of "Weak Measurements" and paradoxes.</h1> Version : 0.5<br> Date : 29/01/2012<br> By : Albert van der Sel<br> Type of doc : Just an attempt to decribe the subject in a few simple weak words. Hopefully, it's any good.<br> For who : For anyone interested. <br> <hr/> <br> <font face="arial" size=2 color="black"> <h2>A few simple notes on some special results of "Weak Measurements" and paradoxes in Quantum Mechanics.</h2> <br> <font face="arial" size=2 color="brown"> <B> &#8658; Before you start, please read this first:<br> <br> First of all, please take notion of the title of this note. It does not make any attempt<br> to describe the full mechanics of "Weak Measurements" in any comprehensive way.<br> <br> The focus will be on some special, but hardly proven, interpretations resulting from those measurements<br> and underlying theory.<br> <br> Especially, if a "time sysmmetric" approach on the measurement theory is followed, it seems to imply that pre-selected<br> results (from the past), and post-selected results (in the future), might have bearing on what<br> you measure <B>presently</B>, under certain circumstances.<br> Mind you, this is quite a statement.<br> <br> Also, some a special classes of experiments produce results which are difficult to understand if not a deviation<br> from the "standard approach" in QM (forward time only) is followed, as is realized in the "Time Symmetrized QM".<br> <br> As already said, this note will only focus on some of the interpretations that might result from the above experiments,<br> and theoretical frameworks.<br> <br> Actually, it's no more than a humble attempt <I>to stimulate curious readers</I>, who are new on the subject,<br> to proceed to real scientific articles (but hopefully, after reading this note first).<br> </B> <br> <br> <font face="arial" size=2 color="black"> <h3>1. Introduction:</h3> An observable (and suppose it's initially is in a superposition of different eigenstates), appears to reduce to a single value,<br> after interaction with an observer (that is: if it's being measured).<br> <br> Now, this key principle of QM, has various interpretations, like the Copenhagen "Collapse of the wave function", or<br> the "Decoherence" theory, which states that only particular pointerstates are "einselected" or leaked into the environment,<br> due to the interaction of the observed system with measuring device (the environment).<br> Especially the "Decoherence" theory, clarifies that the measurable quantities (observables) of the system,<B><br> when actually measured,</B> produces results, which also are influenced by the <B>specifics</B> of the experiment.<br> <br> In some way, in all discussions of the sort like above, we can say that the "art of measuring a system",<br> are <I>"pretty disturbing"</I>: in one way or another, you have forced the system into a <B>reduced state.</B><br> Sometimes, these conventional measurements are called <B>"Strong measurements"</B>, (or "strong perturbative measurements"),<br> as to stress the fact that these types of measurements really influences an observed system (and reduce it in some way).<br> <br> Contrary, although initially it sounds incredable, for certain classes of experimental setups, a method has been devised<br> to perform measurements without hardly disturbing a system. These special setups are called <B>"Weak measurements"</B>.<br> Ofcourse, not every QM experimental setup, is a good candidate for weak measurements. It seems that especially two-state<br> systems lends themselves quite well for these setups. This seems to be a limiting factor, and indeed it is.<br> But not disturbing a system during a measurement, in such a way that the observable does not "collapes" or "projects" onto<br> an eigenvalue, already sounds rather contradictive to QM (but it is not).<br> <br> However, at least in general terms, since the measurement is non-perturbative, what you get back is very weak too.<br> This means that generally you must perform many measurements on systems prepared in the same state (also called<br> an "ensemble").<br> <br> The whole idea is relatively new. It was first published in 1988, by Aharonov, Albert and Vaidman (AAV),<br> although even in 1964, Aharonov, Bergman, and Lebowitz probably layed out the fundamentals of the whole idea.<br> <br> We should probably distinguish two important categories here, but then where both are very related to each other.<br> <br> First, we can say that a (relatively new) general theoretical framework has emerged, of which a <B>time-symmetrical</B><br> approach (fowardtime- and backwardtime evolution) is a key element. The "Two State Vector Formalism", or the<br> "Time Symmetrized QM", are important representatives <br> <br> Secondly, a new class of experiments were devised, which (as we have seen above) are called "weak measurements".<br> <br> As for the latter: only very recently, say as of 2008, startling experiments have been performed, which <B>might</B><br> have an effect on how we must view at certain aspects of QM. I say "might", because as always, different interpretations<br> emerge from newly discovered phenomena. Some indeed are quite spectecular. <br> <br> One special interpretation goes like this: <I>"for what you find at any point in time, it s not just the past<br> that is relevant. It s also the future </I> Put it this way, it would disturb what we usually see as the "arrow of time".<br> <br> Mind you, in a pure classical system, it would not be a surprising statement. But in QM, where predictions are supposed<br> to be statistical in nature, it really is.<br> <br> Sometimes it's not easy to (sort of) qualify theories properly. For example, a theory like Newtonian mechanics can<br> be called a <I>classical theory</I> . Now, the usual Quantum Mechanics we know (like Copenhagen interpretation), is<br> not classical.<br> But let's call the "usual interpretations" (which collectively have resulted in Quantum Mechanics),<br> the <B>"standard views"</B> of Quantum Mechanics. Ofcourse, there really is <B>no</B> standard view of Quantum Mechanics, <br> but this time we really need a term to <B>distinguish</B> the <B>new</B> (rather fragile) interpretations, from the standard ones.<br> Do we need it then?<br> <br> As far as the theory is concerned, most would say "yes", since a time symmetrical approach is quite different from the<br> standard one.<br> <br> <br> <br> <h3>2. Some key notions:</h3> <B><U>Weak Measurement:</U></B><br> <br> Often, a weak measurement is described as follows:<br> <br> If the interaction of a measuring device and a system is made very weak, then the system will be negligibly "disturbed"<br> by the measurement (it should not "collapse" or "decoheres"), and any value measured will be so low, that it is<br> quite meaningless by itself. However, if a large number of measurements is made, then the average will converge to a value<br> defined as the "weak value" of the operator being measured.<br> <br> The measuring device is often called the "pointer" device, with that name (also) choosen as to refer to<br> a traditional analog device, suggesting that the movement of the "output needle", at a weak measurement, is negligable.<br> With a perturbative measurement, the movement of the "output needle" would be very significant.<br> <br> <font face="arial" size=2 color="blue"> As a simple analogy of the above: suppose we have a droplet of some oil on a smooth table.<br> You probably have seen it yourself, that if you <I>very gently touch</I> the droplet with a spoon (or something),<br> the droplet might not be disturbed. However if touch it <I>just a tiny less gently</I>, the droplet "collapses"<br> around your spoon. Ofcourse, this example goes bad if we dig deeper any further. However, it's just to illustrate<br> the difference between perturbative- and weak measurements, if we could call "touching a droplet" a sort of measurement.<br> Well, maybe it could be, if we for example tried to determine the shape of the droplet.<br> <br> <font face="arial" size=2 color="black"> <br> Now, please note that it's very different from the traditional "perturbative" measuerements, which results in<br> finding an eigenvalue.<br> Weak measurement is thus a procedure, performed on pre- and post-selected quantum systems and<br> the coupling to the measuring device is very weak, thereby not disturbing the system.<br> The outcomes of weak measurements, are supposed to be very different, that is, not just a probability for finding<br> the eigenvalues of the measured observable (operator).<br> If measured many times, the weak values converges to a certain value, which could be the "true" value of that observable,<br> (presumably) valid for the ensemble of systems.<br> Actually, the value obtained is interpreted as a complex number. However, the "real" part of that number<br> represents the value of the Observable.<br> <br> <font face="arial" size=2 color="brown"> <B> Why would we perform measurements this way?<br> <br> A "strong" measurement sort of changes the system. However you want to formulate that change, like<br> "collapse of the wavefunction" or "the system decoheres" or "it projects to an eigenvalue" etc..,<br> it's not easy to talk about "a true value" before this measurement.<br> So, what was <I>reality</I> before this measurement? Well, If that would be possible at all,<br> that is: Isn't QM intrinsically stochastic to begin with?<br> <br> Anyway, it is ofcourse very interesting to see what you can get, if it would be possible to measure<br> an observable without disturbing it !<br> An obvious question then is: Would you then have more, or better, information about reality?<br> <br> It seems that "having more information about reality" is probably not exactly what the physicists think<br> what is going on. They rather explain the observed effects, in a "time sysmmetric" framework.<br> If you have a certain pre-selected state (at first), and a certain post-selected state (at a later moment),<br> it sort of influences what can be found "in between" those times.<br> </B> <font face="arial" size=2 color="black"> <br> In a gentle form, a weak measurement might be described this way:<br> <br> <B>&#8658; pre selection on t<sub>0</sub>:</B><br> <br> First, you start with a system in a defined state. It's called a <B>pre-selected state</B>, because we know that <br> the system takes the state |&Psi;<sub>pre</sub> > on on t<sub>0</sub>.<br> <br> Actually, the system is in a defined state with respect to the observable A, while the system usually<br> has many more observables associated with it.<br> <br> It's custom to speak of a "prepared" state with respect to A, which can be achieved by a strong measurement<br> of A, to obtain the state |A=a><br> <br> <B>&#8658; post selection on t<sub>1</sub>:</B><br> <br> At time t<sub>1</sub> we do a "strong" measurement of the system again, with respect to observable B,<br> and only select the cases where |B=b>.<br> <br> This is then called a <B>post-selected state</B><br> <br> <B>&#8658; Perform a weak measurement on t where "t" in t<sub>0</sub> < t < t<sub>1</sub></B><br> <br> Note that we perform the weak measurement <U>in the time between</U> the pre-selection and post-selection.<br> <br> Now, in the time interval "t" in t<sub>0</sub> < t < t<sub>1</sub>, we weakly measure an observable "O".<br> Then, after many measurements on equally prepared (initial) systems, we average the results found on "O".<br> <br> Just to state for the record (we don't do any calculus here):<br> <br> In addition to the above, using certain calculus on bra en ket states, one can find expectation values<br> of O, and additional calculus can be performed on the involved operators, leading to various results.<br> <br> So, what is the essential meaning of all this?<br> Or I should refrase that into: what results are found and what is a possible interpretation?<br> If you are new to the subject, and possibly "somewhat orthodox", then: <I>Are you sure you are seated?</I><br> (at times, I like to be "dramatic")<br> <br> <br> <h3>3. Some rather remarkable interpretations:</h3> Below you will find some interpretations that a few scientists see<br> as possibly valid.<br> <br> If measurements as described above are performed, and a surprising consistent correlation can be found<br> on what you average as (say) O=O<sub>o1</sub> under the condition that A=a and B=B<sub>b1</sub>, and<br> and if you find different values of O if different values of B are (post) selected, then:<br> <br> <B> If you have a certain pre-selected state (at first), and a certain post-selected state (at a later moment),<br> it sort of influences what can be found as the weak value of O "in between" those times.<br> </B> <br> This looks as if the future, and past, have determined the present.<br> <br> In true scientific notes, <I>one does not come away</I> by "just stating an interpretation", and not<br> further elaborate on it, or by not quoting the proper references.<br> <br> Although I honestly do my best to be truthfull, this note can at "maximum" be regarded<br> as a way to stimulate curious readers, who are quite new on the subject, to proceed to more advanced stuff later on.<br> <br> Also, as said before, note that there is <B>not any full consensus</B> about any specific interpretation<br> on weak measurements among physicists. <br> Especially on what's mentioned above, many have reservations, or are quite doubtfull.<br> <br> (1):<br> Remarkably, when a pointer device couples weakly to the observable O of a pre- and post-selected system<br> the average of the "readings" is not to one of the eigenvalues, but to the weak value of O.<br> <br> You may find that quite remarkable by itself.<br> <br> (2):<br> If you started out with initial preparation |A=a>, and post select this state again,<br> you might suspect that both A and O were completely defined at the weak measurements.<br> Especially with non-commuting observables, you might find that quite remarkable (see section 4).<br> This seems to be in conflict with Heisenberg's uncertainty principle.<br> <br> (3):<br> We don't do mathematics here, but some derivations of some real smart folks gives us relations like:<br> <br> <font face="courier" size=2 color="blue"> Forward Time evolution of the Weak operator at t = SomeFunction(t,t<sub>0</sub>)<br> Backward Time evolution of the Weak operator at t = SomeFunction(t,t<sub>1</sub>)<br> <br> <font face="arial" size=2 color="blue"> Ofcourse, these are not the real relations, but that is not the key issue here.<br> <br> <font face="arial" size=2 color="blue"> <B> Both relations just expresses the fact that the events at t, are (at least partly) the result of the<br> post selection in the "future", and the pre selection of the "past".<br> </B> <font face="arial" size=2 color="black"> <br> If this has any touch with reality, you may consider this to be very remarkable !<br> <br> <B><U>Critical remarks:</U></B><br> <br> Some folks argue this way:<br> <br> <I>The results are obtained, using an <B>"ensemble"</B> of entities (the systems<br> that were prepared in the same initial state).<br> So, you still can't say anything about an individual system. It's just statistics.</I><br> <br> This argument cannot be resolved "quicly". A weak measurement, as a neccessity (because it's "weak"),<br> simply needs a lot of identically prepared systems.<br> In general however, the community of scientists involved, does not see this critism as fundamental.<br> <br> Lot's of more interesting stuff can be discussed. True, this simple note leaves out a lot, but<br> if you were indeed new to the subjects, hopefully you are triggered sufficiently by now, to do more explorations.<br> <br> <br> <br> <br> <br> </body> </html>