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Non-orthogonal quantum states are not mutually exclusive

"Interpreters" of quantum mechanics deny nothing less than Born's rule

In classical physics, if we know that the system has generalized coordinates \((x_i,p_i)\), i.e. that it sits at the corresponding point of the phase space, then we may say that it certainly doesn't have generalized coordinates \((x'_i,p'_i)\) if the collections of numbers differ,\[

(x_i,p_i)\neq (x'_i,p'_i).

\] Different points of the phase space are mutually exclusive even if they are very close to each other. This lesson holds in any classical theory, including classical field theory. If two configurations of a classical field differ as functions\[

\Phi(x,y,z)\neq \Phi'(x,y,z),

\] then we may say with certainty that if the system is found in the configuration \(\Phi(x,y,z)\), then it certainly isn't found in the configuration \(\Phi'(x,y,z)\), not even if the latter is close to the former (but not equal).

The people who are incapable of understanding that the quantum revolution has overthrown the general framework of classical physics almost universally assume that the state vector \(\ket\psi\) in quantum mechanics is a form of a classical variable. They're wrong and their being wrong has very dramatic consequences.

Assume that you know that a physical system is found in the pure state \(\ket\psi\) ("psi"), e.g. because it is the eigenstate of a complete set of observables and you have just measured the values of all these observables that uniquely determine \(\ket\psi\), up to an overall normalization factor.

To put it more simply, imagine that you have just measured the value of the observable\[

\hat O = \ket\psi \bra\psi

\] and you got \(\hat O = 1\). This \(\hat O\) is a Hermitian linear projection operator and \(\ket\psi\) is its only eigenstate with the eigenvalue \(+1\) while all states orthogonal to \(\ket\psi\) have the eigenvalue equal to \(0\).

Now, let's ask the question: Is this physical system found in the state \(\ket\chi\) ("chi") which is a different state,\[

\ket\chi \neq \alpha \ket\psi,

\] which is however not orthogonal to \(\ket\psi\)? Our arguments – and the difference between classical physics and quantum mechanics – become extreme if \(\ket\chi\) is very close to \(\ket\psi\) but it is still different. Let's assume that both states are normalized,\[

\langle\psi\ket\psi = \langle\chi\ket\chi = 1.

\] Now, the people who haven't started to think quantum mechanically will imagine that the state vector is a form of a "classical field" in some space, so if we know that this "classical field" has one value, we know for sure that it doesn't have another value. In other words, the physical system in the state \(\ket\psi\) has the (conditional) probability equal to zero\[

P(\ket\chi | \ket\psi) = 0

\] to be in another state \(\ket\chi\). That's how all the anti-quantum zealots, critics, interpreters, Bohmian crackpots, many-world crackpots, GRW crackpots, and members of several other major groups of crackpots will answer the question.

However, the answer provided by quantum mechanics is very different. It is known as Born's rule. If the physical system is in the state \(\ket\psi\), the probability that it is in the state \(\ket\chi\) is equal to\[

P(\ket\chi | \ket\psi) = \abs{ \langle \chi\ket\psi }^2

\] In particular, if the states \(\ket\psi\) and \(\ket\chi\) are "very similar" or "nearly collinear" but still different, the probability that the physical system in the state \(\ket\psi\) is in the state \(\ket\chi\) approaches 100 percent. Recall once again that the answer by the anti-quantum "interpreters" is 0 percent. The difference couldn't be more striking.

As the title has already told you, two non-orthogonal states in quantum mechanics are simply not mutually exclusive. Only orthogonal states are mutually exclusive; the orthogonality is the same thing as the mutual exclusiveness, it is the translation of the mutual exclusiveness to a more precise language of mathematics.

The anti-quantum cranks are not getting this simple point. Because the point is really nothing else than Born's rule and because Born's rule is one of the most important general rules in all of quantum mechanics, you must agree that those people are really not understanding the very basics of quantum mechanics.

It's important that \(\ket\psi\) may emulate – with some nonzero probability – a (non-orthogonal) \(\ket\chi\) in all respects that are in principle measurable. So an anti-quantum zealot could say that the measurements may react to \(\ket\psi\) in similar ways that they would react to \(\ket\chi\), but in principle, one may still say that it is "strictly untrue" for the state to be in \(\ket\chi\) if it is known to be in \(\ket\psi\). They may say that only \(\ket\psi\) is the right description.

But that's not the case. If a system is known to be in the state \(\ket\psi\), the different but non-orthogonal state \(\ket\chi\) also has a nonzero probability to be the "perfectly right description" of the physical system.

The thought experiment involving Wigner's friend makes this point (and other points) clear. In that experiment, Eugene Wigner confines a killing machine containing a hammer, Schrödinger's cat, and his friend to an isolated box. The lethal device measures whether a radioactive nucleus has decayed, and when it sees the decay, it launches the hammer mechanism and kills the cat. Wigner's friend is sitting in the box and observes whether the cat is alive or dead.

(The box contains enough oxygen for all laws of the European Union about the protection of animals to be obeyed. Moreover, the friend is ethnic Russian so if he gets killed by the hammer as well, the current fascist establishment in the U.S. and the EU won't have a moral problem with the experiment.)

After some time, the nucleus evolves in some superposition of "not yet decayed" and "decayed" states,\[

\ket\psi_{\rm nucleus} = 0.6\ket{\rm notdecayed} + 0.8i \ket{\rm decayed}.

\] Because the hammer system detects the nucleus and guarantees a correlation (or entanglement) between the cat and the nucleus, the state of the nucleus+cat system evolves to\[

\ket\psi_{\rm nucleus+cat} &= 0.6\ket{\rm notdecayed}\ket{\rm alive} +\\
&+ 0.8i \ket{\rm decayed}\ket{\rm dead}.}

\] where the products of the ket vectors are really \(\otimes\) tensor products. Similarly, to be sure that quantum mechanics applies to humans as well, the nucleus+cat+friend combined system evolves to\[

\ket\psi_{\rm nucleus+cat+friend} &= 0.6\ket{\rm notdecayed}\ket{\rm alive}\ket{\rm happy} +\\
&+ 0.8i \ket{\rm decayed}\ket{\rm dead}\ket{\rm sad}.}

\] where "happy" and "sad" refer to the state of Wigner's friend if he hasn't seen or has seen a dead cat, respectively. Note that this superposition is the most accurate description of the nucleus+cat+friend system according to Eugene Wigner himself. He is sitting outside the box and knows that quantum mechanics applies to all elementary particles – including those that his friend is composed of.

So Eugene Wigner himself uses this "macroscopic superposition" vector in the Hilbert space to describe the state of the box at some moment – the box with the nucleus, cat, and Wigner's friend. We know what the right state \(\ket\psi_{\rm nucleus+cat+friend}\) is. Again, we may ask the following question: Is it right to describe the physical system by a different state, e.g. by the state below?\[

\ket\psi_{\rm nucleus+cat+friend} = \ket{\rm notdecayed}\ket{\rm alive}\ket{\rm happy}

\] We have erased the coefficient \(0.6\) in order to keep the state normalized.

The anti-quantum zealots would answer "No". If the previous "macroscopic superposition" is the right state to describe the system, another state just can't be right. Except that from Wigner's friend's viewpoint, things are different. By that time, he has made an observation and he has either seen a dead cat (the probability is 64 percent) or he has seen a cat that was still alive (the probability is 36 percent).

The latter possibility – the cat is still alive (because the nucleus hasn't decayed yet) – is a perfectly conceivable option. If this option is right, Wigner's friend obviously uses the simple state vector including just one term because he has made a measurement and has eliminated the possibility that the cat has already died. So from the friend's viewpoint, the "purely alive" state vector is the most accurate element of the Hilbert space to describe the physical system.

The "interpreters" think that there is a paradox in quantum mechanics because the wave function at that moment should either objectively be "collapsed" to the single ("alive") term, or it should still be in the "superposition" state. They don't like either option. If the "right state" is already collapsed, then quantum mechanics has to be supplemented with some non-unitary extra mechanisms that "collapse" the state vector if there are conscious observers, and – they believe – quantum mechanics therefore has to define the soul and becomes unscientific.

Or "the objectively right state" is the superposition which is also bad because it contradicts the fact that people observe "sharp" results – they see that the cat is either alive or dead.

However, quantum mechanics solves this situation completely differently – none of the options advocated by the "interpreters" is right. Quantum mechanics says that the state vector summarizes the observer's knowledge about the system. Eugene Wigner uses the superposition; his friend who has already made the measurement uses the single-term "collapsed" or "purely alive" wave function.

They use different states and the point of Born's rule is that there is no contradiction because these two different states are not mutually exclusive.

By showing that the right wave functions used by the two men at the given moment are different, we haven't shown any paradox in quantum mechanics because the two wave functions don't exclude one another. Their inner product is nonzero – they are non-orthogonal – which means that if we know that the system is found in one of these pure vectors, it may very well be in the other vector, too.

Only if we could prove that a physical system is found in two different and mutually orthogonal states \(\ket{\psi_1}\) and \(\ket{\psi_2}\), there would be a contradiction! A "contradiction in practice" would also emerge if the two states were nearly orthogonal – if the calculated probability were tiny enough so that the option may be "ruled out in practice".

Again, I want to emphasize that the explanation of the non-exclusiveness above isn't open to any "interpretations", "arts", or other forms of bullšiting. It is nothing else than Born's rule. There is no way to express Born's rule that would make two nearby but different state vectors mutually exclusive. The very point of Born's rule is that they are not mutually exclusive and the rule actually quantifies the probability that a physical system in one state is in another state.

One more point. In the thought experiment with Wigner's friend, there seems to be an asymmetry between the two men. Wigner seems to be more free, superior, and (by avoiding "collapses"), he is performing a more accurate calculation that takes the (very small) possibility of future interference effects between the "alive" and "dead" parts of the wave function into account.

One may look at the situation in this asymmetric way. However, one may also see the situation of the two men in a symmetric way – only up to a moment, however. Let's return to the experiment and recall that Eugene Wigner describes the state of the box using the state\[

\ket\psi_{\rm nucleus+cat+friend} &= 0.6\ket{\rm notdecayed}\ket{\rm alive}\ket{\rm happy} +\\
&+ 0.8i \ket{\rm decayed}\ket{\rm dead}\ket{\rm sad}.}

\] If he has very fine gadgets capable of seeing some relative phases in large systems, he may in principle measure the observable\[

\hat O_{\rm Wigner}=\ket\psi_{\rm nucleus+cat+friend} \bra\psi_{\rm nucleus+cat+friend}

\] which is, once again, a linear Hermitian projection operator. Its eigenvalues are zero or one. If he gets "one", then he is sure that the wave function (after the measurement) is indeed the "macroscopic superperposition"\[

\ket\psi_{\rm nucleus+cat+friend}

\] written above (up to an irrelevant complex normalization factor). Now, his friend in the box has already seen that the cat was still alive – he has made the measurement – but he may still ask what would be the result of the measurement of\[

\hat O_{\rm Wigner}

\] performed by Eugene Wigner outside the box. Wigner's friend does the calculation and he finds out that the probability is nonzero for Wigner to obtain \(\hat O_{\rm Wigner}=1\). In fact, the probability is determined by Born's rule i.e. by the inner product so it is equal to 36 percent.

You may say that this result is wrong because Wigner's own "superior" perspective seems to imply that the probability "should be" 100 percent to get \(\hat O_{\rm Wigner}=1\).

However, it's important to realize that there is no contradiction. As long as the predicted probability of the outcomes that actually materialized was nonzero, things are just fine. Wigner's friend's "wrong" prediction of the probability is an artifact of his perspective that completely neglects the possible re-interference of the portions of the wave function. The very fact that he "perceives" one outcome at that time means that he is admitting that all his predictions that depend on the relative phases of the parts of the wave function may be wrong.

You may also try to "measure" the probability of \(\hat O_{\rm Wigner}\) accurately by repeating the same experiment many times. If Eugene Wigner prepares the whole box in the very same state many times and at the end, he measures \(\hat O_{\rm Wigner}\), he will always find that this operator's value is equal to one.

Can Wigner's friend "measure" that the probability is 36 percent by repeating the very same experiment many times? Well, he cannot. If the same experiment with the box is repeated many times, Wigner's friend will sometimes see an alive cat, sometimes he will see a dead cat after the same amount of time.

More importantly, the asymmetry between Wigner and his friend enters at this stage. Eugene Wigner is the master of the whole box so he may be sure that the whole system is prepared in a specific state. But Wigner's friend lives in the box and he has no control over the state of particles outside the box, e.g. electrons in Wigner's brain. The friend can't really measure Wigner's brain, at least not with the required accuracy, so he can't ever be sure that Wigner is found in a pure state etc. He will have to describe all objects outside the box using a mixed state, a density matrix.

A "sufficiently" mixed state predicts that the probability of nearly everything is nonzero. The exactly vanishing probability of some outcome is only possible if the system is an eigenstate of an operator, so it must be pure (or at least not near-maximally mixed).

Wigner's friend's perspective and predictions are inevitably inaccurate due to his limited abilities to measure the physical objects outside the box. However, quantum mechanics gives him a method to predict things in this context, too. There will never be a contradiction – two orthogonal (mutually exclusive) states are always evolving into two orthogonal (mutually exclusive) states – this is mathematically expressed by the unitarity of the evolution operator – so if there are two contradictory assumptions about the initial state of the system, the probability that these assumptions will be proven by the final state (after a sequence of manipulations and measurements) is zero.

At the same moment, quantum mechanics allows one to describe physical systems with an arbitrary accuracy that may be obtained by any gadgets that the theory allows. In particular, Wigner himself is supposed to have a complete control over all the degrees of freedom – and relative phases – defining the state of the box. And with this complete control, he can make perfect probabilistic predictions that just work.

At any rate, the wave function isn't a collection of the "classical data" in any sense. A spectacular manifestation of this insight is that two different state vectors are not mutually exclusive – mutually contradicting – if their inner product is nonzero. Instead, the probability that one state "is" another is given by Born's rule. You can't understand quantum mechanics if you don't accept its logical framework. Different points of the phase space in classical physics were mutually exclusive; different state vectors in the Hilbert space are not mutually exclusive because the Hilbert space has a fundamentally different interpretation than the phase space.

Another important fact is that a value of an observable may only be assumed to be known if it is measured, and a measurement of \(\hat L\) unavoidably disrupts the physical system and especially the (probabilities of) values of all observables \(\hat K\) that don't commute with \(\hat L\), i.e. those for which \(\hat K \hat L -\hat L \hat K \neq 0\). People who are assuming that physical objects may be in pure states even though no measurements are performed are guaranteed to be led to wrong conclusions, too. But that point – measurements inevitably influence the system – may be discussed in a future blog post.

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reader Leo Vuyk said...

the semy pure states of Kaon Long/ Short decay.

Imo, Quarks are compound stringy particles able to decay by the fierce oscillating Higgs field into simpler stringy particles and exchange gluon/photons or even change internal form as the Kaon long/short system seems to do..

It is proposed that the Kaon and anti Kaon Particles (ds)are mixing with themselves and with the Eta (ss) Particle by the continuous changing of one Gluon into a Lepton to change the d Quark into an s Quark and backwards a Lepton into a Gluon from s into a Quark.
(d<=> s)
At the same time it is proposed that the backward changing d into s last longer due to the slight
chirality of the vacuum .
The angle of attack of the Tandem oscillating Higgs Particles to the Quarks, must be slightly different.
This is reason to propose that it is the S, anti-S (Eta Particle) that is the reason for the strange Kaon decay time-differences.
Explanation: the Kaon 0 Particle coded as: d-Quark:
(ORO,LOL,LOL) anti-s Quark: (OLO,OLO,ROR) will change into an Eta (s) Particle: s Quark:
(ORO,ORO,LOL) anti-s Quark: (OLO, OLO, ROR) by the direct Weak attack mechanism of the tandem vacuum Particles to change three hinges of one compound Particle at once (OLO <=> ROR, or ORO <=>LOL).
Then after a while, the intermediate (s anti-s) Eta Particle will be changed into the anti-Kano-0, coded: S: (ORO,ORO,LOL) anti-D: (OLO,ROR,ROR,)
Thus it is assumed that the origin of this strange Long and Short Kaon decay mechanism is to find in the difference in transition speed for the LOL Gluon into the ORO Electron (needed to change the d Quark into an s Quark) which is supposed to be faster due to the chirality of the vacuum than the transition speed to change the OLO Positron into the ROR Gluon (needed to change the anti-s Quark into an anti d Quark, to produce the anti Kaon-0 Particle).

reader john said...

Hi. Why Wigner's friend can't measure the probability ? He can repeat the experiments and only count the ones for which he found the cat is alive. I think it is ok to ask what Wigner will find if I find that the cat is alive.

reader john said...

I believe Wigner's friend shouldn't be able to use his wavefunction for cat+nucleus to find Wigner finds. I think the problem is here, there is no problem with repetition of experiment.

reader Edwin Steiner said...

The friend cannot repeat the whole experiment (the elaborate one that Wigner uses to verify his 100% prediction of the measurement of O_Wigner) because the friend is himself part of this experiment. He would need to prepare his own brain in exactly the same state each time, for example. The friend could run his own private series of little experiments inside the box, but then we are no longer talking about the same experiment that Wigner meant when he calculated the 100%.

reader John McVirgo said...

Leo, your theory looks awesome, but I don't understand it at a glance; could you therefore recommend a beginner/intermediate book on QM that might help?


reader Edwin Steiner said...

If I may, I recommend as an intermediate book "The Principles of Quantum Mechanics" by Dirac. It is by far the clearest QM book I have found so far - written by a true master. (I'm a bit shocked that someone gave this excellent book 1 start on amazon.) If it will help you with the mentioned theory is another question altogether.

reader Humbled said...

WF does the experiment and find the cat dead. Does this mean that W must find the cat dead as well even though his state vector is still a superposition? Could W find the cat alive? I answer from the RF of W yes. From the RF of WF, no. We tend toward a Godlike perspective with these thought experiments.


reader Luboš Motl said...

Hi, first of all, from a practical viewpoint, the agreement between the men is guaranteed, of course. As long as WF works perfectly, his happiness remains entangled/correlated with the nucleus and the cat. This is a statement that Wigner may derive. For this reason, Wigner finds an alive cat iff he finds a happy friend, and vice versa.

However, Wigner's friend is intrinsically an "imperfect", less accurate observer because he isn't equipped with the infinitely many degrees of freedom to make things fully decohere. So the state of a "happy WF" may interfere with the "sad WF" and Wigner may later make a different, less classically intuitive measurement, like the projection on a particular superposition of "happy WF" and "sad WF". The operator performing this measurement does *not* commute with the operator of the happiness of WF, so a particular state of the happiness may turn out to be just an illusion which may always be attributed to WF's imperfection as an observer.

reader Leo Vuyk said...


I would recommend my poster series on Flikr:
as a start.
secondly my essays:
Third: my e-book or paper book:

reader Mikael said...

Dear Lubos,
you always emphasize the asymmetry between Wigner and Wigner's friend. I think even if the whole mankind did nothing else than Schroedingers cat experiments until the end of the universe, Wigner and his friend would agree about the state of the cat in every single case. And if not we would attribute it to some mental disorder of any of them. So with physics being an experimental science I wonder how meaningful these assertions are.

reader ny-ktahn said...

Edwin Steiner , I am sure Leo Vuyk is just messing around with us. There is no way he believes what he is doing although I think he has a degree, and thus vetted power by some institution. Anyways I really hope he is joking. Life is too short to spend time on certain things. There is a world of fun and **sex** out there. There is also joy of study, the act of going through the thoughts of a qualified person, in the form of a text. I personally think Dirac's text is great. Whether or not you understand it at first pass is not relevant. Over time you would surely read other texts and put the pieces together, and have things make sense.

reader John Archer said...

Psst! Edwin!


Shhh!. ;)

reader Yo yo said...

"Quantum mechanics says that the state vector summarizes the observer's knowledge about the system." If I understand it, all the recent results about psi-ontic and psi-epistemic interpretations rule out the idea that the state vector is *only* about an observer's knowledge, e.g. the PBR theorem.

reader Mikael said...

Yo yo, please google the name Pusey on this blog (the P in the PBR theorem).

reader kashyap vasavada said...

@Edwin Steiner and john:

The version I heard is that Wigner’s friend is not in the box
with the cat, but he is in the same room as the box. After he opens the box he is either happy or sad when the wave function collapses or equivalently his knowledge about the system increases. The pre arrangement is that he would call
Wigner who has office in another building in Princeton, about the result. According to Wigner the wave function is still superposition until he receives the phone call from his friend. After that, according to Wigner the wave function has collapsed and both W and WF agree and both are happy or sad and go to a bar!!

reader Gene Day said...

Dirac's book is an absolute wonder of concise writing. Dirac did not waste words either verbally or by pen. It is, indeed, a masterpiece; I have never seen its equal.
You have to read the book very carefully because Dirac does not repeat himself, not ever.
(The above is not quite 100% true for I once found a repetition in the book but, for the life of me, I cannot remember what it was.)
Amazon reviews are often breathtakingly stupid but to give one star to Dirac's textbook sets an entirely new level of ignorance.

reader Leo Vuyk said...

Your "messing around" could mean "very reproductive"
I found this : Andrew Kolhoff loved the ladies,
He always touched them on their boobies,
He was messing around a bit too much,
and so God blessed him with a baby!
by darrr October 06, 2005

reader Gordon said...

Agree absolutely. The first few chapters are lucidly masterful.
(So is the rest.)
There are lots of Dirac stories---one is that during a lecture, someone said they did not understand--could he repeat what he said more clearly. Apparently then Dirac proceded to repeat his lecture from the beginning word for word, and when questioned about this, said that he already was speaking with maximal clarity.
At another seminar, Dirac was writing equations on the board and someone in the audience said,"Professor Dirac, I don't understand the equation you are writing on the upper right."
Dirac ignored the person and kept talking and writing. The person repeated, "Dr. Dirac, I don't follow the equation you are writing." Dirac continued, not acknowledging the comment. At this point, the faculty moderator said, "Dirac, are you not going to answer his question?" Dirac responded, "That was not a question. That was a statement." :)

reader kohn said...

I don't believe this is a valid argument. We can replace humans with machines. Machines can also calculate probabilities and update them when they recieve data.

reader Edwin Steiner said...

Which part you call the "box" is not important. The point is that the friend is part of the state that Wigner prepares for the experiment (by doing some initial measurements). This is very clear to me from Luboš' article:
"In that experiment, Eugene Wigner confines a killing machine containing a
hammer, Schrödinger's cat, and his friend to an isolated box." (my emphasis)
"Eugene Wigner is the master of the whole box so he may be sure that the whole system is prepared in a specific state."

reader Edwin Steiner said...

I don't know what you mean. Neither in the article nor in my post I see any point where a significant difference is made between humans and machines in a way that would affect the argument.

reader Edwin Steiner said...

Re: "There is a world of fun and **sex** out there"
Dang, I think I hit the wrong spot in the landscape! Must be a false vacuum around here or something. ;)

reader Edwin Steiner said...

Yes, there are nice anecdotes about Dirac. There is a long interview with him online, which among other things gives some glimpses of the childhood that shaped the personality of this strange and great man. I found it almost heart-breaking in places:

reader kashyap vasavada said...

@Edwin Steiner and Kohn: OK. But in the version I mentioned, WF has the control. He is outside the box which has the cat and he is competent physicist who can do the experiment. As Kohn says it could be a machine. The mechanism through which he informs W is not important. It could be a telephone or a physical review paper. Machine also can call W. Lubos’ mathematical argument about non orthogonality will go through in either case. Only the interpretation will be little bit different. Different people collapse wave function (or increase their knowledge at different times!). This would say that machine also can collapse a wave function. Mathematics of QM is the same in all the cases.That is never an issue! But about increase of knowledge,it may be human brain at the end of a chain. I think that may be an issue of interpretation.

reader WTF said...

In the comments after the other blog post I was saying something else. Namely that the two states |up> and a|up>+b|down> are different if b is not zero. You seem to be completely incapable of reading what other people write.

reader Leo Vuyk said...

Collapse of the wavefunction ( or Increasing human knowledge at different times)?
IMO we will never come closer to the mysteries of QM and GR if we dont understand how at the quantum level choices are made.
I am convinced that God ( the universal system ) plays dice at multiple entangled pinball machines called (CP(T) symmetric copy universes: equipped with instant entanglement at a long distance, then we don't have to worry about the box and the cat, but our Free will has become a bit smaller than we thought.
see: Democratic Free Will in the instant Entangled Multiverse.

So we are only partly responsible for our own (moral) choices, but not at all for cats inside boxes.

reader Luboš Motl said...

Quite on the contrary, it is you who is the impolite, anonymous person incapable of reading other people's texts - instead of being a Gentleman from Manchester which you could be expected to be. The probability that after so many exchanges, I would be misreading what he had to say was basically zero.

Just if you were asking why you had to be banned as well.

The vectors "up" and "up + b down" are mathematically different vectors, but physical systems in these pure states can't be argued to be "different with certainty". Instead, as Born's rule implies and as I have explained, even though you clearly failed to have read it, the states are different with probability |b|^2/(1+|b|^2), and they are the same with probability 1/(1+|b|^2). The latter probability - that they are the same - is nonzero for any finite "b".

reader Leo Vuyk said...

According to Quantum FFF Theory, the collapse of the wavefunction is assumed to be triggered by particle collisions with instant entanglement between (anti-) COPY universes. So there is Charge and Parity symmetry between the entangled particles living in opposiing universes. Each particle is its own observer!! Each human has to deal with one or more opposing anti-COPY ME-humans.

reader Edwin Steiner said...

Dear Luboš,

Thank you for this excellent article! I think it has improved my understanding of QM. To check: Textbooks always say that the state of a system can be represented by a unit vector |a> and that any normalized multiple of |a> will give the same predictions with 100%, so they are physically the same. What they could (and should) add is that generally any unit vector |b> will give the same predictions as |a> with |<a|b>|^2 probability, so |a> and |b> are physically the same to |<a|b>|^2 - a relation which cannot be expressed well in human languages. ("Probability" IMHO still gives the wrong mental picture that it is secretly either the same or not the same).

This also beautifully explains the under-determination of an ensemble of state vectors by the density matrix, which turns out to be a necessity. It also means that the change proposed by Weinberg in is actually a non-change. There is no physical distinction between ensembles of state vectors and density matrices in the first place, correct?

reader John McVirgo said...

Wow, what's remarkable about the interview is the sheer amount of luck involved that put Dirac in a position where we was able to be a leading player in the development of QM.

He started out with a very good degree in electrical engineering from a good university and a career path in engineering. Then because of the war, he ended up a few years later in Cambridge corresponding with Heisenberg and Born!

reader Edwin Steiner said...

I am always impressed by the sheer amount of luck that you find in the biographies of very successful people. By no means I want to diminish their achievements, but luck is always an additional factor. Fortes fortuna adiuvat. (Not showing off ;} - I googled that looking for a similar German saying.)

reader Stan Liou said...

"Ordinary mechanics must also be statistically formulated: the determinism of classical physics turns out to be an illusion, it is an idol, not an ideal in scientific research." -- Max Born

There's something *extra*-bizarre about denying the Born rule. This goes beyond even the regular "anti-quantum zealotry", in that it would kill classical probabilities as well. The reason is that the standard state vector in a complex Hilbert space formalism is general enough to handle both quantum and classical mechanics *exactly*, and classical mechanics in Hilbert space (Koopman-von Neumann mechanics) incorporates the Born rule.

Some of the main differences between KvN and quantum mechanics are exactly as one should expect: classical observables all commute with each other, and in position/momentum representation, amplitude must be decoupled from phase in order for the double-slit experiment to fail to have to interference. Because QM doesn't bother with those arbitrary-seeming restrictions, this is one of the ways in which QM is a completely natural and more powerful generalization of classical mechanics.

Still, the point is that killing the Born rule would cripple even classical mechanics, because the Born rule is what must happen when one translates talks of probabilities into the language of linear algebra. Attacking the Born rule doesn't even target quantum mechanics specifically; it's a bundle of nonsense that šits indiscriminately.

reader Luboš Motl said...

A very good way to describe it!

reader Crazy Eddie said...

If you actually did this experiment, you'd have trouble keeping Wigner's experiment in superposition after Friend does his measurement. This is the main obstacle for Quantum Computers.

reader Luboš Motl said...

No, it's not true, there isn't any problem like that. Whether "someone" - some collections of electrons and nuclei - makes an "experiment" inside the box has no implications whatsoever on the laws of physics and the fact that the system evolves to the most general superpositions dictated by Schrodinger's equation.

reader john said...

I mean you can in principle build copies of same machine. And those identical machines can do the experiment many times.

reader Florin Moldoveanu said...

Wigner's friend has nothing to do with Born rule because Born rule operates within one Hilbert space and not between Wigner's Hilbert space and Wigner's friend Hilbert space. However I agree that the ontological interpretations of the wavefunctions are deeply misguided. The reason for this is that the Hilbert space representation is just that, a representation, and there are several equivalent representations possible and paraphrasing Orwell one is not more equal than the others. What is going on in Wigner's friend case is that the same quantum system has two different Hilbert space representations. If two representations bother you, take a look in quantum field theory where there are zillions of inequivalent Hilbert space representations.

There is a way to consistently combine Wigner's Hilbert space and Wigner's friend Hilbert space into the same mathematical structure, but I'll explain how later in a paper. There in only unitary time evolution, Bohmian, MWI, GRW are plain wrong. The QM wavefunction is nether psi-ontological nor psi-epistemic. A complex QM wavefunction is just a practical tool to compute probabilities. There are other just as good tools: quaternionic wavefunctions, Wigner functions on phase space and they all give the same experimental predictions. If you want to correctly understand QM, read Peres' and Omnes' books. Peres hit the nail on the head: "quantum phenomena do not occur in a Hilbert space, they occur in a laboratory". This should cure any misguided interpretations of complex wavefunctions.