A Evaluate this paper on the derivation of the Born rule

[Prathyush]

My point was that for a macroscopic system we don't measure properties of single particles but quite "coarse-grained variables".

Sorry about misrepresenting your post. I would be interested to see how Arnold has to respond.

This is clearly a question that I haven't thought about in depth, this discussion was extremely fruitful to me because it brought these issues into the forefront.

This is a crude and preliminary analysis, please treat it that way. I think the important point here is to understand in what situations a coarse grained macroscopic description is applicable. The born's rule does not apply when we talk about measuring a macroscopic system, in the sense that when we extract coarse grained information about the macroscopic system, we don't redefine(or recreate ) a new state for the system. Formally we write $$<A> = tr(\rho A)$$ however it does not have the same meaning when we talk about the same equation for a microscopic system. Which is formally defined as a sum over multiple observations.

This sharp differences in meaning, must lie in the fact that our interaction with a macroscopic system does not appreciably change its state, and we can use extrinsic variables for its description. Any precise observation of our macroscopic system will entail a different experimental apparatus, its meaning of an observation must be changed appropriately and a macroscopic description must be changed into a microscopic one.

It requires more analysis, I don't see right now how to proceed from here. More precisely I think this inherent stability when we talk about a macroscopic system could be formally analysed.

 

[vanhees71]

I disagree. If you take the expectation value of, say the total energy or momentum of a macroscopic system, it's just that, the average of these quantities, and it's done with Born's rule. You contradict yourself, because what you wrote down is Born's rule. Given the state $\hat{\rho}$ (positive semidefinite trace-1 self-adjoint operator) then $\langle A \rangle=\mathrm{Tr}(\hat{\rho} \hat{A})$ is the expectation value of $A$, and that's Born's rule.

 

[Prathyush]

I disagree. If you take the expectation value of, say the total energy or momentum of a macroscopic system, it's just that, the average of these quantities, and it's done with Born's rule. You contradict yourself, because what you wrote down is Born's rule

The point I am making is the following, the sense in which we apply Born's rule, to a macroscopic system is in sharp contrast with the sense in which we apply Born's rule to a macroscopic system. I have explained in post#31 is why this different procedures appear to be in sharp contrast.

Just to be clear when we use Born's rule for a microscopic system, we also insist that the state of system be redefined to highlight the new information that was obtained. However no such thing is done for a macroscopic system. It appears that there are 2 completely independent ways in which we are using Born's rule. This fact needs further clarification.

 

[vanhees71]

I completely disagree with this idea that there are different meanings of the very clear formalism of QT. Born's rule is Born's rule, no matter whether you apply it to a single electron or to a many-body system. I don't know what you mean by the "state of the system is redefined". Do you mean something like a collapse hypothesis? I don't believe in this religion. In almost all measurements the collapse hypothesis is invalid, and where it is valid (a socalled von Neumann filter measurement) it's indeed a preparation procedure (as in my above most simple example of a Stern-Gerlach apparatus).

 

[Prathyush]

I completely disagree with this idea that there are different meanings of the very clear formalism of QT. Born's rule is Born's rule, no matter whether you apply it to a single electron or to a many-body system. I don't know what you mean by the "state of the system is redefined". Do you mean something like a collapse hypothesis? I don't believe in this religion. In almost all measurements the collapse hypothesis is invalid, and where it is valid (a socalled von Neumann filter measurement) it's indeed a preparation procedure (as in my above most simple example of a Stern-Gerlach apparatus).

Now you need to clarify what you mean? Basically yes I am talking about the collapse. When we measure the spin of a system, for any futher measurements we must set newly measured state as the initial state. That is the essence of the born's rule as we learn in textbooks.

Now what don't you believe about this?

 

[A. Neumaier]

define the expectation value of an observable, represented by a self-adjoint operator \hat{A} by

⟨A⟩_ρ=Tr(ρA).​

This is Born's rule.

No. This is not Born's rule. This is just a definition giving a name to a formula. Born's rule is a statement about eigenvalues and the probability of measurement results in an ideal von-Neumann measurement. The above definition has neither a reference to measurement nor to the conditions that make a measurement von-Neumann. SDo how can it be Born's rule? The only connection to Born's rule is that it can be deduced from it in the very special case that the latter is applicable and one averages over a lot of identically prepared measurement results. But the formula holds much generally, and is in fact much more fundamental than Born's rule in that it is very easy to state (no spectral theory needed) and (unlike Born's rule) applies universally.

 

[A. Neumaier]

This work was published in Phys. Rep. 525 (2013) 1-166

Thanks! I updated the reference.

 

[A. Neumaier]

Perhaps the central point is "Measuring a macroscopic system is not governed by Born's rule but by the identification of the expectation value with the measured value (within the intrinsic uncertainty)." which is in direct contrast with the main point of vanhees71's post.

Yes, and vanhees71 is wrong about his claims! He just parrots without justification the same stuff I used to believe as well as long as I only read what is repeated in the textbooks since 1932. But when I looked at the true correspondence of what is measured in a macroscopic system and how it is encoded into the statistical thermoddynamics formalism it was very obvious that the correspondence is the one I gave. The observables to which this applies are all those measured in daily engineering practice - in equilibrium they are for a chemical system total energy (represented by the operator $H$ the Hamiltonian), total mass of each component of a chemical mixture, (represented by the operators $m_iN_i$, where $N_i$ is the number operator of chemical component $i$. These are the fundamental operators in quantum mechanics. Each single measurement of any of these agrees with the expectation value to several digits of relative accuracy.

Thus has nothing at all to do with Born's rule which is completely misplaced when applied in a macroscopic context where single measurements are already significant and probabilities are irrelevant.

 

[A. Neumaier]

So it cannot be treated as an average of individual measurements over the different particles.

It cannot be treated as that for the simple reason that the observable measured is not an average over particles but (for mass) a sum over particles, and the energy in an interacting system is not even additive.

 

[A. Neumaier]

This doesn't make Born's rule superfluous. To the contrary, you need it to define how to evaluate the expectation values in the first place.

There is no need for Born's rule to define expectations values. As stevendaryl had already remarked, the latter is just a name given to a mathematical expression borrowed from statistics It need not have any more relation with the name-giving object as the term state vector has with a 3-dimensional arrow that lead to this name..

One can derive all of equilibrium thermodynamics theory from this definition without ever invoking probability concepts or the notion of measurement. Then one can invoke classical 19th century measurement theory to justify that the expectation value can be identified with the measured equilibrium values - since the formulas are just those that had been in use since 1850. Indeed, this is the way thermodynamics is derived in any physics book that bothers to give a derivation.

Nowhere a single trace of Born's rule!

 

[A. Neumaier]

Any precise observation of our macroscopic system will entail a different experimental apparatus

But if the apparatus does its job well and the apparatus in not coupled to a very sensitive system (such as a single quantum spin) the result will not depend on the apparatus, and always agree with the expectation value.

 

[A. Neumaier]

In almost all measurements the collapse hypothesis is invalid,

And in almost all measurements Born's rule is violated as most measurements are not von-Neumann measurements. All POVM measurements violate Born's rule.

 

[vanhees71]

Now you need to clarify what you mean? Basically yes I am talking about the collapse. When we measure the spin of a system, for any futher measurements we must set newly measured state as the initial state. That is the essence of the born's rule as we learn in textbooks.

Now what don't you believe about this?

The collapse hypothesis is not part of the formal postulates and fortunately not necessary. It causes more problems than it is in any way useful to understand quantum theory.

What happens with the measured system is a question that cannot be part of the general formalism for the simple reason that it depends on how your measurement apparatus is constructed. E.g., if you measure a photon's polarization by checking whether it runs through a polarization foil or not, you absorb it (either by the foil or by the detector telling you that it has gone through the polarizer). In the end you don't have a photon left at all. On the other hand, it's indeed a filter measurement in the literal sense, i.e., you can be (almost) sure that any photon that comes through the polarizer is polarized in the corresponding direction.

 

[vanhees71]

And in almost all measurements Born's rule is violated as most measurements are not von-Neumann measurements. All POVM measurements violate Born's rule.

As far as I know, the definition of POVM measurements relies also on standard quantum theory, and thus on Born's rule (I've read about them in A. Peres, Quantum Theory: Concepts and Methods). It just generalizes "complete measurements" by "incomplete ones". It's not outside the standard rules of quantum theory.

 

[Prathyush]

Arnold may I recommend using 1 or 2 posts per reply as it will be helpful to the reader, and using the multi quote function for this purpose.

On the other hand, it's indeed a filter measurement in the literal sense, i.e., you can be (almost) sure that any photon that comes through the polarizer is polarized in the corresponding direction.

I understand this particular situation is a filter measurement. But that does not appear to be a relevant point.

One can construct experimental apparatus that won't destroy the particle that we intend to measure. Consider the double slit experiment the Born's rule applies in the sense that if we observe which slit the particle when though for making future prediction about it, we need to use this new information to calculate probabilities that is the essence, I don't understand how there can be a disagreement about this. Sure in some situations we destroy the particle, sometime we change the state of the particle but that can be understood from from first principles based on how the apparatus was constructed.

Born's rule is $$p_i = <\psi|A_i|\psi>$$ where Ai is the the projection of the operator onto the nth state. In a suitably constructed apparatus, one can say the final state after measurement is precisely the eigenvalue of the measurement operator. This is what we understand by an ideal measurement. If we have a mixed state it can be written in terms of density matrices etc.
$$<A> = Tr(A \rho )$$ is a straight forward consequence of this rule.

Would you disagree with the above?

Also please tell how to include latex into the main text without going to a new line each time. I used the 2 dollar symbols to add latex.

 

[vanhees71]

There is no need for Born's rule to define expectations values. As stevendaryl had already remarked, the latter is just a name given to a mathematical expression borrowed from statistics It need not have any more relation with the name-giving object as the term state vector has with a 3-dimensional arrow that lead to this name..

One can derive all of equilibrium thermodynamics theory from this definition without ever invoking probability concepts or the notion of measurement. Then one can invoke classical 19th century measurement theory to justify that the expectation value can be identified with the measured equilibrium values - since the formulas are just those that had been in use since 1850. Indeed, this is the way thermodynamics is derived in any physics book that bothers to give a derivation.

Nowhere a single trace of Born's rule!

Well, usual statistics textbook start with equilibrium distributions and define, e.g., the grand canonical operator
$$\hat{\rho}=\frac{1}{Z} \exp[-\beta (\hat{H}-\mu \hat{Q})],$$
to evaluate expectation values using Born's rule, leading to the Fermi- and Bose-distribution functions.

Classical statistical mechanics is also based on probability concepts since Boltzmann & Co. I don't know, which textbooks you have in mind!

 

[vanhees71]

Arnold may I recommend using 1 or 2 posts per reply as it will be helpful to the reader, and using the multi quote function for this purpose.
I understand this particular situation is a filter measurement. But that does not appear to be a relevant point.

One can construct experimental apparatus that don't destroy the particle that we intend to measure. Consider the double slit experiment the Born's rule applies in the sense that if we observe which slit the particle when thought for making future prediction about it, we need to use this new information to calculate probabilities that is the essence, I don't understand how there can be a disagreement about this. Sure in some situations we destroy the particle, sometime we change the state of the particle but that can be understood from from first principles based on how the apparatus was constructed.

Born's rule is $$p_i = <\psi|A_i|\psi>$$ where Ai is the the projection of the operator onto the nth state. In a suitably constructed apparatus, one can say the final state after measurement is precisely the eigenvalue of the measurement operator. This is what we understand by an ideal measurement. If we have a mixed state it can be written in terms of density matrices etc.
$$<A> = Tr(A \rho )$$ is a straight forward consequence of this rule.

Would you disagree with the above?

Also please tell how to include latex into the main text without going to a new like each time. I used the 2 dollar symbols to add latex.

Just to clarify my postings from before: To simplify the discussion, I usually don't dinguish pure and mixed states. For me all states are defined by a statistical operator (positive semidefinite self-adjoint operator with trace 1). A state is by definition pure if it is represented by a projection operator $\hat{\rho}_{\text{pure}}=|\psi \rangle \langle \psi|$ with some normalized vector $|\psi \rangle$. This is much simpler than to talk about unit rays in Hilbert space!

To get inline LaTeX just enclose them by two # instead of two $ signs.

 

[Prathyush]

Just to clarify my postings from before: To simplify the discussion, I usually don't dinguish pure and mixed states. For me all states are defined by a statistical operator (positive semidefinite self-adjoint operator with trace 1). A state is by definition pure if it is represented by a projection operator $\hat{\rho}_{\text{pure}}=|\psi \rangle \langle \psi|$ with some normalized vector $|\psi \rangle$. This is much simpler than to talk about unit rays in Hilbert space!

To get inline LaTeX just enclose them by two # instead of two $ signs.

Sure I will use density matrices as you prefer. Born's rule in this language is, probability $p_i = Tr (\rho A_i)$ where $A_i$ is the projection of the operator that we intend to measure. In a suitably constructed apparatus, the final state is $\rho_i = |\psi_i><\psi_i|$ where $|\psi_i>$ is the eigenvector corresponding to the measurement performed.
Is there a disagreement here?

 

[vanhees71]

I'm not familiar with your terminology. So let me give you mine in a nutshell. Then we can see, whether we understand the same thing when talk about (quantum theory, which I doubt ;-)).

(a) The state of the system is given by a positive semidefinite self-adjoint operator on a (separable) Hilbert space with trace 1, $\hat{\rho}$ (statistical operator). A state is pure iff there exists a normalized vector $|\psi \rangle$ such that $\hat{\rho}=|\psi \rangle \langle \psi |$.

(b) An observable $A$ is represented by a self-adjoint operator $\hat{A}$.

(c) Possible outcomes of precise (complete) measurements of $A$ are the eigenvalues of $\hat{A}$. In the following I use a complete orthonormalized set of eigenvectors of $\hat{A}$, which I denote with $|a,\beta \rangle$, where $a$ is a possible eigenvalue:
$$\hat{A} |a,\beta \rangle=a |a,\beta \rangle, \quad \langle a,\beta|a',\beta' \rangle=\delta_{aa'} \delta_{\beta \beta'}.$$
The label $\beta$ are one or more variables to label the different orthogonal eigenstates to the same eigenvalue. For simplicity I only consider the case that we have discrete spectra of the operators (if you have variables with continuous spectra it becomes only a bit more complicated since you have to use distributions and integrals instead of sums). The eigenvectors form a complete orthonormalized set of vectors,
$$\sum_{a,\beta} |a,\beta \rangle \langle a,\beta|=\hat{1}.$$

(d) If the system is prepared in a state $\hat{\rho}$ the probability to find the value $a$, when observable $A$ is measured precisely on this system is given by Born's rule,
$$P(a)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle.$$

The expectation value of $A$ is given by
$$\langle A \rangle=\sum_{a} a P(a) = \sum_{a,\beta} a \langle a,\beta |\hat{\rho}|a,\beta \rangle = \sum_{a,\beta} \langle a,\beta| \hat{\rho} \hat{A} a,\beta \rangle=\mathrm{Tr} (\hat{\rho} \hat{A}).$$

The probabilities $P(a)$ can indeed also be formulated with the projection operators to the different eigen spaces of $\hat{A}$, because with
$$\hat{P}_a=\sum_{\beta} |a,\beta \rangle \langle a,\beta|$$
obviously we have
$$P(a)=\text{Tr} (\hat{P}_a \hat{\rho}).$$

There is no need to know in which state the system is after measurement. We don't need to complicate this discussion by bringing up the collapse hypothesis, which is in my opinion completely flawed and not commonly assumed anywhere in practitioning QT.

We also don't need to complicate things by thinking about more general incomplete measurements here. In my understanding the socalled "measurement problem" is to somehow explain, why the outcome of a precise measurement is always one and only one eigenvalue of the associated operator $\hat{A}$. For me that's an empty question. What I've written down are the condensed postulates of QT as a theory to describe what's observed in nature by measuring observables (as is also classical physics by the way). The only thing that counts is, whether this theory describes the real-world experiments and observations in nature, and indeed it does with a breathtaking accuracy. So there is no "measurement problem", because the formalism describes everything we have observed so far. There's not more to be expected from a physical theory. The QT we learn today has been formed in the 1st quarter of the 20th century from a careful analysis of observations of the behavior of matter, particularly atomic and subatomic physics, and that's why it works so well (including also the understanding of the "classical" behavior of the macroscopic matter surrounding us with many of its quantitative properties through statistical many-body quantum physics).

You can load QT (as any mathematical model of reality) with some philosophical (not to call it esoterical) questions like, why we always measure eigenvalues of self-adjoint operators, but physics doesn't answer why a mathematical model works, it just tries to find through an interplay between measurements and mathematical reasoning such models that describe nature (or even more carefully formulated what we observe/measure in nature) as good as possible.

 

[Prathyush]

I'm not familiar with your terminology. So let me give you mine in a nutshell. Then we can see, whether we understand the same thing when talk about (quantum theory, which I doubt ;-)).

Fair enough. I use the same formalism.

First a minor point.

The probabilities P(a) can indeed also be formulated with the projection operators to the different eigenspaces of $\hat{A}$, because with $$\hat{P}_a=\sum_{\beta} |a,\beta \rangle \langle a,\beta|$$

I usually don't include the $\beta$ into the definition of the projection operator or born rule, for instance I would write $\hat{P}_a= |a,\beta \rangle \langle a,\beta|$ and other formulas would change appropriately. Ofcourse it has to be used based on the context.

There is no need to know in which state the system is after measurement. We don't need to complicate this discussion by bringing up the collapse hypothesis, which is in my opinion completely flawed and not commonly assumed anywhere in practitioning QT.

In my understanding the so called "measurement problem" is to somehow explain, why the outcome of a precise measurement is always one and only one eigenvalue of the associated operator $\hat{A}$.

This is something you are suggesting that is orthogonal to what most textbooks write. I can understand where you are coming from when you say what happens to the system after measurement is irrelevant, I will critically analyze this statement in a later post.

For now however Consider a 2 slit experiment(for instance feynman's description of it), If we measure which slit the particles went through, the for all future measurements we have to use this information. This is the reason why most textbooks include the collapse postulate. When a measurement is performed this information must be reflected in the state of the particle atleast in this particular context. Ofcourse you can say once the measurement is performed we can move both the particles to the same place(or change it however you want), so the collapse posulate is not a general rule. I will analyze this "rule" carefully based on your response.

We can talk about macroscopic and microscopic descriptions once we resolve the collapse stuff.

 

[Ddddx]

There really is no such thing as collapse. It is as meaningless as saying that the probability P = 1/2 "collapses" once the coin lands heads.

"Probability amplitudes, when squared, give the probability of a complete event. Keeping this principle in mind should help the student avoid being confused by things such as the 'collapse of the wave function' and similar magic" - Richard Feynman

 

[stevendaryl]

There really is no such thing as collapse. It is as meaningless as saying that the probability P = 1/2 "collapses" once the coin lands heads.

The first sentence might be true, but it's not obvious that the second sentence is. That's really the whole point of Bell's theorem. We don't worry about collapse with coin flips because we assume that after a coin settles down, there is a "fact of the matter" about whether it is heads or tails. So even if we flip the coin with the lights out, and don't see the result, we believe that there is a result, we just don't know what it is. The probabilities reflect our ignorance about what that result its.

In the case of quantum mechanics, if you have a particle that is in a superposition of spin-up and spin-down, it's not that it's either spin-up or spin-down, we just don't know which. It's neither spin-up nor spin-down until we measure the spin.

 

[vanhees71]

Fair enough. I use the same formalism.

First a minor point.

I usually don't include the $\beta$ into the definition of the projection operator or born rule, for instance I would write $\hat{P}_a= |a,\beta \rangle \langle a,\beta|$ and other formulas would change appropriately. Ofcourse it has to be used based on the context.

That's wrong, because then your probabilities won't sum up to 1 (except if the spectrum of $\hat{A}$ is not degenerate, and each eigenspace is exactly 1D).

This is something you are suggesting that is orthogonal to what most textbooks write. I can understand where you are coming from when you say what happens to the system after measurement is irrelevant, I will critically analyze this statement in a later post.

For now however Consider a 2 slit experiment(for instance feynman's description of it), If we measure which slit the particles went through, the for all future measurements we have to use this information. This is the reason why most textbooks include the collapse postulate. When a measurement is performed this information must be reflected in the state of the particle atleast in this particular context. Ofcourse you can say once the measurement is performed we can move both the particles to the same place(or change it however you want), so the collapse posulate is not a general rule. I will analyze this "rule" carefully based on your response.

We can talk about macroscopic and microscopic descriptions once we resolve the collapse stuff.

Again, you have to define the measurement done clearly. Then you won't need a collapse hypothesis. It is not enough to say you gained some which-way information in the double-slit experiment, but you have to say how you definitely measure it to describe the setup of your experiment completely. Then you can say which state is prepared and which probabilities for detecting the particles on the screen you expect from the QT formalism.

One example is to use linearly polarized photons in the double-slit experiment. To gain which-way information you can put quarter-wave plates into each slit, the one in $+45^{\circ}$, the other $-45^{\circ}$ orientation relative to the polarization direction of the photons. Then a photon running through the first (second) slit will be left-circular the other right-circularly polarized and thus you can exactly distinguish through which way the photons went through the slits. At the same time since the polarization states are exactly perpendicular to each other there is no more interference and thus the interference pattern vanishes. You can also decide to gain incomplete which-way information by distorting the angles of the quarter-wave plates a bit. Then you get partial interference, i.e., an interference pattern with less contrast than without the quarter-wave plates.

The very simple message of this example is that of course the outcome of a measurement depends on the preparation of the measured observable. This is not very profound and is as valid in classical physics as in quantum theory.

 

[vanhees71]

The first sentence might be true, but it's not obvious that the second sentence is. That's really the whole point of Bell's theorem. We don't worry about collapse with coin flips because we assume that after a coin settles down, there is a "fact of the matter" about whether it is heads or tails. So even if we flip the coin with the lights out, and don't see the result, we believe that there is a result, we just don't know what it is. The probabilities reflect our ignorance about what that result its.

In the case of quantum mechanics, if you have a particle that is in a superposition of spin-up and spin-down, it's not that it's either spin-up or spin-down, we just don't know which. It's neither spin-up nor spin-down until we measure the spin.

Indeed, it depends on the preparation whether $\sigma_z=+1/2$ (up) or $\sigma_z=-1/2$ (down) or whether it's indetermined. In the latter case you know, provided you know the quantum state of the measured spin, probabilities for the two possible outcomes. What happens when measuring $\sigma_z$ depends on the used measurement apparatus. There's no general rule like collapse describing what's going on. In my above description of the SG experiment it's clear that you can use the magnetic field to prepare (almost exactly) a pure $\sigma_z=+1/2$ or $-1/2$ state by filtering out the wanted beam since through the magnetic field position and spin get (almost precisely) entangled. But still you don't have a collapse, changing any entity simultaneously everywhere but it's just filtering out one partial beam by blocking the other with some "beam dump". The interaction of these particles with the beam dump is quite local. There's nothing collapsing instaneously in the entire universe as claimed by the collapse proponents.

 

[mikeyork]

In the case of quantum mechanics, if you have a particle that is in a superposition of spin-up and spin-down, it's not that it's either spin-up or spin-down, we just don't know which. It's neither spin-up nor spin-down until we measure the spin.

I don't think your second sentence is quite right. It's true only in the sense that the particle might not even be there at all. Are you saying that Schrodinger's cat is neither alive nor dead until we open the box?

To be pedantic:

An observational apparatus imposes a context -- of a representation (what is to be measured) and an associated frame of reference. We could, in principle, impose those to describe a particle we knew was there without ever observing it. The "probability amplitudes" would have the same meaning in terms of the theoretical asymptotic relative frequencies if a sequence of observations were made. Making one, two or a 100 observations makes no difference to that since we can never make an infinite number of observations. So it is quite possible that the particle was already prepared in a spin eigenstate before we observe it -- as if the notion of "collapse" had already happened when the particle was produced. It's just that we can't start actually measuring those relative frequencies until we start detecting the state and repeating multiple times.

The content of QM then, is not that the particle is not in a spin eigenstate, but that there are multiple representations and frames of reference we could use to describe it before we actually decide what context to use for an actual observation. It is the incompatibility of different contexts that an observer might impose that makes QM differ from classical statistics.

 

[stevendaryl]

I don't think your second sentence is quite right. It's true only in the sense that the particle might not even be there at all. Are you saying that Schrodinger's cat is neither alive nor dead until we open the box?

I'm saying (actually, I did say) that an electron that is in a superposition of spin-up and spin-down is neither spin-up nor spin-down until we measure it. What this implies about dead cats is complicated.

 

[Prathyush]

That's wrong, because then your probabilities won't sum up to 1 (except if the spectrum of $\hat{A}$ is not degenerate, and each eigenspace is exactly 1D).

When I add up probabilities I do a sum over $\beta$. Its mostly a personal preference, it is equivalent to the formula you wrote. I like my projection operators to have a trace of 1.

There really is no such thing as collapse. It is as meaningless as saying that the probability P = 1/2 "collapses" once the coin lands heads.

When measurements are performed new information comes to light and it must be reflected in the new of description of the state. This change in the description as new information becomes available is basically what people call collapse.

There's nothing collapsing instaneously in the entire universe as claimed by the collapse proponents.

This kind of thinking happens because people seem to associate some kind of physical attributes to information, when it does not exist. The wavefunction is basically the same as information available. I want to avoid using the word collapse as it seems to imply things that I don't intend.

Again, you have to define the measurement done clearly. Then you won't need a collapse hypothesis. It is not enough to say you gained some which-way information in the double-slit experiment, but you have to say how you definitely measure it to describe the setup of your experiment completely. Then you can say which state is prepared and which probabilities for detecting the particles on the screen you expect from the QT formalism.

I can construct a detailed experiment, but that would require time. Would you agree with the following statement, when a measurement is performed, the state of the system(meaning information available to us about it) in general changes to reflect the outcome of the measurement.

 

[mikeyork]

I'm saying (actually, I did say) that an electron that is in a superposition of spin-up and spin-down is neither spin-up nor spin-down until we measure it. What this implies about dead cats is complicated.

In the case of the cat, being dead or alive is part of what we mean by it being a cat. So it must be either. In the case of a particle, if we impose a descriptive context that says it must have a definite spin component in a specific direction then that is what it will have (with appropriate probabilities), regardless of whether we measure it.

So I would revise your statement to say "neither spin-up nor spin-down until we choose to describe it as either, whether we measure it or not and if we don't (or can't) choose to describe it as either, then we have nothing to say about it being either". Superpositions tell us only how to switch between incompatible descriptive choices. They don't have any other meaning.

(Edited)

 

[stevendaryl]

In the case of the cat, being dead or alive is part of what we mean by it being a cat. So it must be either. In the case of a particle, if we impose a descriptive context that says it must have a definite spin component in a specific direction then that is what it will have, regardless of whether we measure it.

So I would revise your statement to say "neither spin-up nor spin-down until we choose to describe it as either, whether we measure it or not and if we don't choose to describe it as either, then we have nothing to say about it being either". Superpositions tell us only how to switch between incompatible descriptive choices. They don't have any other meaning.

I don't think that's true. I should say more definitely: it is not true. Superpositions are not a matter of descriptive choices. To say that an electron is in a superposition \alpha |u\rangle + \beta |d\rangle implies that a measurement of the spin along axis \vec{a} will yield spin-up with a probability given by (mumble..mumble---I could work it out, but I don't feel like it right now). So there is a definite state \alpha |u\rangle + \beta |d\rangle, and it has a definite meaning. It's not just a matter of descriptions.

 

[Prathyush]

In the case of quantum mechanics, if you have a particle that is in a superposition of spin-up and spin-down, it's not that it's either spin-up or spin-down, we just don't know which. It's neither spin-up nor spin-down until we measure the spin.

Basically when we say we have a particle in a state of superposition, we are saying something about its preparation procedure.

I don't think we should discuss cats here. However for the sake of completeness.

I don't think your second sentence is quite right. It's true only in the sense that the particle might not even be there at all. Are you saying that Schrodinger's cat is neither alive nor dead until we open the box?

If you can construct an experiment that can interfere between dead and alive states of a cat you will realize what stevendaryl is saying is correct. However in practice it is impossible to do so.