A Difference Between Collapse and Projection

[vanhees71]

I suspect that @vanhees71 refers to a closed system, because he believes that we can in principle include the observer and measurement apparatus in the quantum state, so that there is only unitary evolution. This is also my reading of what Ballentine means in his textbook, given his criticism of standard QM. I believe that postulating unitary evolution without state reduction is not correct unless one introduces additional postulates (eg. as attempted by many worlds, hidden variables, which also remain non-standard).

That's precisely the question. Which additional postulates do you mean? Why precisely do you think that the minimal statistical interpretation is incomplete? You always claim Ballentine's textbook is fundamentally wrong, but there's no convincing argument for that claim. It's just an opinion.

I think it's a philosophical standpoint rather than a valid critique of QT as a physical theory. You seem to consider QT as incomplete in the same sense as Einstein did, i.e., Einstein indeed seems to have accepted QT in its minimal (ensemble) interpretation but then considered it incomplete, because for Einstein a complete theory should be deterministic in the sense that, provided you have "complete knowledge" about the system's state, you should be able to precisely predict the outcome of all possible measurements on the individual system and not only about the statistics for these outcomes when measured on an ensemble of equally prepared systems.

Einstein's postulate what "complete knowledge" in the sense of a more comprehensive theory than QT means was that there are unkown "hidden variables" which we just ignore or don't have discovered yet to exist. In other words QT was incomplete in the same sense as the description of a classical mechanical in terms of a coarse-grained probabilistic description using only partial information about the system in statistical mechanics, because you simply cannot keep track of the full phase-space trajectories of $10^{24}$ particles, i.e., it's the step from the full Liouville equation (describing full knowledge about the phase-space evolution of the entire system) to some transport equation, throwing away a lot of information in terms of correlations by cutting the BBGKY hierarchy at a certain order. In the Boltzmann equation you only consider single-particle phase-space distribution functions and throwing away correlations already at the two-particle level, introducing the "molecular-chaos assumption" and at this point throwing away a lot of detailed information about the system, leading to increasing macroscopic entropies and dissipation, implying the H-theorem (with the thermodynamic time arrow identical with the causal time arrow implicit in the full mechanical dynamics from the beginning).

 

[atyy]

That's precisely the question. Which additional postulates do you mean? Why precisely do you think that the minimal statistical interpretation is incomplete? You always claim Ballentine's textbook is fundamentally wrong, but there's no convincing argument for that claim. It's just an opinion.

I think it's a philosophical standpoint rather than a valid critique of QT as a physical theory. You seem to consider QT as incomplete ...

The question is not whether the minimal statistical interpretation is complete. My claim is that Ballentine does not give the minimal statistical interpretation. He gives a wrong theory due to omission of state reduction. The minimal statistical interpretation contains state reduction.

 

[vanhees71]

But the minimal statistical interpretation doesn't contain a state-reduction postulate. That's the main point distinguishing it form some flavors of Copenhagen interpretations including a state-reduction postulate. So you are of the opinion that you cannot use QT without a state-reduction postulate. That's an opinion I have to respect, but I don't understand where you need the state reduction to apply the formalism to real-world experiments nor do I understand, why I should accept a postulate contradicting the other postulates and is not generally consistent with what's done in experiments. Projective measurements are very rare and very hard to realize!

 

[atyy]

Projective measurements are very rare and very hard to realize!

The elementary description of Bell experiments uses projective measurements, and these experiments are realized routinely nowadays. If the measurements are spacelike separated, there will be a frame in which you can consider the measurements simultaneous. But that means in another frame, the measurements are sequential, and the projection postulate ensures consistency of outcomes regardless of what frames are used to calculate.

 

[Demystifier]

if there were a detector after the first magnet, it would absorb the atom and make it unavailable for the other magnets.

Is it possible to detect an atom without absorbing it?

 

[Demystifier]

Concerning physics, never listen to mathematicians

Von Neumann was much more than just a mathematician. He was the first who understood what physically happens with the state of the closed system during the measurement, using non-rigorous physical arguments. It's now called Von Neumann theory of measurement. He was also the first (or maybe the second, after Landau) who introduced the concept of density matrix in quantum physics, in a physical fashion.

Of course, as a mathematician he also contributed significantly to rigorous functional-analysis style formulation of quantum mechanics, but that's another story.

 

[Lord Jestocost]

Nowadays the measurement data are stored in some computer file and read by a "conscious observer" months after the data are taken. The measured system is long gone. For me von Neumann's interpretation is esoterics and has nothing to do with science.

Please, stay with the mathematics of the purely quantum-mechanical von Neumann measurement chain (and, please, avoid terms like "esoterics" as scientific arguments).

Regarding the Heisenberg cut (or “collapse” or “state reduction” or whatever one might chose to call it), it is a “purely epistemological move without any counterpart in ontology” (as N.P. Landsman characterizes Heisenberg’s and Bohr’s reasoning with respect to the Heisenberg cut in his paper “Between classical and quantum”). Or, to elaborate it a little bit more (Klaas Landsman in “Foundations of Quantum Theory / From Classical Concepts to Operator Algebras”):

“Summarizing the preceding discussion, 'our' measurement problem states that:
• Certain pure post-measurement states of an (ontologically quantum-mechanical!) apparatus coupled to a microscopic quantum object induce mixed states on the apparatus (and on the composite) once the apparatus is described classically.
This is a precise version of Schrödinger’s Cat problem (rather than von Neumann’s purely quantum-mechanical measurement problem), making it clear that at heart the problem does not lie with the (dis)appearance of interference terms (which is a red herring) but with the inability of quantum mechanics to predict single outcomes.” [Italics and bold in original, LJ]

It is extremely important to understand this point: In case you want to avoid this epistemological move (one has to grasp that it's - so to speak - a mental decision where to arbitrarily interrupt the purely quantum-mechanical von Neumann measurement chain; epistemologically without any ontological basis), you end up at the end of the von Neumann measurement chain. There is no esotericism, it's pure mathematics.

Statements like “Nowadays the measurement data are stored in some computer file and read by a "conscious observer" months after the data are taken.“ exactly mirror the fundamental misunderstanding of the messages send by quantum mechanics. Such statements are nothing but "classical common sense" imaginations, but the mathematics of the purely quantum-mechanical von Neumann measurement chain speaks something else.

 

[Demystifier]

You always claim Ballentine's textbook is fundamentally wrong, but there's no convincing argument for that claim.

In the book he explicitly makes a testable prediction that the quantum Zeno effect does not exist. Experiments prove him wrong.

 

[Demystifier]

In general I don't think that you have a pure state after the measurement, because the measured system is entangled with the measurement device ("immediately after the measurement") and thus the system's state given by the partial trace (tracing out the system), and this leads to a mixed state for the system rather than a pure state after the measurement.

If so, then the state of the closed system (the measured system + the measurement device) is in the macroscopic superposition. It's a superposition of different possible measurement outcomes. Yet only one outcome actually realizes, we never observe superpositions of different possible outcomes. Do you agree that, when we learn what the actual outcome is, we can update our knowledge by using the state (2.93) in post #43?

 

[vanhees71]

Von Neumann was much more than just a mathematician. He was the first who understood what physically happens with the state of the closed system during the measurement, using non-rigorous physical arguments. It's now called Von Neumann theory of measurement. He was also the first (or maybe the second, after Landau) who introduced the concept of density matrix in quantum physics, in a physical fashion.

Of course, as a mathematician he also contributed significantly to rigorous functional-analysis style formulation of quantum mechanics, but that's another story.

Yes, and the latter is his great achievement concerning quantum theory. His part on "interpretation" is not so brillant. That's known since Herman's work in the 30ies.

 

[vanhees71]

If so, then the state of the closed system (the measured system + the measurement device) is in the macroscopic superposition. It's a superposition of different possible measurement outcomes. Yet only one outcome actually realizes, we never observe superpositions of different possible outcomes. Do you agree that, when we learn what the actual outcome is, we can update our knowledge by using the state (2.93) in post #43?

Of course, we update our knowledge, but whether after that the system is in the state (2.93) cannot be answered without knowing the specifical experiment. If a photon is absorbed by the measurement (2.93) is not the state of the em. field after the measurement but it's the vacuum state.

 

[vanhees71]

Statements like “Nowadays the measurement data are stored in some computer file and read by a "conscious observer" months after the data are taken.“ exactly mirror the fundamental misunderstanding of the messages send by quantum mechanics. Such statements are nothing but "classical common sense" imaginations, but the mathematics of the purely quantum-mechanical von Neumann measurement chain speaks something else.

But von Neumann claims the state collapse only occurs after a conscious being hat taken note of the measurement result. This doesn't make scientific sense! This has nothing to do with "classical common sense".

 

[vanhees71]

In the book he explicitly makes a testable prediction that the quantum Zeno effect does not exist. Experiments prove him wrong.

That's true. Where is this statement in his book? One has to see the context.

The quantum Zeno effect is simply due to the interaction with some measurement device stabilizing the lifetime of an unstable "state". There's nothing contradicting quantum-mechanical dynamics.

 

[Demystifier]

Of course, we update our knowledge, but whether after that the system is in the state (2.93) cannot be answered without knowing the specifical experiment. If a photon is absorbed by the measurement (2.93) is not the state of the em. field after the measurement but it's the vacuum state.

The vacuum state is a state of EM field. Therefore (2.93) is correct even in this case. Indeed, the theory of measurement around (2.93) is the general theory of measurement. As far as we know, all measurements (not just projective ones) satisfy those principles.

 

[Demystifier]

That's true. Where is this statement in his book? One has to see the context.

I already quoted his exact statement several times to you before. I'll not do it again, but I'm sure you can find it in the book by yourself.

The quantum Zeno effect is simply due to the interaction with some measurement device stabilizing the lifetime of an unstable "state". There's nothing contradicting quantum-mechanical dynamics.

Perhaps, but there is something contradicting Ballentine.

 

[vanhees71]

The vacuum state is a state of EM field. Therefore (2.93) is correct even in this case. Indeed, the theory of measurement around (2.93) is the general theory of measurement. As far as we know, all measurements (not just projective ones) satisfy those principles.

No (2.93) claims there's still a single-photon state collapsed to the state according to the measurement outcome! I say that all you can say from the formalism are the probabilities (2.92) for the outcome of measurements not in which state the measured quantum system is after the measurement. For the letter you need the details about the measurement apparatus and the interactions between the apparatus and the measured system.

 

[Demystifier]

No (2.93) claims there's still a single-photon state collapsed to the state according to the measurement outcome!

Sorry but you are wrong, in a way that has nothing to do with interpretations and philosophy. You must have misunderstood something purely physical about POVM measurements.

Let me explain how POVM works for photon detection. Let $\{|m\rangle\}$, $m\neq 0$, be a basis of 1-photon states and let $|m=0\rangle$ be the photon vacuum. The Kraus operators for ideal (perfectly efficient) photon detection can be taken to be
$$M_m=|0\rangle\langle m|$$
They satisfy
$$\sum_m M_m^{\dagger}M_m=\sum_m |m\rangle \langle m|=1$$
If the state before measurement is a superposition
$$|\psi\rangle=\sum_{m'} c_{m'}|m'\rangle$$
then after measurement (that is, when the result of measurement is known), the updated state is proportional to
$$M_m|\psi\rangle=c_m|0\rangle$$
So after the measurement we always have the vacuum, that is, the photon is destroyed (absorbed) by the detector.

@atyy I have significantly edited this post that you already liked. I hope you will still like it, perhaps now even more.

 

[Fra]

Von Neumann was completely aware of that, so to avoid the contradiction he postulated two types of evolutions. One is valid when conscious observations are present, the other when conscious observations are not present.

The strategies that try to resolve the problem of introducing a "cut", by moving the cut around - or typically move it out of FAPP reach and think its solved by making bigger and bigger hilbertspaces. It somehow IMO misses the troublesome point and is like cheating. Unfortunately as long as we stick to particle physics (ie looking and small subsystems from distance) this seems to pass and seems like the least problematic way to interpret things as is.

But for those that try to "interpret" or revise QM for purposes of unifications, this trick does not seem to suffice. At some point the cut is even way out of reach to the "original observer"; so I see now way to escape the original problems.

/Fredrik

 

[Lord Jestocost]

But von Neumann claims the state collapse only occurs after a conscious being hat taken note of the measurement result.

To my mind, nothing else remains to be done. I would like to reformulate your sentence: In case you cannot show by means of a rigorous mathematical proof where to apply the projection postulate in the von Neumann measurement chain, a conscious observer cannot be regarded as pure physical (ontologically quantum mechanical) system. As Nick Herbert (in his recommendable book “Quantum Reality: Beyond the New Physics”) describes John von Neumann's reasoning:

“In other words, where in fact is a quantum measurement actually accomplished?

While searching for a natural place to break his chain, von Neumann proved an important mathematical fact that deepens the mystery of measurement. Von Neumann showed that as far as final results are concerned, you can cut the chain and insert a collapse anywhere you please. This means that the results themselves can offer no clues as to where to locate the division between system and measuring device….

.…On each side of the wave function collapse, von Neumann erects impeccable mathematical structures familiar to quantum physicists — the world expressed as proxy waves. However, separating these two sides of the argument — the world unmeasured and the measured world — is a logic gap in which von Neumann effectively writes, ‘And then a miracle occurs.’

Von Neumann could not find a natural place to locate his ‘miracle.’ Everything, after all, is made of atoms: there’s nothing holy about a measuring instrument. Following the von Neumann chain, driven by his own logic, in desperation von Neumann seized on its only peculiar link: the process by which a physical signal in the brain becomes an experience in the human mind. This is the only process in the whole von Neumann chain which is not mere molecules in motion.”

 

[PeterDonis]

Is it possible to detect an atom without absorbing it?

I don't know. I do know that the detector used in the Stern Gerlach experiment did absorb the atoms. If you are proposing a different kind of detector in your thought experiment, it's up to you to specify what kind of detector it is.

 

[Demystifier]

I don't know. I do know that the detector used in the Stern Gerlach experiment did absorb the atoms. If you are proposing a different kind of detector in your thought experiment, it's up to you to specify what kind of detector it is.

Electron microscope can detect a single atom without absorbing it. But electron microscope probably can't detect an atom that moves, so it's still not adequate for measuring spin with a SG magnet. Any ideas? You are a nuclear engineer, right?

 

[PeterDonis]

Any ideas? You are a nuclear engineer, right?

Nuclear engineering doesn't really deal with manipulating single atoms (or single nuclei) at a time, unfortunately.

 

[vanhees71]

Sorry but you are wrong, in a way that has nothing to do with interpretations and philosophy. You must have misunderstood something purely physical about POVM measurements.

Let me explain how POVM works for photon detection. Let $\{|m\rangle\}$, $m\neq 0$, be a basis of 1-photon states and let $|m=0\rangle$ be the photon vacuum. The Kraus operators for ideal (perfectly efficient) photon detection can be taken to be
$$M_m=|0\rangle\langle m|$$
They satisfy
$$\sum_m M_m^{\dagger}M_m=\sum_m |m\rangle \langle m|=1$$
If the state before measurement is a superposition
$$|\psi\rangle=\sum_{m'} c_{m'}|m'\rangle$$
then after measurement (that is, when the result of measurement is known), the updated state is proportional to
$$M_m|\psi\rangle=c_m|0\rangle$$
So after the measurement we always have the vacuum, that is, the photon is destroyed (absorbed) by the detector.

@atyy I have significantly edited this post that you already liked. I hope you will still like it, perhaps now even more.

Ok, then it's fine for this case, and I stand corrected. So $\hat{M}_m$ describes effectively a transition matrix element between states of the measured object after interacting with the measurement device. Then it's fine, because you don't assume something outside the time-evolution formalism in the sense that the $\hat{M}_m$ can be derived in principle by the time evolution of an open quantum system.

This misunderstanding could have been avoided by clearly defining the meaning of the operators $\hat{M}_m$ in the beginning!

 

[Demystifier]

Ok, then it's fine for this case, and I stand corrected. So $\hat{M}_m$ describes effectively a transition matrix element between states of the measured object after interacting with the measurement device. Then it's fine, because you don't assume something outside the time-evolution formalism in the sense that the $\hat{M}_m$ can be derived in principle by the time evolution of an open quantum system.

This misunderstanding could have been avoided by clearly defining the meaning of the operators $\hat{M}_m$ in the beginning!

So we agree on physics. But there is one interpretation issue that, I think, you was still not completely clear about. If we consider a closed system, including the measuring apparatus, is everything deterministic?

 

[vanhees71]

The state evolution is deterministic but the state doesn't determine all possible observables but provide and only provide probabilities for the outcome of measurements of any observable, for whose test you need an ensemble of equally prepared individual systems.

 

[Demystifier]

only provide probabilities for the outcome of measurements

Suppose that there is no measurement at the time $t$. Consider two quantities
$$|\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle \; \; \; \; (1)$$
$$p(t)=\langle\psi(t)|\pi|\psi(t)\rangle \; \; \; \; (2)$$
where $\pi$ is a projector (not associated with a measurement because, as I said, there is no measurement at time $t$). Since there is no measurement at $t$, does probability (2) have any physical meaning? If not, does the state (1) have any physical meaning?

A related question. How the fact that there is a measurement expressed mathematically? I would like an answer in the form: We say that an observable is measured when
$$some \;\; equation$$

 

[vanhees71]

If $\pi$ is a projector, it's of the form $|a \rangle \langle a|$ with a unit-vector $|a \rangle$. If you interpret $|a \rangle$ to be the eigenstate of some operator $\hat{A}$ that represents some observable $A$, $p(t)$ is of course the probability to find the value $a$ when you measure $A$ and the system is prepared in the state $\hat{\rho}=|\psi(t) \rangle \langle \psi(t)|$.

The physical meaning is the usual one $\hat{\rho}$ is the statistical operator in the Schrödinger picture of time evolution and $p(t)$ is the probability to get $a$ as the result of a measurement of $A$ when measured at time $t$.

The whole point is that you cannot give an answer to your related question in terms of a general postulate. It depends on the individual experiment, how the system evolves when interacting with the measurement device.

 

[vanhees71]

Yes, but assuming a measurement causes a collapse, i.e., a change of the state, implies a causal influence of the measurement on the state, and that's the problem particularly in this context. It's contradicting the very assumptions you make about the dynamics of the system (microcausality condition), which by construction cannot violate causality, i.e., space-like separated events cannot be causally connected.

 

[Demystifier]

Yes, but assuming a measurement causes a collapse, i.e., a change of the state, implies a causal influence of the measurement on the state, and that's the problem particularly in this context. It's contradicting the very assumptions you make about the dynamics of the system (microcausality condition), which by construction cannot violate causality, i.e., space-like separated events cannot be causally connected.

You reject that measurement causes collapse of the state, but you accept that measurement entails update of the state, am I right? But "causing collapse" and "entailing update" are described by the same mathematics and no experiment can distinguish one from the other. So when you insist that they are different, you are doing philosophy, not physics. And it would not be a problem if it was a consistent philosophy, but it's not. It's inconsistent because the update of the state also violates the Schrodinger equation (or its relativistic QFT analog), so you both accept and don't accept violation of the Schrodinger equation. So you do philosophy, and you do it inconsistently, but you are not disturbed because "that's just philosophy", so who cares.