Measurement Theory - Quantum Mechanics Q&A by VVS

[VVS]

Hey,
I have a question about Measurement Theory.
I am reading Quantum Mechanics by Ballentine chapter 9 Measurement and Interpretation of States.
Please check the following pdf for my question:
View attachment Measurement Theory Ballentine.pdf

thank you
VVS

 

[Incnis Mrsi]

Could you check the middle expression in the equality (5)? Ī’m confused by (outer product of) three ket-vectors “⟩”, whereas in other contexts states of the composite systems consist of two factors.

 

[VVS]

Yes you are right the last one shouldn't be there.
I corrected it now.

 

[Incnis Mrsi]

Ī am astonished that this measurement theory describes posterior states simultaneously as pure states and independent of the result. If a theory ignores the result (non-selective measurement), then it must have mixed states (density matrices). If a theory uses pure states exclusively, then it must have some kind of result-dependent state reduction. Could somebody elucidate this?

 

[strangerep]

I have a question about Measurement Theory.
I am reading Quantum Mechanics by Ballentine chapter 9 Measurement and Interpretation of States.
Please check the following pdf for my question: [...]

For future reference, it would be easier to reply helpfully if you included more specific references to Ballentine, and matched your equation numbers to Ballentine's. I can only guess that you're referring to his section 9.2?

It would also be better to put your actual questions in your post rather than the pdf, so I don't have to cut, paste and correct the latex as below.

Shouldn’t the $r$ in $|\alpha_r, m\rangle$ actually be $r'$ because the apparatus would actually be measuring the disturbed value.

No. The idea is that final state of the apparatus should be correlated with the initial state of the object system (else it's not a measurement in any sensible meaning of that term).

Moreover I don’t understand the meaning of the indices $rm$ in $u^{r'm'}_{rm}$ if we are only summing over $r' m'$.

We are summing over $r$. The $m$ is not summed because it was just a shorthand to denote all the other quantum numbers of the apparatus apart from the one which becomes correlated with the initial state of the object system after the interaction.

 

[bhobba]

Ī am astonished that this measurement theory describes posterior states simultaneously as pure states and independent of the result. If a theory ignores the result (non-selective measurement), then it must have mixed states (density matrices). If a theory uses pure states exclusively, then it must have some kind of result-dependent state reduction. Could somebody elucidate this?

The answer is right at the beginning of Ballentine.

States are positive operators of unit trace - period. Pure states are simply a subset. So QM does not use pure states exclusively. Indeed, if you study probability models in general, and QM is an example of a probability model, mixed states are always part of such a model. Indeed it must be because it describes the situation of presenting a pure state randomly for observation.

Standard probability theory is simply the theory of mixed states of basic vectors [0,0,... 1,0,0,...], which are its pure states, where the 1 is in the position of the i'th outcome. You get QM when you want to have continuous transformations between pure states and you naturally end up with pure states that are complex numbers. Then you use the Born rule to interpret such weird pure states. Quantum states are, just like probability theory, the mixed states of the theory.

Thanks
Bill

 

[strangerep]

Ī am astonished that this measurement theory describes posterior states simultaneously as pure states and independent of the result. If a theory ignores the result (non-selective measurement), then it must have mixed states (density matrices). If a theory uses pure states exclusively, then it must have some kind of result-dependent state reduction. Could somebody elucidate this?

Er,... but,... have you actually studied Ballentine's ch9, or are you just commenting on VVS's brief pdf file?

(Later in ch9, Ballentine does consider more general states.)

 

[bhobba]

Er,... but,... have you actually studied Ballentine's ch9, or are you just commenting on VVS's brief pdf file? (Later in ch9, Ballentine does consider more general states.)

Drats - I was just going to mention Chapter 9 - you beat me to it.

Indeed the studying of Chapter 9 is a must.

Thanks
Bill

 

[Incnis Mrsi]

Er,... but,... have you actually studied Ballentine's ch9 … ?

Ī do not know anything about Ballentine. Ī learned quantum measurements (those that are more general than envisaged by Max Born) from A. S. Holevo, physicists’ common knowledge (likely rooted in ideas of J. von Neumann, but including later ideas of quantum decoherence), and my own exercises. What Ī read in the PDF seems to be an impossible measurement theory.

 

[strangerep]

Ī do not know anything about Ballentine. [...] What Ī read in the PDF seems to be an impossible measurement theory.

In part, Ballentine's textbook "QM -- A Modern Development" was intended to correct some of the old fictions surrounding QM.

Further discussion of the items in the thread should probably be suspended until you get a chance to study Ballentine in detail -- if your interests lie in that direction.

 

[atyy]

Ī do not know anything about Ballentine. Ī learned quantum measurements (those that are more general than envisaged by Max Born) from A. S. Holevo, physicists’ common knowledge (likely rooted in ideas of J. von Neumann, but including later ideas of quantum decoherence), and my own exercises. What Ī read in the PDF seems to be an impossible measurement theory.

Ballentine's chapter 9 is in my opinion somewhere between misleading and wrong, whereas Holevo gives the standard theory. However, I think one can take what is presented in the pdf as the state at the end of "pre-measurement", ie. the state from which the probabilities of classical readings of the apparatus can be calculated using the Born Rule and an observable defined on the apparatus. It is not the "collapsed" post-measurement state (unlike what was stated in the pdf), which should be a pure state if the result is known, or a mixed state if the result is unknown (as you stated).

I put "collapsed" in quotes because if the result is known, state reduction is required, but if the result is unknown, one can imagine that the mixed state is obtained by decoherence.

 

[VVS]

Could you please refer me to a specific book where the measurement theory is outlined nicely? I googled "measurement theory Holevo" but couldn't find anything.

 

[atyy]

Here are some references that I've found useful.

http://arxiv.org/abs/1110.6815
The modern tools of quantum mechanics
Matteo G. A. Paris

http://arxiv.org/abs/0706.3526
"No Information Without Disturbance": Quantum Limitations of Measurement
Paul Busch

http://arxiv.org/abs/0810.3536
Guide to Mathematical Concepts of Quantum Theory
Teiko Heinosaari, Mario Ziman

The following are standard references.

http://books.google.com/books?id=-s4DEy7o-a0C&source=gbs_navlinks_s
Quantum Computation and Quantum Information
Michael A. Nielsen, Isaac L. Chuang

http://books.google.com/books?id=uGl188JPxdQC&dq=holevo+statistical&source=gbs_navlinks_s
Statistical Structure of Quantum Theory
Alexander S. Holevo

http://books.google.com/books?id=anL-mDHBHQcC&source=gbs_navlinks_s
Operational Quantum Physics
Paul Busch, Marian Grabowski, Pekka Johannes Lahti

http://books.google.com/books?id=1YO9tQ4mFY8C&source=gbs_navlinks_s
The Quantum Theory of Measurement
Paul Busch, Pekka Johannes Lahti, Peter Mittelstaedt

The traditional textbook measurement theory has two limitations.

Let's consider first single measurements, so that we do not need to consider state reduction. By considering apparatus-system interactions, and executing projective measurements on the joint system, one can derive a more general class of obsevables called POVMs. Some people prefer stating POVMs as more fundamental than projective measurements (in Copenhagen/operational/instrumental/shut-up-and-calculate viewpoints), but since POVMs can be derived from projective measurements, it is possible to postulate projective measurements as fundamental (as usually assumed in decoherence-based viewpoints).

For successive measurements, we do need some form of state reduction if we use a picture in which states evolve in time. However, the projection postulate cannot apply to continuous variables, and a more general state reduction rule is needed for quantum systems with continuous variables. In fact, there is not a unique state reduction rule corresponding to an observable. One can see this even if one assumes the projection postulate in a finite-dimensional system by defining the post-measurement state produced by an apparatus to be projection followed by a unitary transformation. So in modern theory, one defines the state reduction rule via an "instrument", which in turn defines an observable. An even more specific notion than "instrument" is "measurement model" in which one specifies the Hamiltonian governing the interaction between apparatus and system. A measurement model defines an instrument which defines an observable. However, one can define an observable without defining a particular instrument, and one can define an instrument without defining a particular measurement model.

There are formalisms in which state reduction is done away with, or at least hidden very well. You can find discussions in these references.

http://arxiv.org/abs/quant-ph/0209123
Do we really understand quantum mechanics?
Franck Laloe

http://books.google.com/books?id=ZNjvHaH8qA4C&source=gbs_navlinks_s
Quantum Measurement and Control
Howard M. Wiseman, Gerard J. Milburn

A very interesting related topic to measurement theory is decoherence. It is important to note that decoherence alone does not solve the measurement problem, and does not remove the need for state reduction. For that, one needs additional assumptions like those in Bohmian Mechanics (which works for non-relativistic quantum mechanics) or Many-Worlds (which is an interesting approach, although there is no consensus if it works).

http://arxiv.org/abs/quant-ph/0306072
Decoherence and the transition from quantum to classical -- REVISITED
Wojciech H. Zurek

http://arxiv.org/abs/quant-ph/0312059
Decoherence, the measurement problem, and interpretations of quantum mechanics
Maximilian Schlosshauer

 

[Demystifier]

Great post atyy!

 

[VVS]

Yes thanks!