Collapse and Peres' Coarse Graining

[billschnieder]

But without collapse, how can measurement be used as a means of quantum state preparation, where we use the classical result obtained to figure out the quantum state of the selected sub-ensemble? (I do understand there is a more general collapse rule than projective measurements, but let's keep things simple here, since there is still collapse in the more general rule.) Does this mean that measurement cannot be used as a form of state preparation in the minimal interpretation?

In classical probability, a probability distribution represents alternate "possibilities". A measurement "actualizes" a sub-ensemble. Would you say there is collapse involved?
The sub-ensemble could still be described with a different probability distribution in which case you could say the measurement "prepared" the new state by selecting a subset of the possibilities. However, if probability distributions are understood as information and not physical, there is absolutely no need to invent a concept of "collapse".

 

[TrickyDicky]

It seems the collapse and quantum-classical cut you refer to is just the non-commutativity of observables, the quantum hypothesis itself, so it should be present in any interpretation of QM as they all must obey the HUP. A minimum volume is given to the classical phase space that assures irreversibility of measured observables whether one postulates it formally or not the math is there.

 

[stevendaryl]

In classical probability, a probability distribution represents alternate "possibilities". A measurement "actualizes" a sub-ensemble. Would you say there is collapse involved?
The sub-ensemble could still be described with a different probability distribution in which case you could say the measurement "prepared" the new state by selecting a subset of the possibilities. However, if probability distributions are understood as information and not physical, there is absolutely no need to invent a concept of "collapse".

Yes, if it's just a subjective way to describe our information about the system, then whether you call it a collapse or not, there is nothing very weird about it. But consider an example such as EPR with anti-correlated twin pairs (electron/positron). Alice measures spin-up for the electron, then she knows that Bob will measure spin-down for the corresponding positron. So, she just acquired the information, and there's nothing weird about that. But if it's only a matter of acquiring information, then one would think that Bob's particle was spin-down in the z-direction before Alice's measurement. So the view of "collapse" as being purely information would (it seems to me) to imply pre-existing values for such things as spin, which is basically hidden variables, which is ruled out by Bell's theorem.

The corresponding "collapse" in the classical case really does mean hidden variables. You have two pieces of paper, one white and one black. You put each into an envelope and mix them up, and give one to Alice to open and another to Bob to open. The second that Alice opens her envelope and finds a white piece of paper, she knows that Bob's envelope contains a black piece of paper. In that case, it is completely consistent (and perfectly natural) for Alice to assume that Bob's envelope contained a black piece of paper even before either of them opened their envelopes.

 

[billschnieder]

I like the terms "absolute reality" and "relative reality" that Tsirelson uses in his discussion of the measurement problem

I think the wave function is "possible reality" while experimental results are "actual reality". What selects one of the "possibilities" to be actualized, then is what some people call "collapse", but this is obviously not restricted to quantum mechanics.

 

[atyy]

It seems the collapse and quantum-classical cut you refer to is just the non-commutativity of observables, the quantum hypothesis itself, so it should be present in any interpretation of QM as they all must obey the HUP.

In the minimal interpretation, the notion of measurement is fundamental. A measurement is the interaction between a classical measurement apparatus and a quantum system, resulting in a definite classical outcome. So the classical/quantum cut is fundamental to the minimal interpretation, unless the notion of measurement can be removed as fundamental in quantum mechanics.

A minimum volume is given to the classical phase space that assures irreversibility of measured observables whether one postulates it formally or not the math is there.

There is no minimum volume or even a classical probability distribution over the classical phase space in quantum mechanics. It is true that some books such as Reif use this concept, but although it is useful, I don't think (maybe I'm wrong) it is fundamental, more a very good heuristic like wave-particle duality.

 

[billschnieder]

The corresponding "collapse" in the classical case really does mean hidden variables. You have two pieces of paper, one white and one black. You put each into an envelope and mix them up, and give one to Alice to open and another to Bob to open. The second that Alice opens her envelope and finds a white piece of paper, she knows that Bob's envelope contains a black piece of paper. In that case, it is completely consistent (and perfectly natural) for Alice to assume that Bob's envelope contained a black piece of paper even before either of them opened their envelopes.

It is not as easy as you make it sound. If we move away from the Bertlmann's socks type variables of "paper color" to the more relevant space-type dynamically changing variables then it is impossible for Alice to say by opening her envelope, what Bob will see. For example, let the papers by dynamically changing colors between black and white (opposite sequence between the two) at a given hidden frequency such that once you open the envelope, the exposure to light stops the dynamics instantly and locks it to one color. Your claim that Alice will know the result of Bob's paper by opening her envelope is false. However, we know for a fact that the two pieces of paper have perfectly anti-correlated colors. Often when we discuss these things, we easily gloss over the fact that the discussion requires that Alice and Bob opened their envelopes at the exact same time. But it is impossible to test this experimentally without post-selection. You have to take time-tags on both sides and use the time at one end to filter the results at the other end. Only then will the experiment match what is actually happening. But post-selection invalidates the derivation of the Bell inequalities since the joint probability distribution for post-selected experiments is non-factorable. Not surprising that all experiments to date, claiming violation of inequalities, employ one form of post-processing or another.

 

[atyy]

@billschneider, please discuss your postselection issue by starting another thread, not here.

 

[microsansfil]

However, if probability distributions are understood as information and not physical, there is absolutely no need to invent a concept of "collapse".

And if it is understood as formal mathematics measure (http://en.wikipedia.org/wiki/Measure_(mathematics)) based on an independent axiomatic of any application, leaving aside any semantic notion (just a formal writing game) ?

Patrick

 

[TrickyDicky]

In the minimal interpretation, the notion of measurement is fundamental. A measurement is the interaction between a classical measurement apparatus and a quantum system, resulting in a definite classical outcome. So the classical/quantum cut is fundamental to the minimal interpretation, unless the notion of measurement can be removed as fundamental in quantum mechanics.
There is no minimum volume or even a classical probability distribution over the classical phase space in quantum mechanics. It is true that some books such as Reif use this concept, but although it is useful, I don't think (maybe I'm wrong) it is fundamental, more a very good heuristic like wave-particle duality.

Measurement is fundamental to any empirical science, not specifically to the minimal
interpretation of QM, if you define it as interaction classical apparatus/quantum system you are already introducing a specific heuristic or interpretation as fundamental when the measurement problem is basically the lack of consensus about what measurent in QM entails.
Sometimes I think it would be more productive to turn to the Schrodinger for puzzlement instead of being surprised by measuring classical observables and getting classical outcomes

 

[atyy]

Measurement is fundamental to any empirical science, not specifically to the minimal
interpretation of QM, if you define it as interaction classical apparatus/quantum system you are already introducing a specific heuristic or interpretation as fundamental when the measurement problem is basically the lack of consensus about what measurent in QM entails.

In classical physics (Newtonian physics, special and general relativity), measurement is not a fundamental concept. Historically, Einstein did postulate measurement as fundamental in special relativity: the speed of light measured by any inertial observer is the same. However, we have removed that, and nowadays we say that special relativity means the laws have Poincare symmetry. Historically, measurement was also important in the genesis of general relativty: test particles follow geodesics. The test particle is a sort of measurement apparatus that is apart from the laws of physics because it does not cause spacetime curvature, in contrast to all other forms of matter. However, in the full formulation of general relativity, test particles are not fundamental. So quantum mechanics is different from classical physics in needing to specify measurement as a fundamental concept.

I am using a particular interpretation to define QM, but it is the minimal interpretation. The measurement problem is that we have to put this classical/quantum cut to define the minimal interpretation. The other interpretations are then approaches to solving the measurement problem by removing the need for measurement to be a fundamental concept in the mathematical specification of a theory. Examples of such interpretations are consistent histories (flavour of Copenhagen), hidden variables (generally predicting deviations from QM), or Many-Worlds.

Sometimes I think it would be more productive to turn to the Schrodinger for puzzlement instead of being surprised by measuring classical observables and getting classical outcomes

Another way of stating the measurement problem, is that if everything is quantum and we have a wave function of the universe, how can we make sense of such an idea? The minimal interpretation cannot make sense of such an idea, and always needs a classical/quantum cut. Bohmian Mechanics and Many-Worlds are two approaches to solving the measurement problem, in which the wave function of the universe is proposed to make sense.

 

[billschnieder]

However, in the full formulation of general relativity, test particles are not fundamental. So quantum mechanics is different from classical physics in needing to specify measurement as a fundamental concept.

Measurement is needed to "actualize" one of the "possibilities" (or a small set of the non-mutually exclusive possibilities -- aka commuting observables). Without measurement, there is nothing actual to talk about, just possibilities. Measurement will not be so important in a theory that does not rely on a device which represents simultaneously all the "possible realities", like general relativity. In probability theory however, measurement is very important.

I am using a particular interpretation to define QM, but it is the minimal interpretation. The measurement problem is that we have to put this classical/quantum cut to define the minimal interpretation.

We do not. In the minimal interpretation, the classical/quantum cut is simply an unnecessary fiction, which only appears once you choose to interpret the wave function as a real physical thing. But the minimal interpretation is that it is a device for cataloging information about possible states within an ensemble.

Another way of stating the measurement problem, is that if everything is quantum and we have a wave function of the universe, how can we make sense of such an idea?

Understanding the wave function as a catalog of information about possible realities of the universe, there is no difficulty to make sense of. It already makes sense, in the minimal interpretation. No classical quantum cut is needed. MWI and BM are attempts to solve a problem introduced because the proponents insist on interpreting the wavefunction as a real physical thing. MWI by suggesting that the possibilities are all actualities (including the mutually exclusive ones), BM by suggesting that the possibilities exist as "guiding waves" to orchestrate observations.

 

[atyy]

Understanding the wave function as a catalog of information about possible realities of the universe, there is no difficulty to make sense of.

If one actually tries to construct the catalogue of all the possible realities in the wave function, one ends up with the consistent histories approach.

 

[zonde]

Say we have two orthogonal polarizers and we shine a light trough them. There is practically no light passing through them.
Then we put in additional polarizer at 45 deg. between first two and we get quarter of light trough.
Considering this I don't understand how one can view measurement as information update.

 

[atyy]

@zonde and @billschneider, could you start a new thread about whether collapse is update or physical? I agree it's an interesting question, but not so relevant at the moment. I would like to focus on the hard science question I wrote in post #47. Is Weinberg's Eq 2.1.7 in Vol 1 of his QFT text part of quantum theory? If it is, is it postulated, or derivable from only {unitary evolution + Born rule without collapse}? If you'd like to discuss whether Eq 2.1.7 represents information update or a physical process in this thread, why don't we wait a bit until we understand vanhees71's view of Weinberg's Eq 2.1.7?

 

[Ilja]

In classical probability, a probability distribution represents alternate "possibilities". A measurement "actualizes" a sub-ensemble. Would you say there is collapse involved?
... However, if probability distributions are understood as information and not physical, there is absolutely no need to invent a concept of "collapse".

In this case, of course, there would be no need for a physical process identified with a collapse. But this picture is incompatible with the violation of Bell's inequalities.

And interpreting something as "information" always requires an answer to the question "information about what?".

 

[Ilja]

Well, it's at least not a physical process, as claimed by some collapse proponents in the case of quantum theory. It's then even less needed in classical than in quantum theory.

Indeed, in classical statistics we can explain the uncertainty of the statistics as completely being a problem of insufficient information. Getting more information about the real process does not mean that the process is physically influenced.

In quantum theory such an explanation is impossible.

You have an eigenstate of A with eigenvalue a1. You can repeat the measurement of A as much as you like, the result is always a1, never a2, a3, ...

Now you measure some non-commuting B. Then, you choose the subgroup of those with result b1. This operation differs from choosing a subgroup in classical statistics given some additional information that B has value b1, as can be easily seen: Measuring A gives now, with nonzero probability, other values than a1. Instead, restricting to a subensemble of those with value b1 would not modify the result of A being a1.

 

[zonde]

@zonde and @billschneider, could you start a new thread about whether collapse is update or physical? I agree it's an interesting question, but not so relevant at the moment. I would like to focus on the hard science question I wrote in post #47. Is Weinberg's Eq 2.1.7 in Vol 1 of his QFT text part of quantum theory? If it is, is it postulated, or derivable from only {unitary evolution + Born rule without collapse}? If you'd like to discuss whether Eq 2.1.7 represents information update or a physical process in this thread, why don't we wait a bit until we understand vanhees71's view of Weinberg's Eq 2.1.7?

Born rule without collapse could work only if filtering measurement represents information update. So it's relevant to your question. Besides didn't vanhees71 express already his viewpoint in post #45?

 

[TrickyDicky]

In classical physics (Newtonian physics, special and general relativity), measurement is not a fundamental concept. Historically, Einstein did postulate measurement as fundamental in special relativity: the speed of light measured by any inertial observer is the same. However, we have removed that, and nowadays we say that special relativity means the laws have Poincare symmetry. Historically, measurement was also important in the genesis of general relativty: test particles follow geodesics. The test particle is a sort of measurement apparatus that is apart from the laws of physics because it does not cause spacetime curvature, in contrast to all other forms of matter. However, in the full formulation of general relativity, test particles are not fundamental. So quantum mechanics is different from classical physics in needing to specify measurement as a fundamental concept.

I am using a particular interpretation to define QM, but it is the minimal interpretation. The measurement problem is that we have to put this classical/quantum cut to define the minimal interpretation. The other interpretations are then approaches to solving the measurement problem by removing the need for measurement to be a fundamental concept in the mathematical specification of a theory. Examples of such interpretations are consistent histories (flavour of Copenhagen), hidden variables (generally predicting deviations from QM), or Many-Worlds.
Another way of stating the measurement problem, is that if everything is quantum and we have a wave function of the universe, how can we make sense of such an idea? The
minimal interpretation cannot make sense of such an idea, and always needs a classical/quantum cut. Bohmian Mechanics and Many-Worlds are two approaches to solving the measurement problem, in which the wave function of the universe is proposed to make sense.

But you don't seem to be talking about what is usually known as the minimal or ensemble interpretation, you are giving too much reality to the quantum state, not even Copenhagen is so realistic about the state, it only considers it well defined in the context of observation, that's why they need a well defined collapse of the wave function at the moment of measurement. If you don't consider physical states at all then collapse doesn't arise in such explicit form because there is no wave function to collapse in any actual sense to begin with, but of course you still have irreversibility and the Born rule in your post 47 form, by definition the basic math of QM must be the same for all interpretations, otherwise it would be a different theory.
This doesn't mean the ensemble interpretation solves the measurement problem anymore than magical collapse, or decoherence or many-worlds do, they all leave the preferred-basis side of it basically untouched, but as we saw recently in the thrashing thread's paper, that problem is inherent to the first postulate so it can't be cured just looking at measurement.

 

[vanhees71]

Collapse is the statement of the Born rule in the form $P(\phi) = |\langle \phi | \psi \rangle|^{2}$, which is what happens in a filtering measurement. This is Eq 2.1.7 in volume 1 of Weinberg's QFT text. I think you agreed that measurement can be used as a means of state preparation, so in that sense I thought you said that collapse exists. If collapse does not exist, then are you saying that Weinberg's Eq 2.1.7 does not exist? If collapse does exist, then it seems you are saying that collapse can be derived, ie. Weinberg's Eq 2.1.7 can be derived.

The only thing I need is the statement that, if a quantum system is prepared in a state, represented by a normalized vector $|\psi \rangle$, the probability (density) to find the value $a$ of the observable $A$ in the discrete (continuous) part of the spectrum of its representing self-adjoint operator $\hat{A}$, is given by
$P(a)=\sum_{\beta} |\langle a,\beta|\psi \rangle|^2,$
where $\beta$ labels a complete set of orthonormalized eigenvectors of $\hat{A}$ to the eigenvalue $a$. Of course, $\beta$ can also be continuous. Then the sum has to be substituted with the corresponding integral.

Where do I need a collapse for Born's postuate?

 

[atyy]

The only thing I need is the statement that, if a quantum system is prepared in a state, represented by a normalized vector $|\psi \rangle$, the probability (density) to find the value $a$ of the observable $A$ in the discrete (continuous) part of the spectrum of its representing self-adjoint operator $\hat{A}$, is given by
$P(a)=\sum_{\beta} |\langle a,\beta|\psi \rangle|^2,$
where $\beta$ labels a complete set of orthonormalized eigenvectors of $\hat{A}$ to the eigenvalue $a$. Of course, $\beta$ can also be continuous. Then the sum has to be substituted with the corresponding integral.

Where do I need a collapse for Born's postuate?

So you disagree with Weinberg that his Eq 2.1.7 is a postulate?

 

[dextercioby]

What is equation 2.1.7?

 

[atyy]

What is equation 2.1.7?

Here is my transcription of Weinberg's Eq 2.1.7 in http://books.google.com/books?id=doeDB3_WLvwC&source=gbs_navlinks_s (p50).

$P(\mathscr{R} \rightarrow\mathscr{R_{n}}) = |(\Psi,\Psi_{n})|^{2}$

It is the probability that a system prepared in state $\mathscr{R}$ is found in state $\mathscr{R_n}$, if a test is done to find out whether the system is in one of several orthogonal states $\{ \mathscr{R_1}, \mathscr{R_2}, .. \}$, and $\Psi$ is a (unit) vector representing the state $\mathscr{R}$.

 

[dextercioby]

Oh, you mean his QFT book, I thought his regular QM book. OK, that makes sense now, even though it's odd it appears in his QFT book.

 

[microsansfil]

Here is my transcription of Weinberg's Eq 2.1.7 in http://books.google.com/books?id=doeDB3_WLvwC&source=gbs_navlinks_s (p50).

$P(\mathscr{R} \rightarrow\mathscr{R_{n}}) = |(\Psi,\Psi_{n})|^{2}$

It is the probability that a system prepared in state $\mathscr{R}$ is found in state $\mathscr{R_n}$, if a test is done to find out whether the system is in one of several orthogonal states $\{ \mathscr{R_1}, \mathscr{R_2}, .. \}$, and $\Psi$ is a (unit) vector representing the state $\mathscr{R}$.

Is it applicable for a dissipative system, that is put in a mixed quantum state ?

Patrick

 

[atyy]

Is it applicable for a dissipative system, that is put in a mixed quantum state ?

When the state is mixed, it has to be represented by a density operator. The Born rule, and collapse or state reduction for a density operator is given in
http://arxiv.org/abs/1110.6815 (Eq II.3, II.4 on p9)
http://arxiv.org/abs/0706.3526 (Eq 2, 3 on p4)

 

[vanhees71]

I guess, you mean Eq. (2.1.7) in Quantum Theory of Fields Vol. 1? Then, of course it's Born's rule and thus one of the postulates of quantum theory. I still don't see, where you need a collapse for its statement.

There's no difference when you are in a general (mixed ) state. Also there no collapse is needed. You have a mixed state, whenever in addition to the irreducible indeterminacy of observables (which are present even when the state of the system is completely determined) there's also incomplete knowledge about in which state the system is prepared.

A pure state is of course also uniquely and equivalently described by a statistical operator, not only a ray in Hilbert space. A statistical operator represents a pure state if and only if it is a projection operator.

 

[atyy]

I guess, you mean Eq. (2.1.7) in Quantum Theory of Fields Vol. 1? Then, of course it's Born's rule and thus one of the postulates of quantum theory. I still don't see, where you need a collapse for its statement.

Yes, in Weinberg's QFT volume 1. Eq 2.1.7 is the definition of collapse, because it gives the probability for the system to jump from state R to Rn. For example, given a Bell state |00> + |11>, when Alice gets the measurement outcome "0", the state will transition to |00>. That is the collapse, just a mathematical statement that is postulated.

 

[vanhees71]

The formulation in Weinberg may be a bit misleading, but there's no collapse implied as far as I understand him. Also one should be more careful in formulating Born's rule to make it compatible with the dynamics. You have to distinguish between the kets representing the state (rays in Hilbert space) and the (generalized) eigenvectors of observables. In a general picture (Dirac picture) of time evolution they evolve with different unitary time-evolution operators
$$|\psi,t \rangle=\hat{C}(t) |\psi,0 \rangle, \quad |\vec{a},t \rangle=\hat{A}(t) |\vec{a},0 \rangle.$$
Here $\vec{a}$ is in the common spectrum of a complete compatible set of observables, and the time-evolution operators obey the equations of motion
$$\frac{\mathrm{d}}{\mathrm{d} t} \hat{C}(t)=-\mathrm{i} \hat{H_1}(t) \hat{C}(t), \quad \frac{\mathrm{d}}{\mathrm{d} t} \hat{A}(t)=\mathrm{i} \hat{H_2}(t) \hat{C}(t)$$
with the initial conditions
$$\hat{C}(0)=\hat{A}(0)=\hat{1}.$$
The operators $\hat{H}_1$ and $\hat{H}_2$ are local in $t$ and self-adjoint operators, related to the Hamiltonian of the system by
$$\hat{H}=\hat{H}_1+\hat{H}_2.$$
Then the probability (density) to find the outcome $\vec{a}$ at time $t$ when measuring the complete set of observables on the system, prepared in the state represented by $|\psi,t \rangle$ is
$$P(\vec{a},t|\psi)=|\langle \vec{a},t|\psi,t \rangle|^2.$$
By construction the time evolution of this probability (density) is independent of the choice of the picture of time evolution.

This does not imply that the quantum system automatically somehow collapses into a state represented by $|\vec{a},t \rangle$ at the time of the measurement. In the case of observables with continuous spectrum this even contradicts the "kinematical" part of the quantum postulates, because then this is not a normalizable state in Hilbert space but a generalized state in the dual space of the nuclear space, where the observable operators are densely defined. Already this formal argument shows that the collapse hypothesis makes no proper sense.

 

[stevendaryl]

The only thing I need is the statement that, if a quantum system is prepared in a state, represented by a normalized vector $|\psi \rangle$, the probability (density) to find the value $a$ of the observable $A$ in the discrete (continuous) part of the spectrum of its representing self-adjoint operator $\hat{A}$, is given by
$P(a)=\sum_{\beta} |\langle a,\beta|\psi \rangle|^2,$
where $\beta$ labels a complete set of orthonormalized eigenvectors of $\hat{A}$ to the eigenvalue $a$. Of course, $\beta$ can also be continuous. Then the sum has to be substituted with the corresponding integral.

Where do I need a collapse for Born's postuate?

I think we already went around once about this. The question is: How does measurement constitute a preparation procedure for a particular state? Take the simplest example of preparing an electron in the state with spin-up in the z-direction. The usual assumption is that measuring the spin, and finding it to be spin-up means that afterward, the electron is in the spin-up state. Isn't that basically the collapse hypothesis? Without some assumption along those lines, how would you prepare an electron in the spin-up state?

 

[vanhees71]

I think we already went around once about this. The question is: How does measurement constitute a preparation procedure for a particular state? Take the simplest example of preparing an electron in the state with spin-up in the z-direction. The usual assumption is that measuring the spin, and finding it to be spin-up means that afterward, the electron is in the spin-up state. Isn't that basically the collapse hypothesis? Without some assumption along those lines, how would you prepare an electron in the spin-up state?

I answered this already, but again:

Take a Stern-Gerlach (SG) apparatus to "measure" the spin-z direction. A particle running through this apparatus is deflected in one of two possible directions. After this deflection, which is described by unitary time evolution for a particle with spin running through an appropriate inhomogeneous magnetic field (with a large homogeneous component in z direction, which sorts out the spin-z components as measured observables), particles at the respective are with in principiple arbitrary accuracy prepared in the corresponding spin state. Nowhere do you need a collapse. It's simply unitary time evolution, in this example even of a single-particle Schrödinger-Pauli equation.