ÿþ<html> <head> <title>Albert van der Sel : Intro Interpretations in Quantum Mechanics</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> </head> <body> <h1>A few notes on Entanglement, Decoherence, and other aspects of Quantum Mechanics (QM).</h1> Version : 0.8 <br> Date : 25/01/2012<br> By : Albert van der Sel<br> Type of doc : Just an attempt to decribe the subject in a few simple words. Hopefully, it's any good.<br> For who : For anyone interested. <br> <hr/> <br> Ofcourse this is not a "great" or "well written" document. But fortunately, the Internet has many<br> excellent articles on the interpretation of QM.<br> The sole purpose of this doc, is to quickly introduce some concepts, that hopefully "clarifies"<br> a few important concepts. For example, many articles also deal on the subject of "entanglement" or "decoherence".<br> This doc touches on those concepts, without going too "deep" ofcourse, but at the same providing some clarification,<br> hopefully sufficiently, for understanding the basic concepts.<br> <br> <font face="arial" size=2 color="blue"> Contents:<br> <br> 1. A few words on "entanglement<br> 2. A few words on "non-locality"<br> 3. A few words on (classical) "Copenhagen interpretation"<br> 4. A few words on "Decoherence"<br> 5. A few words on the "Many Worlds" and the "Many Minds" Interpretations<br> 6. A few words on Weak Measurements and TSVF<br> <font face="arial" size=2 color="black"> <br> <br> <font face="arial" size=2 color="blue"> <h3>1. A few words on "entanglement":</h3> <font face="arial" size=2 color="black"> Erwin Schrödinger came up with the expression  entanglement , and called it <I> the"</I> characteristic trait of quantum mechanics.<br> <br> Many of us probably have heard of it, but often it is felt as a rather "strange" effect.<br> <br> Indeed, what is it ? Let's consider a two "particles" system. In a special case, their common state can be in such a way,<br> that we may only describe it effectively as being two parts of <I>the same entity</I>.<br> Like the word "entanglement" already suggests, it's a sort of "correlation". In better words:<br> you need both particles in order to fully 'describe' the state of the system.<br> <br> Fig 1.<br> <img src="entanglement1.jpg" align="centre"/> <br> The &Psi;(r,t) function of a single particle in QM, describes the system as a distribution in space (r) and time (t).<br> When you would consider a two particle system, the state of the two-particle system would described by the wave function<br> &Psi;(r1,r2,t). <br> If we would leave out time for a moment, and if the particle one is in &Psi;A(r) state and particle two is in &Psi;B(r) state,<br> then the total state can be written as the product &Psi;(r1,r2)=&Psi;A(r1) . &Psi;B(r2)<br> <br> But the "expression" as shown in figure 1, is quite different. Here it is not possible to seperate the entangled state<br> into product states.<br> <br> In figure 1, you see up and down arrows. This is so, because in this example, we are looking at a special property<br> that a particle might have, which is called 'spin", which can be thought of as some sort of "angular momentum".<br> Note that many articles use entangled photons, where the "polarization" is used in a similar sense.<br> <br> The expression <B> &Psi;a,b=1 / &#8730; 2 (&#8593;a &#8595;b + &#8595;a &#8593;b) </B> is quite remarkable.<br> <br> Such a state means that <B>if</B> the spin of one particle "a" is up, <B>then</B> the spin of the other particle "b" must be down.<br> The thing is, that this state can not be separated into the product state, as neither particle is in a <B>definite state</B><br> of being spin up or spin down. So, as long as we don't measure anything, we just don't know.<br> <br> Suppose that the particles initially were in close vincinity. Now, suppose that both particles "fly away" in opposite directions.<br> The weird thing is, that from a QM viewpoint, both "particles" (of the same entity) have both full "freedom"<br> in attaining any spin of up or down. We just don't know.<br> Now, if you would measure particle 1 in a detector on the left, and you would find "up", then in a detector on the right,<br> you would find "down".<br> This in itself is remarkable. Quantum Theory namely says that before you measure that particular observable,<br> it can have no defined value for this observable - it is in a superposition state.<br> The upper statement is very typical for Quantum Mechanics.<br> And now, suddely, if we find "up" on the left, then we (must) find "down" on the right. Actually, it's strange.<br> <br> You might say, "Well, ain't it how you defined the system in the first place?". Yes.., but No, not according to<br> the general findings in QM.<br> If you look again at the expression of the entangled state above, we find superposition of "up" and "down", and "down" and "up",<br> and for just one of the individual particles, we cannot say anything beforehand.<br> You might ask: then are the particles, actually are "distinguishable"? This is a good question, and I pospone it a bit.<br> <br> A key thing is: the state is in a superposition of all possible states, and only when a measurement is done, the state<br> "collapses" (a term of the Copenhagen interpretation) into a certain value.<br> Remember, both a and b have both the same freedom to be up or down, as long as as they are "opposite".<br> <br> <br> <font face="arial" size=2 color="blue"> <h3>2. A few words on "non-locality":</h3> <font face="arial" size=2 color="black"> Now, intuitively, if particle a and particle b are really close together, we might not be too disturbed, if on the left,<br> we measure "up", and on the right we measure "down" (or vice versa), although it doesn't feel right acording to the superposition<br> principle of substates, as expressed in QM.<br> <br> But now, suppose that the detectors "left" and "right" are really quite far apart, and if we then still observe the same<br> results, it might get more puzzling.<br> <br> Suppose partice a is found to be "up". Now suppose particle "a" has a nifty "means" to inform "b", that it was measured as "up", and thus<br> "b" now knows that it has to be "down". Now suppose that really such a fantastic mechanism would exist.<br> But there is a problem if the distance between "a" and "b" is so great, that "a" would have to use a mechanism that exceeds the<br> speed of light.<br> Indeed, certain experiments suggests that the correlation still remains, even if there is no way that "a" can inform "b", or "b"<br> can inform "a", how it was measured, unless a superluminous signalling mechanism would be at work.<br> Clearly, that is not generally acceptable, since it conflicts with Special Relativity Theory of Einstein.<br> Also, this is what Einstein called "the spooky action at a distance".<br> <br> You know, there are many "interpretations" out there, to explain the effect. <br> When you would believe in <B>"Local reality"</B>, you would say the <B>state</B> of an observable<br> would be clearly defined. Now, you could <I>measure</I> some observable and obtain a certain result.<br> It's quite accepted, that when you measure some observable, in many cases you would interact with that system, <br> and thus you would have exterted some influence on that system.<br> It's "pretty close" to what the idea of "local realism" actually is, but not entirely.<br> Local realism is somewhat opposed to what QM seems to "look like": <B>QM is probabilistic in nature.</B><br> If you would favour the idea of local realism, you probably would not like QM too much, and you might argue that<br> QM can't be a full and complete theory.<br> <br> The essential meaning of Local realism then would be, that a system truly has some defined state, and is not "fuzzy" at all.<br> Even if it would just "appear" to be probabilistic (fuzzy), then below the surface, something like "hidden variables"<br> are at work, (which we don't know about, yet) which would make it "exact".<br> <br> The "hidden variables" would then also be the mechanism to explain the seemingly non-local effect, as we have seen at those<br> two particles, seperated over a long distance.<br> <br> However, most physicists accept that "non-locality" is a fundamental aspect of Nature.<br> Some scientists have come up with some other very interesting arguments. In effect they say:<br> We are used to the fact that <B>space</B> and <B>time</B> are our defining "metrics" to valuate experimental results.<br> But (as they say), quantum non-locality shows us that there has to be more, something that we are just not sure about yet.<br> <br> <font face="arial" size=2 color="brown"> Note:<br> There are some ideas though, on the "new" structure of space and time, to explain nonlocality.<br> One idea is the concept of "prespace". Partly based on original ideas of Bohm, Hiley (and others) formulated the concept<br> of "prespace", which essentially says that the "usual" space-time manifold emerges from a more fundamental level of physical space.<br> Using a "Quantum Potential" and a new revised idea of "time", they formulated a framework which might explain nonlocality.<br> See "antapex.org/quantumphysics.htm" for links exploring that concept further.<br> <font face="arial" size=2 color="black"> <br> <br> <font face="arial" size=2 color="blue"> <h3>3. A few words on the (classical) "Copenhagen interpretation"</h3> <font face="arial" size=2 color="black"> The measurement of an "observable" of a system like a particle, can be described in several ways.<br> One method which "seems" to be quite natural in QM, is to to describe the system as a superposition of "eigenfunctions",<br> just like you can describe a vector in n-space as being composed of n orthogonal eigenvectors.<br> (Note: Below is a very simple example of a socalled a "bra" "ket", that is,"|>" equation.)<br> <br> Fig 2.<br> <br> <img src="qm1.jpg" align="centre"/> <br> In this way, it's also described, what the "probabilities" of the possible values of the observable could be.<br> These probabilities are namely <I>related</I> to the <I>coefficients</I> of the above equation.<br> <br> So, let's suppose that some sort of "measurering apparatus" finds value "c3", then at least one important question arises.<br> Why did the system collapsed into the state c3|3>, and what happened to all the other eingenstates?<br> <br> This is an example of the socalled "collapse of the wavefunction". If an actual measurements is done,<br> out of many possible results, just one is "selected" in "some way"<br> An observable, initially in a superposition of different eigenstates, appears to reduce to a single value<br> of the states, after interaction with an observer (that is: if it's being measured).<br> <br> It has puzzled many people for years, and different interpretations have emerged.<br> Although "Quantum decoherence" (more on that later) is quite accepted by most folks nowadays,<br> other interpretations exist as well.<br> <br> The so-called "Copenhagen" interpretation, accepts the wavefunction (or state vector) as a <I>workable</I> solution,<br> and the collapse that happens at a measurement, is a way to describe why a particular value is selected.<br> As a simple example, they would say that a system can for example be described as &Psi; = a|1> + b|2>,<br> and we have probability |a| to find the observable to be in state |1>.<br> <br> So, it says that a quantum system, before measurement, doesn't exist in one state or another,<br> but in all of its possible states at once.<br> <br> The "Copenhagen" interpretation is most often "associated" with the principle of the <B>"collapse of the wave function"</B>,<br> when a measuerment is done. But it does not strictly mean that all who favour this description,<br> also believe in the physical reality.<br> <br> Bohr seemed to have gone one step further (in a later phase), by essentially saying that the wavefunction<br> is not equal to a "true" <B>pictorial</B> description of reality, but instead is (just) a symbolic representation.<br> <br> With a litle "bending and twisting", you might call this view and methodology, something like the "early"<br> cornerstone of QM. The theory is succesfully applied at many physical systems.<br> For example at the Hydrogen atom, any eigenstate of the electron in the hydrogen atom is described fully<br> by a number of quantum numbers. Also, the actual state of the electron may be any superposition of eigen states.<br> <br> In the Copenhagen interpretation, the wavefunction is a superposition of eigenstates. In slightly different words:<br> It says that a quantum system doesn't exist in one state or another, but in all of its possible states at once.<br> A very simple example is this: &Psi; = a|1> + b|2><br> It's only when we observe its state, that a quantum system is forced to choose one "probability", and collapse<br> into a certain eigenstate (which produces a certain value for an observable)<br> So, When we observe an object, the superposition collapses and the system is forced into <B>one of the states</B><br> of its wave function.<br> <br> In figure 3, you see a few other examples of "superposition", and ways to denote the wavefunction:<br> <br> Fig 3.<br> <br> <img src="entanglement2.jpg" align="centre"/> <br> Many people have disliked the postulate of a wavefunction that collapses, without a associated physical reality, as was<br> implied by the Copenhagen Interpretation.<br> <br> One philosophical question is this: How and why does the unique world of our experience, at measurement, then emerge<br> from the multiplicities of <B>all alternatives</B> available in the superposed quantum world?<br> Also, as said before, the Copenhagen interpretation states that a quantum system, before measurement,<br> doesn't exist in one state or another, but in all of its possible states at once.<br> <br> One classical "thought experiment" tries to illustrate the shortcomings of the Copenhagen interpretation.<br> It's the famous "Schrodinger Cat" thought experiment.<br> <br> <B><U>Schrodingers Cat:</U></B><br> <br> When taken literaly, and to the extreme, a nice (and famous) paradox can be constructed. Suppose a cat is trapped<br> inside a completely sealed box. Inside the box, a radioactive source is present with a certain QM probability to decay.<br> If that happens, some mechanisme activates, and it releases a deadly poison, that will kill the cat immediately.<br> Now consider an external observer, who does not know the state of the cat.<br> Under the Copenhagen description, the observer whould describe the cat as: <B>|state of cat> = a|alive> + b|dead></B>.<br> Only when a "measurement" is done, that is, the observer opens the trap and checks the state of the cat, that state<br> would collapse to either "dead" or "alive". This is ofcourse somewhat absurd.<br> <br> Now, this "macroscopic" example, is really not quite comparable to the microscopic world where QM seems to dominate.<br> The point seems to be, that the paradox nicely illustrates how hard it is to deal with the description<br> of a superposition of states, and the collapse of the wavefunction.<br> <br> <br> Among many other interpretations, two nice refinements, or alternatives, emerged.<br> <br> <ul> <li>The <B>"Many Worlds Interpretation"</B> already dates from 1957.<br> This theory, although it's quite consistent, never had <I>too</I> many supporters (at that time).<br> By some, it is percieved as being too "fantastic". But that ofcourse has nothing to do with the possible validity of the theory.<br></li> <li>The other one, <B>"Decoherence"</B> stems from around 1980-1990 (or so), and received quite much acceptance in the physics community.<br></li> </ul> <br> <br> <font face="arial" size=2 color="blue"> <h3>4. A few words on "Decoherence":</h3> <font face="arial" size=2 color="black"> This rather new interpretation, has it's origins in 1980s and 1990s. In a sense, this theory "de-mystifies" certain<br> formerly unsolved mysteries, for example, why and how a quantum system interacts with the "environment", and<br> the "measuring device".<br> Also, the Copenhagen phrase "Collapse of the wavfunction" is replaced by quite a solid theoretical framework.<br> <br> Many physicists have embraced the theory, and it seems that only a minority of sceptics remain.<br> Especially technical and experimental oriented physicst, for example working on the field Quantum Computing,<br> see the theory as one of their primary working tools.<br> (For example, sometimes they want to preserve a qubit or nqubit for some time, <I>before it de-coheres</I>).<br> <br> However, in general, we all must be carefull in associating "truth" to any theory at all. It's just characteristic<br> of science, that theories ever evolve, and once in a while, even get completely replaced by better ones.<br> But the "decoherence" framework, is might well be, the best interpretation of QM we have right now.<br> <br> Decoherence is important in the area of "the measurement problem", possibly also to "the flow of time", and<br> above all it's very appealing for most physicists because it seems to solve the question on how the "classical world"<br> emerges from quantum mechanics.<br> <br> Key to the "Decoherence" interpretation, is that it was realized that the "environment" and the quantum system (like a particle)<br> are really tied into a temporary "entangled" system, for a certain duration.<br> Ofcourse, it was already long known that a measurement will influence any system, but this time, a whole framework<br> emerged from "the deep intertwinement" of the quantum system and the environment.<br> <br> In a nutshell, the central idea of the theory goes (more or less) like this:<br> <br> As usual, initially, an undisturbed quantum system, is a superposition of coherent states.<br> When the quantum system and the environment (the environment as a whole, or a measuring device) starts<br> to interact, the coherent states will <B>"decohere"</B> into socalled "pointerstates" which are really<br> determined by the environment. It means the loss of coherence or ordering of the phase angles between the components<br> of the quantum superposition. Effectively, the Copenhagen "phrase" <I>"collapse of the state vector"</I>,<br> this time has a real physical basis or "explanation".<br> After the system has decohered, these "pointerstates" corresponds to the eigenstates of the observables.<br> In other words, only selected components of the wavefunction are decoupled from a coherent system, and are "leaked" out<br> into the environment.<br> This selection of pointerstates is also called "einselection" in various articles describing decoherence.<br> <br> Again rephrased in slightly other words: the wavefuctions of the environment and the quantum system, will interact<br> in such a way, that the coherent superposition of the quantum system will <B>de-cohere</B> and "ein-selected" states<br> remain, which resembles a classical system again.<br> This too is quite important: before the decoherence phase, the system is truly a Quantum sytem, that is, a superposition.<br> Then, after the "perturbation", or de-coherence phase, the system behaves much like a classical system.<br> So, for the first time, a clear boundary has emerged between a pure Quantum state, and what is believed to be "classical".<br> <br> Note that the theory provides for a better explanation for what happens at a measurement, or interaction,<br> with the environment, compared to the Copenhagen interpretation.<br> <br> For example, do you notice that "Quantum decoherence" only gives the <B>appearance</B><br> of wave function collapse?<br> <br> In the interpretation using the principle of "decoherence", is not possible to separate an object being measured,<br> from the apparatus performing the measurement. Or, to seperate the object from the "environment".<br> In this interpretation, in any interaction between the system and the environment, decoherence takes place,<br> and not just when you use some measurement device. <br> Generally speaking, the environment and the particle are bound, or entangled, in such a way, that only a subset <br> of superimposed waves (the "pointerstates) is "selected". <br> After the de-coherence phase, the actual value of an observable is selected from the pointerstates.<br> <br> This new interpretation has quite some philosophical impact as well. The theory further describes how the environment<br> sort of "monitors" the de-cohered systems, and it affects our human perception as well.<br> There is a sort of redundancy of the pointer states in the environment, simply because there is "a lot" of environment<br> out there. So to speak, the environment "bans" arbitrary quantum superpositions.<br> Once a system has decohered, and only pointerstates remain (determined by the environment), and we as observers<br> come on the second place. The environment "observes" the system, and determines what "we get to see" from the system.<br> Ofcourse, any apparatus is part of the environment too, so it makes sense.<br> <br> What might be nice to notice, is that the "decoherence" theory implies that the quantum wavefunction is a physical reality<br> rather than a mere abstraction, or postulate, as in many other interpretations.<br> <br> <B><U>Non-locality and Decoherence:</U></B><br> <br> Nonlocality is hardly seen at macroscopic level because of "hard" decoherence, caused by so many interactions<br> with the environment.<br> But what happens with two entangled microscopic particles, flying away in different directions, and keeping their "correlation"?<br> In section 2, we touched on that subject. First of all, non-locality still seems to be a fundamental feature of QM.<br> Non-locality is ofcourse more than just the example of the two entangled particles of section 2.<br> However, it's still a "strong" example of nonlocality, that is, the absence of a local agent (see section 2).<br> <br> But decoherence is a "powerfull mechanism" to destroy quantum features, so what about "non-locality"?<br> It's not easy to answer this one. Much experimental work, and theoretical studies have been done, and it's<br> probably fair to say that a <B>very conclusive</B> answer is not reached yet.<br> At first sight, decoherence might look like a local mechanism (like a quantum system that interacts with the local environment),<br> but probably it is not, according to some studies.<br> <br> However, other studies have shown that non-locality seems to be very "robust".<br> <br> <br> <font face="arial" size=2 color="blue"> <h3>5. A few words on "Many Worlds" and "Many Minds" Interpretations: </h3> <font face="arial" size=2 color="black"> QM is absolutely astonishing by itself, but when we go to the Many Worlds interpretation, the excitement-level "shifts gear"...<br> <br> We know what the Copenhagen Interpretation globally means:<br> <br> An a quantum system, initially in a superposition of possibly many states, appears to reduce to a single value<br> of the states, after interaction with an observer (that is: if it's being measured).<br> <br> So, at the moment of measurement, the wave function describing the superposition of states,<br> appears to collapse into just one member of the superposition.<br> <br> In figure 3 above, we already have seen a few simple forms of a superposition of states.<br> <br> Take a look at figure 3, example1, again. Now, suppose, that in a subtle manner, we rephrase the former statements like so:<br> <br> <font face="arial" size=2 color="red"> <B>A superposition expresses all possible <U>"alternatives"</U> for finding the value of a certain observable.</B><br> <br> <font face="arial" size=2 color="black"> Expressed in <I>this way</I> it get's somewhat more plausible, that <I>all the other</I> "alternatives" may have realized<br> in other "branched off" Universes.<br> <br> Above is not exactly the way that Everett originally formulated it.<br> His theory goes "more or less" like described below. But whatever your opinion will be of the MWI, it really <I>does</I> resolve some paradoxes in QM,<br> like the strange "collapse of the wave function", and even provides a solution for the non-locality puzzle (section 2).<br> And..., some core aspects of the "Decoherence interpretation" (section 4), looks remarkably to the central statement of MWI.<br> <br> In MWI, there is no "collapse of the state vector". Suppose you do a measurement on a quantum system.<br> Before you actually perform the experiment, the system resides in superposition, meaning in all possible states at once.<br> This is what we already new from an undisturbed quantum system.<br> <br> Now you perform the measurement, and you find a certain value. In MWI, different versions of you will have found<br> all the other possible values of the observable. It is as if the current Universe has "branched", or "forked", into<br> multiple Universes where each version of you, is happy with it's own private value.<br> <br> I hear you think.. <I>"Will then the Universe be rebuild N times, at each ocurrence of such an event and so on and so on..?</I><br> No. The trick is in the superposition. The alternatives "are already there", so to speak.<br> <br> Everett realized, that the observer (with measurement device), and the system, forms a deep intertwined system.<br> <B>Each</B> "wave" of the superposition will "sort of interact" with the observer-measurement device.<br> From a mathematical perspective, this pair will split off, and further evolve independently from all the other possible "pairs" of<br> the {wave elements of superposition - observer}.<br> Hence, a number of independtly forked Universes will occur.<br> <br> Do you see the resemblence with the Decoherence theory? Only this time, no environment driven pointerstates are created,<br> but contrary all "relative states" will "start an existence of their own".<br> <br> If you were indeed new to this stuff: <I>don't you say "Wow!" ???</I><br> <br> <br> <font face="arial" size=2 color="blue"> <h3>6. A few words about Weak Measurements and the Two State Vector Formalism: </h3> <font face="arial" size=2 color="black"> If there is something else really stunning in recent findings in QM, it must be the implications of the socalled<br> "Weak Measurements" and the Two State Vector Formalism.<br> <br> Ofcourse, again different <I>interpretations</I> are possible, but one implication seems to be that<br> if you measure an observable without (hardly) disturbing it (and then ofcourse you need to repeat that many times<br> to get any significant result to standout above the noise), then it seems that;<br> <br> Pre-selected results (in the past), <U>and</U> post-selected results (in the future), may have impact<br> on what you measure presently.<br> <br> Take a closer look at those last few words: it's not the normal causality that we are used to.<br> <br> These new formalisms use a time-symmetric model, instead of the usual "standard" approach used in QM.<br> <br> If you want to read more, take a look at:<br> <br> <a href="weak_measurements.htm">a few notes on "weak measurements" and QM paradoxes.</a><br> <br> <br> <br> <br> <br> </body> </html>