B Does the EPR experiment imply QM is incomplete?

[vanhees71]

An apparatus never "indicates the eigenstate of the system". I don't even know what "the eigenstate of the system" means. What a good measurement apparatus does is precisely what its name suggests, it measures the values of observables defined by (an equivalence class of) measurement procedures. In the same sense a quantum state is (an equivalence class) of preparation procedures.

 

[Mentz114]

An apparatus never "indicates the eigenstate of the system". I don't even know what "the eigenstate of the system" means. What a good measurement apparatus does is precisely what its name suggests, it measures the values of observables defined by (an equivalence class of) measurement procedures. In the same sense a quantum state is (an equivalence class) of preparation procedures.

The system I refer to is the one being measured. Clearly I misunderstand everything about the Born rule and measurement. For instance what is written here https://en.wikipedia.org/wiki/Born_rule

The Born rule states that if an observable corresponding to a Hermitian operator A {\displaystyle A}

with discrete spectrum is measured in a system with normalized wave function | ψ ⟩ {\displaystyle |\psi \rangle }

(see bra–ket notation), then

• the measured result will be one of the eigenvalues λ {\displaystyle \lambda }
  
  of A {\displaystyle A}
  
  , and
• the probability of measuring a given eigenvalue λ i {\displaystyle \lambda _{i}}
  
  will equal ⟨ ψ | P i | ψ ⟩ {\displaystyle \langle \psi |P_{i}|\psi \rangle }
  
  , where P i {\displaystyle P_{i}}
  
  is the projection onto the eigenspace of A {\displaystyle A}
  
  corresponding to λ i {\displaystyle \lambda _{i}}
  
  .

 

[vanhees71]

Well, Wikipedia is not very accurate here... There's no "king's path" to understanding quantum physics, at least it's not provided by Wikipdia. You have to study the real thing as written in good textbooks like Dirac's.

 

[stevendaryl]

Well, Wikipedia is not very accurate here...

What is inaccurate about it?

 

[atyy]

Well, Wikipedia is not very accurate here... There's no "king's path" to understanding quantum physics, at least it's not provided by Wikipdia. You have to study the real thing as written in good textbooks like Dirac's.

Dirac's book has collapse.

 

[vanhees71]

Dirac's book first of all has a very good exposition of the formalism in Dirac's own bra-ket formulation. The collapse is not overemphasized, and of course, to read about interpretation (which you shouldn't be too worried about as a beginner anyway, because it's a side subject for specialists; the real success of QT as a physical theory simply needs the minimal statistical interpretation and not philosophical details which should be postponed to be studied by the interested student after s/he has mastered the math, because you cannot talk about QT without the math) should be studied from more modern textbooks. E.g., for the minimal statistical interpretation Ballentine, for Bohmian mechanics the writings of Dürr, consistent histories by Griffiths or Omnes. I've no clue of which value the socalled many-world interpretation should be, but I guess there Deutsch is the main proponent with the best books.

 

[vanhees71]

About Wikipedia

https://en.wikipedia.org/wiki/Born_rule

What is inaccurate about it?

Is this a joke? You'd have to rewrite the entire article from the beginning to the end to answer this question. To answer the opposite question is simple: What's accurate about it? Nothing!

 

[Mentz114]

About Wikipedia

https://en.wikipedia.org/wiki/Born_ruleIs this a joke? You'd have to rewrite the entire article from the beginning to the end to answer this question. To answer the opposite question is simple: What's accurate about it? Nothing!

From Jochen Rau, Statistical Physics and Thermodynamics (Oxford, 2017), page 10

A measureable physical quantity -an observable - is represented by a Hermitian operator. When measured the result will be one of the eigenvalues of the operator.

Maybe that is an oversimplification but it is what Wiki and most textbooks say.

 

[stevendaryl]

Is this a joke? You'd have to rewrite the entire article from the beginning to the end to answer this question. To answer the opposite question is simple: What's accurate about it? Nothing!

It summarizes the Born rule in about the same way that I've always heard it summarized by every physicist that's ever tried to explain it. So I have no idea what you are talking about.

The main points, which I think the Wikipedia gets across, are (1) a measurement of an observable produces an eigenvalue of the corresponding operator, and (2) the probability of getting an eigenvalue is the square of the corresponding amplitude (which Wikipedia gives in terms of projection operators).

That's almost the same as I would have described it. If it's completely inaccurate, then I don't see it. You need to spell it out.

 

[stevendaryl]

Googling for "Born Rule" gives as the first result after Wikipedia this article: http://www.math.ru.nl/~landsman/Born.pdf

It describes the rule this way:

Let $\hat{a}$ be a quantum-mechanical observable, mathematically represented by a self-adjoint operator on a Hilbert space $H$ with inner product denoted by $( , )$. For the simplest formulation of the Born rule, assume that $\hat{a}$ has non-degenerate discrete spectrum: this means that $\hat{a}$ has an orthonormal basis of eigenvectors $(e_i)$ with corresponding eigenvalues $\lambda_i$ , i.e. $\hat{a}\ e_i = λ_i\ e_i$ . A fundamental assumption underlying the Born rule is that a measurement of the observable a will produce one of its eigenvalues $\lambda_i$ as a result. In what follows, $\Psi \in H$ is a unit vector and hence a (pure) state in the usual sense. Then the Born rule states: If the system is in a sate $\Psi$, then the probability $P(a = \lambda_i | \Psi)$ that the eigenvalue $\lambda_i$ of $\hat{a}$ is found when $\hat{a}$ is measured is $P(a = \lambda_i | \Psi) = |(e_i , \Psi)|^ 2$.

I would say that that's substantially the same as what Wikipedia says. Wikipedia uses the projection operator $P_i$ defined to be $|e_i\rangle \langle e_i|$, but that's equivalent.

 

[Mentz114]

@stevendaryl .
The point I was leading up to is about your 'soft contradiction'. It seems to me that (1) something is prepared in a superposition (2) it interacts with the apparatus to form a superposition with it (as I think you spelt out) (3) something happens (X) and the system and the apparatus are left in the same (or highly correlated) state which indicates one of the members of the superposition.

Isn't this just a (oldish) statement of the measurement paradox ? We don't know what X is but it has to be non-unitary. I don't see a contradiction anymore because the two options (in your contradiction) are not mutually exclusive but different stages in a process.

 

[stevendaryl]

Isn't this just a (oldish) statement of the measurement paradox ? We don't know what X is but it has to be non-unitary.

Right. If you allow for something non-unitary to be happening during a measurement, then there is no paradox. But unless that something is spelled out, then the formalism is not complete, because part of the dynamics (the nonunitary part) is not spelled out.

I don't see a contradiction anymore because the two options (in your contradiction) are not mutually exclusive but different stages in a process.

I wouldn't call them stages. If you analyze the measurement process from the point of view of the Born rule, then you get a different answer than if you analyze the same process considering it a quantum mechanical interaction like any other. One or the other analysis has to be wrong.

If the analysis that uses unitary evolution applied to the measurement device (and environment) is the one that's wrong, then to me, it shows that unitary evolution is incorrect, and is only approximately true (it's only true in the limit of small systems).

 

[Mentz114]

... (allow) something non-unitary to be happening during a measurement...

Surely this is tautological. Can you call something that is unitary a measurement ? My point is that measurement is always non-unitary and a untary interaction is not a measurement.

I think you've always asserted that measurement is not an ordinary (unitary ?) interaction. I agree but I don't see it as a problem.

I wouldn't call them stages. If you analyze the measurement process from the point of view of the Born rule, then you get a different answer than if you analyze the same process considering it a quantum mechanical interaction like any other. One or the other analysis has to be wrong.

What other point of view is there ? Projective measurement cannot be analysed otherwise, can it ? You probably will say evolution, but evolution is a process and the Born rule is a constraint which has an implicit definition of 'measurement'.

If the analysis that uses unitary evolution applied to the measurement device (and environment) is the one that's wrong, then to me, it shows that unitary evolution is incorrect, and is only approximately true (it's only true in the limit of small systems).

It is incorrect for dissipative processes. Even if only one photon escapes to infinity unitarity is gone.

If this looks like hair-splitting I apologise. I broadly agree with your points.

 

[stevendaryl]

Surely this is tautological. Can you call something that is unitary a measurement ?

I suppose you could define a measurement in such a way that it must be non-unitary, but then it's an open question whether a measurement is possible. According to both the Bohm and Many-Worlds interpretations, evolution is always unitary.

 

[stevendaryl]

I suppose you could define a measurement in such a way that it must be non-unitary, but then it's an open question whether a measurement is possible. According to both the Bohm and Many-Worlds interpretations, evolution is always unitary.

It's sort of like (and I'm not sure whether this analogy is deep, or not) defining a measurement in classical mechanics so that only irreversible changes can be measurements. Then presumably you could prove from Newton's laws that no measurements are possible.

 

[vanhees71]

That's already way better than Wikipedia, and that's how you indeed start to explain it when you start introducing QT, but it's not the final word. First of all, indeed the operators representing observables in QT have to be self-adjoint; Hermitean is not enough to guarantee the consistency of Born's rule. Another important point is that not the Hilbert-space vectors represent (pure) states but rays in Hilbert space (or more conveniently and equivalently by projection operators $|\Psi \rangle \langle \Psi|$, with $|\Psi \rangle$ normalized. Of course, the special case of entirely non-degenerate spectra to degenerate ones is also an important point but easily generalized.