Understanding the Cat in a Box Paradox

[Ableman]

I don't understand what the significant difference would be between microscopic and macroscopic. If the hammer is broken, then the wavefunction of the cat would be in the alive state no matter what the Geiger counter registers.

 

[mn4j]

Two-slit interference patterns with electrons is a case that is similar, but where it is more obvious. If you look through which hole the electrons go through, the pattern disappears, if you don't there is one.

Do you know of anyone that has ever ACTUALLY done this experiment as you describe? I would like to read it. I've been looking everywhere for such an experiment but came up short.

In fact I found real experiments disproving the above claim:

Time-Resolved Diffraction and Interference: Young's Interference with Photons of Different Energy as Revealed by Time Resolution
N. Garcia; I. G. Saveliev; M. Sharonov
Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Vol. 360, No. 1794, Interference: 200 Years after Thomas Young's Discoveries. (May 15, 2002), pp. 1039-1059. http://www.jstor.org/view/1364503x/sp020018/02x0298l/0

The authors say:

In #3 we present interference experiments with photons of different energies going
through different slits. We know at which slit the blue and red, high- and low-energy,
photons arrived, although we cannot distinguish these photons when they reach the
screen. This observation may indicate that some modification is needed in the books
that state that interference is produced only if we do not know through which slit each
photon goes. In our experiments, the indeterminacy is after the slits.

I also hear claims about how wavefunction collapse has been experimentally proven. How can that be. The wavefunction by definition can NEVER be measured experimentally because it supposedly collapses as soon as you measure it. How then can a person claim with a straight face that wavefunction collapse has been demonstrated experimentally, I will really like to see the article in which a wavefunction was measured before it collapsed, and after it collapsed. There is no other way to experimentally prove that there is such a physical process as wavefunction collapse.

As you can see wavefunction collapse is a non-falsifiable theory which tells you a lot about it's validity.

 

[Anonym]

The Geiger counter's "click" is an irreversible process. It's that which fixes one of the two possible values of the result. In that moment wavefunction has collapsed and there are no more superpositions.

You make me crazy. I put the cat in the box alive and then take him out alive (the hammer is broken). Where you see an irreversible process?

Regards, Dany.

 

[Anonym]

I don't understand what the significant difference would be between microscopic and macroscopic.

Read E. Schrödinger, “THE PRESENT SITUATION IN QUANTUM MECHANICS”. When you will understand the significant difference between microscopic and macroscopic, we will continue the discussion.

Regards, Dany.

 

[OOO]

Do you know of anyone that has ever ACTUALLY done this experiment as you describe? I would like to read it. I've been looking everywhere for such an experiment but came up short.

In fact I found real experiments disproving the above claim:

Time-Resolved Diffraction and Interference: Young's Interference with Photons of Different Energy as Revealed by Time Resolution
N. Garcia; I. G. Saveliev; M. Sharonov
Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Vol. 360, No. 1794, Interference: 200 Years after Thomas Young's Discoveries. (May 15, 2002), pp. 1039-1059. http://www.jstor.org/view/1364503x/sp020018/02x0298l/0

The authors say:



I also hear claims about how wavefunction collapse has been experimentally proven. How can that be. The wavefunction by definition can NEVER be measured experimentally because it supposedly collapses as soon as you measure it. How then can a person claim with a straight face that wavefunction collapse has been demonstrated experimentally, I will really like to see the article in which a wavefunction was measured before it collapsed, and after it collapsed. There is no other way to experimentally prove that there is such a physical process as wavefunction collapse.

As you can see wavefunction collapse is a non-falsifiable theory which tells you a lot about it's validity.

There has never been a single experiment where the electromagnetic field has been measured in whole space or even in every point of a finite volume. Yet most of us believe somehow in the reality of the EM field. Likewise the wave function. It's clearly an abstraction but it works. If we are able to predict what comes out in the end, we may use any abstraction. If disproven we will finally end up with another abstraction.

I see no point in fighting against the confused religious ideas of various dead persons. So I repeat what I have said before: if you have something better, tell us about it.

 

[Ableman]

I think a thought experiment is adequate to answer this, though it is true that I have not seen the experiment described (though my professors spoke of it, and it is described in my physics book. There though the explanation is that when you put a detector on, it adds a random phase to the electron, which makes the interference pattern disappear, but to me that seems superfluous, because it would lead to very strange, though certainly not impossible results, if the pattern did not disappear).

Well, first, I'd like to say that the interference pattern is real, I've done this experiment with light in my physics class. Also, I think we can all agree that light is quantized (though I spoke of electrons earlier, light should work just as well). Now, please imagine what would happen if the wavefunction does not collapse when you detect through which slit the photon passed. You would get an interference pattern still. But once the electron is on the other side of the slits, they should have no effect on it. (This is one part where I could be wrong, but it seems it would be strange if a slit the electron never even interacted with could affect it). So, since the other slit never interacted with the photon, it would be very improbable that the interference pattern would emerge.

To resolve this you would need to allow one of three choices, as far as I can see. 1. The electron interacts somehow with the slit that it didn't go through. 2. It is possible to build a sort of improbability machine that makes normally unlikely outcomes likely again. 3. The wavefunction collapses when you detect which position the object is in. I prefer the third one.

Also, as I understand, you can't measure a wavefunction, it is just a mathematical convenience. Heisenberg had equally accurate results using matrices. We use the wavefunction because it is more familiar to most physicists.

The experiment you linked to is very interesting and does appear to disprove what I'm saying. As such it makes me think that either option 1 or 2 is true, however those go even further against instinct than Quantum Mechanics does (though granted that doesn't make them untrue). Although something else to me seems out of the ordinary in that experiment, mainly that they can trace which slit the electron went through but not where it landed on the screen. So, it doesn't exactly disprove it, it merely says that the indeterminacy can come in somewhere other than which slit did the photon go through, it requires some modification. That is, the experiment implicitly says that if they had been able to trace the photon to the point it landed on the screen, it would not have created an interference pattern.

In fact, upon reading further of the experiment it becomes much more interesting, because "The same results have been obtained when slits were discarded and interference of the two beams emerging from the fibers occurred" italics theirs. To me this seems to say that the slits in this experiment are unnecessary and thus it doesn't disprove the collapse of the wavefunction when it is measured.

 

[reilly]

I also hear claims about how wavefunction collapse has been experimentally proven. How can that be. The wavefunction by definition can NEVER be measured experimentally because it supposedly collapses as soon as you measure it. How then can a person claim with a straight face that wavefunction collapse has been demonstrated experimentally, I will really like to see the article in which a wavefunction was measured before it collapsed, and after it collapsed. There is no other way to experimentally prove that there is such a physical process as wavefunction collapse.

As you can see wavefunction collapse is a non-falsifiable theory which tells you a lot about it's validity.

Think about the following. QM is about probabilites; trying to predict what will happen, when there is a portfolio of possible outcomes. It was not until the birth of QM that reality was ascribed to a probability function. It's very convenient in practice to equate wave- function- generated probabilities as mental states -- your brain can carry knowledge of each outcome. Once you know the outcome, your knowledge collapses, hence the wave function collapses. -- via neural phenomena --. This is simple, and recognizes that probability is just that, probability, whether quantum or classical. Sir Rudolph Peierls was a strong proponent of this approach.

This approach, by no means precludes interference effects, or other odd quantum phenomena.

Regards,
Reilly Atkinson

 

[Ableman]

Read E. Schrödinger, “THE PRESENT SITUATION IN QUANTUM MECHANICS”. When you will understand the significant difference between microscopic and macroscopic, we will continue the discussion.

Regards, Dany.

Very well, will come back later, thanks for directing me to a book that may satisfy my curiosity.

 

[Anonym]

Very well, will come back later, thanks for directing me to a book that may satisfy my curiosity.

It is not a book. It is a paper, for example, in J.A. Wheeler and W.H.Zurek,”Quantum Theory and Measurement”, p.152 (1983). After it take G. Greenstein and A.G. Zajonc “The Quantum Challenge” Jones and Bartlett Pub., Sudbury MA. (1997).

Regards, Dany.

 

[mn4j]

I think a thought experiment is adequate to answer this

I'll tell a story:)

A man was very depressed to the point he started believing he was dead. His family tried for weeks to convince him he was alive, but he kept saying "I'm dead". So they took him to an expert who explained to him and succeeded in convincing him that "dead men don't bleed". He believed it and kept repeating it to himself "Yes, dead men don't bleed. Yes, dead men don't bleed. .." At that moment the expert pierced him with a needle and he started to bleed. The hope was that the man will draw the obvious conclusion that he was alive. Instead, he shouted "Oh oh, I guess dead men do bleed!"​

There has never been any real evidence of any physical entity such as a wavefunction. None. A thought experiment can never be evidence for the existence of a physical entity such as a wavefunction.

There though the explanation is that when you put a detector on, it adds a random phase to the electron, which makes the interference pattern disappear, but to me that seems superfluous, because it would lead to very strange, though certainly not impossible results, if the pattern did not disappear).

The experiment I quoted to you disproves the idea that knowing where the electron passes collapses the wavefunction. The authors set out to measure the duration of the wavefunction collapse. The found no evidence of a wavefunction. They state as much in their results.

Well, first, I'd like to say that the interference pattern is real, I've done this experiment with light in my physics class. Also, I think we can all agree that light is quantized (though I spoke of electrons earlier, light should work just as well). Now, please imagine what would happen if the wavefunction does not collapse when you detect through which slit the photon passed.

Which wavefunction. Show me an experiment that proved the existence of the wavefunction. Your question is similar to the guy in my story above explaining to the expert "Since I am dead and I bleed therefore dead men bleed".

You would get an interference pattern still. But once the electron is on the other side of the slits, they should have no effect on it. (This is one part where I could be wrong, but it seems it would be strange if a slit the electron never even interacted with could affect it).So, since the other slit never interacted with the photon, it would be very improbable that the interference pattern would emerge.

It is not strange, see this paper (http://docto.ipgp.jussieu.fr/IMG/pdf/Couder-Fort_PRL_2006.pdf) for a completely classical experiment experiment showing that double slit diffraction can occur even when the particle only passes through one slit. It is wrong to assume that the only way to have interference is through a probability amplitute wavefunction.

To resolve this you would need to allow one of three choices, as far as I can see. 1. The electron interacts somehow with the slit that it didn't go through.

If you read the article I quoted above, this seems to be the most likely scenario.

2. It is possible to build a sort of improbability machine that makes normally unlikely outcomes likely again. 3. The wavefunction collapses when you detect which position the object is in. I prefer the third one.

This makes no sense. Probability is an epistemological property not an ontological one. By definition, probability wavefunctions are never real "things".

Also, as I understand, you can't measure a wavefunction, it is just a mathematical convenience. Heisenberg had equally accurate results using matrices. We use the wavefunction because it is more familiar to most physicists.

EXACTLY! Then stop trying to explain "WHEN" or "HOW" the wavefunction collapses. Those are ontological questions which make no sense when dealing with epistemological issues like you just admitted. The mathematics works but it does not represent any real physical entity.

That is, the experiment implicitly says that if they had been able to trace the photon to the point it landed on the screen, it would not have created an interference pattern.

No it does not. It says knowing which slits the photons went through does not disturb the interference pattern contrary to popular claims that it should.

In fact, upon reading further of the experiment it becomes much more interesting, because "The same results have been obtained when slits were discarded and interference of the two beams emerging from the fibers occurred" italics theirs. To me this seems to say that the slits in this experiment are unnecessary and thus it doesn't disprove the collapse of the wavefunction when it is measured.

Interesting point. If that were the case, it means the wavefunction collapsed the first time when they determined which slits the photons passed through, then expanded again after the slits only to collapse again at the detector!?

BTW, What wavefunction? The authors never saw evidence of any wavefunction! Nor has anyone else.

 

[lightarrow]

You make me crazy. I put the cat in the box alive and then take him out alive (the hammer is broken). Where you see an irreversible process?

Regards, Dany.

Sincerely I don't know if the irreversible process is in the interaction between the particle and the counter or in the next amplification of the signal or in both, but, however, the detection is irreversible, or it wouldn't be a detection at all; maybe this could be stated as definition of "measurement". If the hammer is broken or there is a tube with poison and a cat or what you want, it's the same for me, the detection has already happened.

 

[Anonym]

I'll tell a story:)

A man was very depressed to the point he started believing he was dead. His family tried for weeks to convince him he was alive, but he kept saying "I'm dead". So they took him to an expert who explained to him and succeeded in convincing him that "dead men don't bleed". He believed it and kept repeating it to himself "Yes, dead men don't bleed. Yes, dead men don't bleed. .." At that moment the expert pierced him with a needle and he started to bleed. The hope was that the man will draw the obvious conclusion that he was alive. Instead, he shouted "Oh oh, I guess dead men do bleed!"​

There has never been any real evidence of any physical entity such as a wavefunction. None. A thought experiment can never be evidence for the existence of a physical entity such as a wavefunction.

Are you describing yourself? The collapse is nothing more than real life realization of the Spectral Decomposition Theorem. 99.999… % of our knowledge is based on the indirect evidence. The most famous example is the adequate kinematical description of the EM field (AB phenomenon). From here the way to Yang-Mills and to all fundamental interactions in nature is almost straightforward. By the way, the statement "dead men don't bleed" is based on the indirect evidence.

Your question is similar to the guy in my story above explaining to the expert "Since I am dead and I bleed therefore dead men bleed".

Take a mirror. For me, you are the indirect evidence that the human brain is the quantum computer. We (you and me) call that the circular argument and the circular logic.

It is not strange, see this paper…

Come on! Ableman is an undergraduate physics student!

Regards, Dany.

 

[Anonym]

Sincerely I don't know if the irreversible process is in the interaction between the particle and the counter or in the next amplification of the signal or in both, but, however, the detection is irreversible, or it wouldn't be a detection at all.

That is A.Einstein statement. I doubt. It is not so with Rabi oscillations.

If the hammer is broken or there is a tube with poison and a cat or what you want, it's the same for me, the detection has already happened.

E. Schrödinger Cat is not a gedanken experiment, it is a thought experiment. The only reason that prevent it performance at least in my country is that it considered crime. In addition, it is unnecessary; the result is obvious to everybody.To emphasize that, E. Schrödinger introduced apparently too detailed description. However, I don’t know, may be also it was intentional.You are right.The description contains several collapses (measurements) and the consequent recoveries. But after all the central item is a cat and you may drop the intermediate details.Then you are right, the cat is the detector. That also was my statement in post #112.

Regards, Dany.

 

[Anonym]

By the way, to the best of my knowledge there are no paradoxes in quantum mechanics -- only pseudoparadoxes. Much like the twin pseudoparadox of special relativity, you only run into problems if you make unwarranted assumptions.

I completely agree with you (post#62) and want to illustrate your statement using Schrödinger Cat paradox. I only consider it surrealistic to discuss that in the session entitled “I won't debate on the "wavefunction collapse".

E. Schrödinger made the following assumptions:

1) The cat may be described microscopically: wrong, the collapse is the experimental demonstration that the cat is macroscopic object; otherwise the cat should disappear, it is “blurred”. (“That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks”.);

2) The transition is continuous: wrong, h is a constant, it is completely wrong mathematically to apply lim operation h->0(remember the definition of lim!); it is precisely the discontinuous gap of h/2 in the lowest bound of HUP that make all difference between the micro and the macro. (“It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy”);

3) The collapse occurs after the transition took place: wrong, when? What happens within the Newtonian mechanics that the object suddenly behaves in the previously unknown way? (“which can then be resolved by direct observation”).

Regards, Dany.

 

[shaun_o_kane]

mn4j: I stand corrected on ET Jaynes. I would never have guessed for the selection of papers I read. And I'm not being funny.

I will make one more comment. Copenhagen does not make the assertion.

(My own ignorance) => (Nature is indeterminate)

That has too many "hidden assumptions". A better but somewhat glib version might be.

(Positivism): Nature is what I can interact with.
(HUP): Nature stops me from knowing too much.
My ignorance = everyone else's ignorance => I cannot predict events with certainty => My experience of nature = everyone elses' experience of nature = indeterminate.

I strongly dispute your view of what the Copenhagen interpretation of Schrodinger's Cat is. However I have pretty much given up the will to live agruing over this topic, so I merely refer you to previous comments.

 

[IMP]

I agree. If the interaction of any/all particles just creates bits of information, those bits are created at the moment of interaction. I do not think the bits are waiting on an observer before being 'finalized'. The cat is either alive or dead, which state is already 'written in bits' at the very moment of interaction.

 

[Dadface]

Let common sense prevail and leave the cat to the metaphysicians.

A metaphysician is a man who goes into a dark cellar at midnight without a light looking for a black cat that isn't there anon

 

[ZapperZ]

I think that you should have let this thread alone, since it is more than a year old.

Zz.

 

[Vals509]

Well firstly remember that the situation is hypothtical and that it is not real. Secondly we say that the cat is dead and alive and that means not literally dead and alive but a superposition of both assuming that the cat is a particle. This is pretty much what the others have said. Secondly all the events that ocuur in the box are closed and this means that we are unaware of what is going on inside whether the outcome affects us or not. Finally if the cat is killed by the particle we still don't know that it has happened since the box is closed>

 

[klaymen]

the problem i see with this experiment is that there is no such thing as random. there will always be some measurable way of knowing when the vile will break. the only thing that matters is the physics that are making the poison vile "randomly" crack open. i would love to see exactly how a randomly activated device like that would work. because it seems to me that anything we perceive as random merely means it is something we cannot yet measure or do not yet understand.

that's not to say that someone's body heat couldn't possibly change the air currents and cause the time to alter if the 'random' breaking device used the weather somehow to determine when it is triggered. but i hardly think that has anything to do with the emotions that the observer has towards the cat.

any thoughts?

 

[ZapperZ]

the problem i see with this experiment is that there is no such thing as random. there will always be some measurable way of knowing when the vile will break. the only thing that matters is the physics that are making the poison vile "randomly" crack open. i would love to see exactly how a randomly activated device like that would work. because it seems to me that anything we perceive as random merely means it is something we cannot yet measure or do not yet understand.

that's not to say that someone's body heat couldn't possibly change the air currents and cause the time to alter if the 'random' breaking device used the weather somehow to determine when it is triggered. but i hardly think that has anything to do with the emotions that the observer has towards the cat.

any thoughts?

You have not understood the difference between the classical, statistical random events like tossing a coin, versus the "random" events in QM, i.e. the measured outcome of a superposition of states.

Zz.

 

[klaymen]

You have not understood the difference between the classical, statistical random events like tossing a coin, versus the "random" events in QM, i.e. the measured outcome of a superposition of states.

Zz.

tossing a coin is not random. physics can determine which side it will land on, you just have to know the exact measurements and location of everything affecting it. just because that would be a ridiculously tedious thing to measure, doesn't mean that it can not be done, therefore it is not random, but dependent on factors such as: the movement of the coin flipper, the chemical makeup and flow of the air that the coin is being flipped in, the weight and size of the coin, etc.

you're right, i don't understand the measured outcome of a superposition of states as i haven't looked into it. what resources would you recommend?

 

[Monitor16807]

Radioactive decay, is 50% chance.

I'd say that's pretty random.

 

[ZapperZ]

tossing a coin is not random. physics can determine which side it will land on, you just have to know the exact measurements and location of everything affecting it. just because that would be a ridiculously tedious thing to measure, doesn't mean that it can not be done, therefore it is not random, but dependent on factors such as: the movement of the coin flipper, the chemical makeup and flow of the air that the coin is being flipped in, the weight and size of the coin, etc.

you're right, i don't understand the measured outcome of a superposition of states as i haven't looked into it. what resources would you recommend?

Then maybe you shouldn't be making such definitive statement before you understand the physics.

Coin tossing is known to not be a "random" process. We lump our ignorance of the fine details of the mechanics into the randomness. But it is still deterministic, and the physics is there to verify that.

This is NOT the case in quantum mechanics. Even Einstein had tried to include "hidden variables" that so far have no empirical support. So as far as we know, there is no physics on why a measured superposition of state will produce one of the possible outcome. To say otherwise is to make an unsupported speculation.

What resources? Any standard college QM text.

Zz.

 

[moving_on]

Anyone listen to BBC Radio 4 this morning (depending where you are in the world!)
http://www.bbc.co.uk/radio4/history/inourtime/inourtime.shtml.
'The measurement problem in physics'.
Basil Hiley, Simon Saunders and Roger Penrose explaining 'the cat' to Melvyn Bragg (bless him) and Radio 4 listeners in general.
Shows how hard it is to get the subject across to beginners in my opinion.
It did start to get interesting just as I got to work, typical.
I'll have to wait to catch up later.

 

[ZapperZ]

Anyone who has been on PF long enough can testify to the fact that the issue of the "Schrodinger Cat" comes up extremely frequently. Just do a search on that and on superposition of states. The Schrodinger Cat is nothing more than an illustration of the superposition phenomenon in QM. That must always be the starting point, i.e. the superposition principle, and NOT the "cat", which is merely a "visual" example.

I've given, over the years, many references and experiments that clearly illustrates this principle, and it appears that, even with this latest series of necroposting in a very old thread, that maybe the references might come in handy again. So here it is:

These are the papers that clearly show the Schrodinger Cat-type states (alive+dead, and not alive or dead). All the relevant details are there and anyone interested should read them. Also included is the reference to a couple of review articles which are easier to read, and the reference to two Leggett's papers, who was responsible in suggesting this type of experiments using SQUIDs in the first place. Again, the papers have a wealth of citations and references.

The two experiments from Delft and Stony Brook using SQUIDs are:

C.H. van der Wal et al., Science v.290, p.773 (2000).
J.R. Friedman et al., Nature v.406, p.43 (2000).[ArXiv version can be found here]

Don't miss out the two review articles on these:

G. Blatter, Nature v.406, p.25 (2000).
J. Clarke, Science v.299, p.1850 (2003).

However, what I think is more relevant is the paper by Leggett (who, by the way, started it all by proposing the SQUIDs experiment in the first place):

A.J. Leggett "Testing the limits of quantum mechanics: motivation, state of play, prospects", J. Phys. Condens. Matt., v.14, p.415 (2002).

A.J. Leggett "The Quantum Measurement Problem", Science v.307, p.871 (2005).

This paper clearly outlines the so-called "measurement problem" with regards to the Schrodinger Cat-type measurements.

Zz.

 

[klaymen]

Then maybe you shouldn't be making such definitive statement before you understand the physics.

Coin tossing is known to not be a "random" process. We lump our ignorance of the fine details of the mechanics into the randomness. But it is still deterministic, and the physics is there to verify that.

This is NOT the case in quantum mechanics. Even Einstein had tried to include "hidden variables" that so far have no empirical support. So as far as we know, there is no physics on why a measured superposition of state will produce one of the possible outcome. To say otherwise is to make an unsupported speculation.

so because humans have not found any definitive patterns, it means there aren't any?

i'm sorry, the real problem i have with this whole debate is the implication that the human mind has any effect on the time that the cat will be poisoned. i believe it is purely determined by the activating device, whether we understand the physics of it or not.

 

[ZapperZ]

so because humans have not found any definitive patterns, it means there aren't any?

i'm sorry, the real problem i have with this whole debate is the implication that the human mind has any effect on the time that the cat will be poisoned. i believe it is purely determined by the activating device, whether we understand the physics of it or not.

Luckily, physics cannot be challenged simply based on a matter of tastes, or what you find appealing. It can only be challenged either based on logical inconsistencies, or experimental observation, neither of which is something you have offered here.

The principle of superposition is alive and well, both in chemistry and in material science. That's why we have bonding-antibonding states, why NH3 molecule behave as it is, and why the Delft/Stony Brook SQUID experiments detected the coherence gap.

Zz.

 

[lightarrow]

i'm sorry, the real problem i have with this whole debate is the implication that the human mind has any effect on the time that the cat will be poisoned.

But you know that this is just one of the *interpretations* and nothing more than this.

i believe it is purely determined by the activating device, whether we understand the physics of it or not.

Maybe you prefer the line of thought of "decoherence".

 

[Count Iblis]

What about this paradox?

Let |X_r> denote a member of a complete set of states for the cat being alive and |Y_s> a member of a complete set of states for the cat being dead (I know, this is not very rigorous...).

Let Q be some arbitrary mapping of the set of "alive" states into the set of "dead" states, so for any arbitrary |X_r> there exists an s such that Q|X_r> = |Y_s>. We may then choose the labeling of the dead states such that Q|X_r> = |Y_r>. The total number of alive states N will, of course, be much smaller than the total number M of dead states.

Now, consider measuring the observable

$$A = \sum_{r=1}^{N}\left[|X_r><Y_r| + |Y_r><X_r|\right] +\sum_{r=N+1}^{M}|Y_r><Y_r|$$