Difference Between Collapse and Projection

[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 Schrödinger equation (or its relativistic QFT analog), so you both accept and don't accept violation of the Schrödinger equation. So you do philosophy, and you do it inconsistently, but you are not disturbed because "that's just philosophy", so who cares.