B How to understand unitarity in QM?

[Grinkle]

But quantum mechanics, as far as we know, has a property called unitarity: if I know the exact state of the system at time T, then I can, given enough computer power, calculate the precise state of the system at any other time, no matter what.

Quote lifted from a thread in the cosmology forum.

What does it mean to know the exact state of a QM system? QM predicts probabilities that particles will be in one of multiple states when the particles are observed, and when observed, not all properties of a particle are simultaneously knowable to an exact degree (eg position and momentum).

Does knowing the exact state mean I know the probability functions for each particle in a given system, or is it different than that?

 

[dextercioby]

If you know the state of the system, you have encoded in this state all probabilities of measuring the value a of the quantum observable A (the set of all this possible values is the spectrum of the operator associated to A). This is the content of the so-called Born rule (or the probabilistic interpretation of QM). But knowing the exact state of the system is a tricky business, since you'd have to know the Hamiltonian and how to solve the time evolution equation. Only for a time-independent Hamiltonian, the set of possible system states is expressible in an exact manner in terms of a set of solutions to a partial differential equation.

 

[Grinkle]

since you'd have to know the Hamiltonian and how to solve the time evolution equation.

Thanks for the reply. Now I may get into trouble because I don't know how to make my follow up very precise.

If one could know and solve respectively the above, would the solution be a time-varying set of probabilities?

As opposed to Newtonian physics providing a time-varying set of state values each with probability 1, I mean.

Another question - is it in theory possible to discover and solve the Hamiltonian for a time dependent system? Is this why @kimbyd said 'given enough compute power'?

 

[bhobba]

To answer you query you really need to see an axiomatic derivation of the Born Rule:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

See post 137.

Thanks
Bill

 

[atyy]

If one could know and solve respectively the above, would the solution be a time-varying set of probabilities?

No, it is a time varying many sets of probabilities. There are many sets of incompatible probabilities, because one can make many incomptible measurements.

Also, when a measurement is made, unitarity fails.

 

[PeroK]

Quote lifted from a thread in the cosmology forum.

What does it mean to know the exact state of a QM system? QM predicts probabilities that particles will be in one of multiple states when the particles are observed, and when observed, not all properties of a particle are simultaneously knowable to an exact degree (eg position and momentum).

Does knowing the exact state mean I know the probability functions for each particle in a given system, or is it different than that?

A simple answer is that in QM the state of a system evolves deterministically - according to the Schrödinger equation. But a measurement produces probabilistic outcomes.

There are no probabilities, as such, in the time evolution of the state of a system.

 

[Jilang]

A simple answer is that in QM the state of a system evolves deterministically - according to the Schrödinger equation. But a measurement produces probabilistic outcomes.

There are no probabilities, as such, in the time evolution of the state of a system.

Is that correct? I understood that the state is a probabilistic construct itself?

 

[PeroK]

Is that correct? I understood that the state is a probabilistic construct itself?

Please define "probabilistic construct".

 

[Jilang]

Please define "probabilistic construct".

the wavefunction (of the state) being something that tells you about the probability of finding something in that state.

 

[PeroK]

the wavefunction (of the state) being something that tells you about the probability of finding something in that state.

What part of that definition precludes the state from evolving deterministically?

How does your definition preclude a state from remaining constant, for example? That would be the simplest form of deterministic time evolution.

 

[PeroK]

the wavefunction (of the state) being something that tells you about the probability of finding something in that state.

I should add, however, that that is not a valid definition of a wavefuction.

 

[Jilang]

I should add, however, that that is not a valid definition of a wavefuction.

How would you define it?

 

[PeroK]

How would you define it?

You could Google for wavefunction definition. The Wikipedia entry looks good enough.

 

[Jilang]

You could Google for wavefunction definition. The Wikipedia entry looks good enough.

OK it says..
"A wave function in quantum physics is a mathematical description of the quantumstate of a system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it."
It certainly seems to lead to probabilities.

 

[PeroK]

OK it says..
"A wave function in quantum physics is a mathematical description of the quantumstate of a system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it."
It certainly seems to lead to probabilities.

The results of measurements are probabilistic but the way the wavefunction itself evolves over time is not probabilistic.

A crude analogy is a coin. If you toss a coin you get heads or tails with 50% probability - that's the measurement. But the coin itself and the probabilities of getting heads and tails do not change over time. That represents the state.

In other words, the toss of a coin is probabilistic, but the coin itself is always the same.

Note that this is a rough analogy and not meant to be precisely related to QM.

 

[Jilang]

Perok, what do you think the wavefunction represents, if not the outcome of a measurment?

 

[bhobba]

Perok, what do you think the wavefunction represents, if not the outcome of a measurment?

Did you read my post 4.

I will give its outline.

First we need to define a Positive Operator Value Measure (POVM). A POVM is a set of positive operators Ei ∑ Ei =1 from, for the purposes of QM, an assumed complex vector space.

Elements of POVM's are called effects and its easy to see a positive operator E is an effect iff Trace(E) <= 1.

Now we can state the single foundational axiom QM'

An observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

Note - nothing said at all about a state.

I will evoke a very beautiful theorem which is a modern version of a famous theorem you may have heard of called Gleason's, and in the link prove it.

It says a positive operator of unit trace P exists such that the probability of Ei occurring in the POVM E1, E2 ... is Trace (Ei P).

This is called Born's Rule. P by definition is called the state of the system. It is simply an aid in calculating the probability of Ei. That's it - that's all. Its nothing 'mystical' etc etc. Its simply something used as an aid in calculating the probability of a certain outcome derived from the fundamental axiom that outcomes can be mapped to POVM's.

Thanks
Bill

 

[PeroK]

Perok, what do you think the wavefunction represents, if not the outcome of a measurment?

I don't think, I know what it represents. You may be getting confused between the time evolution of the wavefunction between measurements (deterministic, according to Schrödinger equation) and the probabilistic "collapse" of the wavefunction upon measurement.

In simple terms this means that if you leave a system alone (perhaps after an initial measurement) its wavefunction evolves deterministically until you make a further measurement, at which point it randomly collapses into a new wavefunction, which then evolves deterministically again.

 

[tom.stoer]

One question and one comment:

The fundamental assumption in this thread is that quantum mechanics is a probabilistic theory dealing with measurements. How is "measurement" defined, especially in contrast to unitary time evolution, i.e. when does a system evolve unitarily, and when does it collapse?

I guess all this does not apply to the "modernized" Everett's interpretation making use of decoherence; in this context the POVM must be there as well b/c we know that they fit to our observations, but they should not be introduced by an axiom but be a derived or emergent result or theorem.

 

[Simon Phoenix]

What does it mean to know the exact state of a QM system? QM predicts probabilities that particles will be in one of multiple states when the particles are observed, and when observed, not all properties of a particle are simultaneously knowable to an exact degree (eg position and momentum).

Does knowing the exact state mean I know the probability functions for each particle in a given system, or is it different than that?

You're not really using the terminology correctly here, but reading between the lines you seem to have the right idea, or at least the essence of it. It might be worthwhile to revisit the basic axioms of QM that can be found in a lot of textbooks. I'm going to add the caveat here that this is just a place to start - think of it like watching an Avengers movie; temporarily suspend one's disbelief and just try to enjoy the ride. So your mindset should be "OK, not too sure about these, but let's run with them and see what happens". The axioms I'm going to write ultimately need all sorts of refinements, additions and details added - but we have to start somewhere.

1. The state of physical system (eg an electron) is represented by a vector in a complex Hilbert space
2. This state evolves according to the Schrodinger equation
3. Observables are represented by linear Hermitian operators
4. The possible results of a measurement of an observable $\hat {\mathbf A}$ are the eigenvalues of $\hat {\mathbf A}$
5. If the initial state is $| \psi \rangle$ then the probability of getting the eigenvalue $a_i$ as a result of the measurement of $\hat {\mathbf A}$ is given by $| \langle a_i | \psi \rangle |^2$ where $| a_i \rangle$ is the eigenstate of $\hat {\mathbf A}$ associated with the eigenvalue $a_i$
6. Immediately after the measurement of $\hat {\mathbf A}$ in which the eigenvalue $a_i$ was obtained as a result of the measurement, the new state of the system is given by $| a_i \rangle$

Holy Gotham City Batman It's no wonder that students, exposed throughout their education to classical physics, see these axioms and have a very serious "WTF?" moment. Once we've had a chance to rest in a darkened room for several hours in order to calm down, we can try to use these frankly bizarre set of rules. The surprising thing is that they work, and they work very well indeed (by work I mean successfully allow us to calculate experimental predictions).

Now these axioms should be taken alongside a whole bunch of warning flags and alerts - they're just someplace to start. They're not necessarily the best set of axioms we could pick, or the most elegant, and on reflection we can see there are some gaping holes (or at least some major questions). So with the proviso that we may need to swap these out later on for a much more elegant and 'better' set of rules (that are equivalent) let's try to answer your question.

So, typically, in an experiment we might prepare our system in some known state (as best we can). So suppose we want to prepare a bunch of systems in some state $| a_i \rangle$ then we'd take a collection of systems and make measurements of $\hat {\mathbf A}$ and select all of those systems for which we got the result $a_i$. So we 'filter' out the states we want. Now we can experiment on these systems which we've prepared in a known state.

We might want to know what happens to our systems, prepared in the state $| a_i \rangle$, if we apply an electric field. So we work out what the Schrodinger equation would be in this situation and solve it to give us the new state that $| a_i \rangle$ evolves to when we apply the electric field. Then we decide what property we're going to measure (energy?, angular momentum? etc) and work out the probabilities of the results we should get.

In the absence of the measurement everything is evolving smoothly and reversibly according to the Schrodinger equation - and this is the 'unitary' bit. It's actually essential to make the probabilities all sum to 1. Notice that, in general, if we measure something like energy we'll get a particular set of possible results (the energy eigenvalues) with associated probabilities, but if we choose to measure angular momentum instead we'll get a different set of possible results (the angular momentum eigenvalues) with a different set of associated probabilities.

Now there are several issues with these 'beginning' set of axioms that I've presented (and I've written them out from memory - so apologies for any mistakes which I hope others will correct). Principal amongst them is the slightly 'magical' character of this thing I've blithely termed "measurement". Whilst it is obvious operationally what a measurement is ("Oh look, the intensity reads such and such") it's not really clear in the theoretical framework above what a measurement is. Whatever it is it would appear to be different from the nice smooth evolution dictated by the Schrodinger equation. But surely my measurement device is made up of all sort of bits and pieces (atoms and such) that also obey the QM evolution equation? Therein lies at least one of the thorny issues that QM presents us.

Another big issue is what is actually meant by the thing I've called a 'state'. You'll notice the seductive language that is difficult to avoid when I've talked about the system being "in a state". This, not too subtly, leads us to suppose that there is some real, objective thing we're talking about that somehow changes when we do this mystical thing called measurement. That's another decidedly vexing issue.

Yet another is this notion that measurements are so clean cut. In a typical experiment, say in a quantum optics lab, we'd end up destroying the thing we're measuring - photons are absorbed by photodetectors, for example. So we really need a different formalism to cope with what happens when our measurements aren't of this nice projective character implied by the axioms above. This is the POVM formalism that Bill mentioned but in my view you need a fair degree of sophistication to appreciate that, so it's not the best place to start (again in my view, but I'm sure Bill would disagree here). But even here the POVM formalism is equivalent to adding an ancillary system and doing these nice ideal measurements (whatever they are within the theory) on this ancillary system.

I'm sorry if by now I've thoroughly confused you - there is a certain sense in which QM is confusing - and it just takes practice and patience and effort to fool oneself that actually it isn't confusing at all. That might take a few months or even years

 

[tom.stoer]

Another big issue is what is actually meant by the thing I've called a 'state'. You'll notice the seductive language that is difficult to avoid when I've talked about the system being "in a state". This, not too subtly, leads us to suppose that there is some real, objective thing we're talking about that somehow changes when we do this mystical thing called measurement [or unitary time evolution]. That's another decidedly vexing issue.

It is - if you subscribe to the positivistic hairsplitting reasoning introduced into quantum mechanics. If you don't - as I suspect is the case in your everyday world - this vexing issue goes away but is replaced by a new but more reasonable problem.

If you get up in the morning, you never wonder if and how the bathroom you are just entering has existed while you were sleeping. You never ask yourself whether you performed a measurement when entering.

The positivistic ideas not to discuss the existence of the bathroom at all, to talk about measurements only, the problem that orthodox quantum mechanics is talking about measurements w/o being able to explain - even in principle - what a measurement really is and w/o being able to tell when a measurement is performed or when the system evolves unitarily is fundamentally misguided. Any interpretation talking about measurements w/o being able to define it is pointless, i.e. it falls back to shut-up-and-calculate.

From my perspective, the only reasonable way out is to understand physics as a representation of reality, comprising the existence of a “world”, its observation, its observers etc. - w/o introducing any cut between the quantum system, the measurement device and the observer, and even w/o distinguishing in a fundamental or axiomatic way, simply b/c this distinction does not exist in nature.

If physics shall represent nature, then we must follow this rule. If we accept physics not to represent nature or “reality” but to provide cooking recipes to calculate values we may do this as well, but then we should stop any discussion about interpretations, simply b/c we simply interpret our artifacts we introduced at the fundamental level.

Back to my example: when and why does entering the bathroom collapse the bathroom? or when does the bathroom and the atoms of my body evolve unitarily? Discussing these questions is pointless b/c we are asking the wrong questions. We are asking questions about our misconception to split nature in time evolution and measurement.

We can either calculate how the bathroom may look like w/o understanding what this means, or we are aiming to understand how quantum mechanics represents the bathroom while we are sleeping and what happens when we are entering, w/o distinguishing between unitary time evolution and measurement (collapse).

If we want to calculate, we can just calculate.

If we want to understand we must introduce reasonable axioms enabling us to understand. Collapse, Born’s rule and everything like that is useless at the axiomatic level.

(I do not agree with the MWI in all details, but it seems to be the only interpretation taking this train of thought seriously)

 

[bhobba]

this vexing issue goes away but is replaced by a new but more reasonable problem.

Indeed - our modern knowledge has not solved the issue - simply morphed it. If its a worry or not is entirely interpretation dependent.

Thanks
Bill

 

[Simon Phoenix]

If you don't - as I suspect is the case in your everyday world - this vexing issue goes away but is replaced by a new but more reasonable problem.

Yes - I've not seen a version (interpretation?) of QM that really works for me. That may well be a function of my own inadequacy rather than any fundamental issue with the theory, of course. I would say that certain interpretations are better at hiding the fundamental problems than others

 

[vanhees71]

1.-5. are pretty close to correct (although some subtle points are not; e.g., in 1. you have to use rays instead of vectors; otherwise you'd miss the existence of half-integer spin, which is a pity since the matter around us consists of a lot of spin-1/2 particles; in 3. Hermitean is not sufficient it should be self-adjoint; otherwise you run in contradictions).

6. is at least very problematic; I consider it as even wrong. In my opinion this socalled collapse postulate is neither necessary for anything relevant to physics and it's contradicting the very foundation of relativistic QFT, leading to the very successful Standard Model of elementary particle physics.

 

[Simon Phoenix]

1.-5. are pretty close to correct (although some subtle points are not; e.g., in 1. you have to use rays instead of vectors; otherwise you'd miss the existence of half-integer spin, which is a pity since the matter around us consists of a lot of spin-1/2 particles; in 3. Hermitean is not sufficient it should be self-adjoint; otherwise you run in contradictions).

Agreed - I was actually thinking of these very things when I said that the axioms needed further refinement

6. is at least very problematic; I consider it as even wrong. In my opinion this socalled collapse postulate is neither necessary for anything relevant to physics and it's contradicting the very foundation of relativistic QFT

Interesting. I'm not sure I understand you here.

Are you saying that if I have some spin-1/2 particle prepared in the state $|+>_z$ and measure ${\hat {\mathbf \sigma_x} }$, and obtain the positive eigenvalue the spin-1/2 particle is NOT in the state $|+>_x$ after the measurement?

So, for example, $N$ measurements of ${\hat {\mathbf \sigma_x} }$, on the same spin-1/2 particle does not give us the $N$-bit string 111...1 with unit probability if the first measurement yields 1?

So if I measure ${\hat {\mathbf \sigma_x} }$ for my spin-1/2 particle and obtain the result +1 what do you think the state of it is immediately after measurement?

Or is it simply that you think QM does not apply to single systems?

 

[Jilang]

Interesting, I would think 6 is fine but 1 needs some refinement. Something along the lines of "our knowledge of the state of the system is represented by a wavefunction in complex Hilbert space which evolves deterministically"

 

[PeroK]

Interesting, I would think 6 is fine but 1 needs some refinement. Something along the lines of "our knowledge of the state of the system is represented by a wavefunction in complex Hilbert space which evolves deterministically"

That is already covered by Axiom 2.

And, don't you think you ought to learn QM first, before pronouncing on how its axioms need to be changed?

 

[vanhees71]

Agreed - I was actually thinking of these very things when I said that the axioms needed further refinement
Interesting. I'm not sure I understand you here.

Are you saying that if I have some spin-1/2 particle prepared in the state $|+>_z$ and measure ${\hat {\mathbf \sigma_x} }$, and obtain the positive eigenvalue the spin-1/2 particle is NOT in the state $|+>_x$ after the measurement?

So, for example, $N$ measurements of ${\hat {\mathbf \sigma_x} }$, on the same spin-1/2 particle does not give us the $N$-bit string 111...1 with unit probability if the first measurement yields 1?

So if I measure ${\hat {\mathbf \sigma_x} }$ for my spin-1/2 particle and obtain the result +1 what do you think the state of it is immediately after measurement?

Or is it simply that you think QM does not apply to single systems?

I can't say in which state something is after a measurement if I don't know the measurement apparatus. If you use a Stern-Gerlach apparatus, then you can prepare $\sigma_z$ eigenstates by entangling position and spin component due to the inhomogeneous magnetic field, but there nothing like the collapse happens but just unitary time evolution.

QT applies to single systems, but measuring something on one single system doesn't tell you much. All you know, given the state (represented by a self-adjoint positive semidefinite operator with trace one $\hat{\rho}$, the Statistical Operator), are probabilities for the outcome of measurements. So you need an ensemble to verify that your preparation procedure leads to the state you claim to prepare. That's the minimal statistical interpretation, and practice in the lab uses just this. There's no need for additional assumptions like a collapse, many worlds, Bohm trajectories and what not the more philosophy oriented quantum physicists like to invent.

 

[tom.stoer]

Indeed - our modern knowledge has not solved the issue [what is actually meant 'state' and 'measurement'] - simply morphed it. If its a worry or not is entirely interpretation dependent.

I don't think that it's interpretation-dependent.

Any interpretation that does not or cannot define the terms used and cannot explain the relations between the mathematical formulation and the concept of "reality" addressed in the interpretation is in trouble. That means - at least to me - most interpretations are not better than "shut-up-and-calculate" b/c they cannot define what a 'measurement' is in terms of the mathematical formulation, and what distinguishes a measurement from an ordinary interaction - in terms of the mathematical formulation.

That is not an issue if one admits explicitly that this is "shut-up-and-calculate". But to me the term "interpretation" is misleading; introducing words w/o meaning and having an agnostic view on what's going on is not interpreting or explaining anything.

 

[vanhees71]

What a measurement is is defined by the apparati in the lab and not by mathematics!

 

[tom.stoer]

That's the minimal statistical interpretation, and practice in the lab uses just this. There's no need for additional assumptions like a collapse, many worlds, Bohm trajectories and what not the more philosophy oriented quantum physicists like to invent.

I hope that my example of the "existence of the bathroom while not being observed" makes it clear that this has not been invented by some physicists but is there - and has always been there. A question does not vanish by declaring its irrelevance; doing this does not answer the question but simply narrows down the (own) area of interest. Questions outside will persist, even when being ignored.

 

[tom.stoer]

What a measurement is is defined by the apparati in the lab and not by mathematics!

And why does the apparatus not work according to the Schrödinger equation when used in the context of a measurement? What is the relation of this equation to the measurement performed in the lab?

I am referring mainly to the so-called Maudlin trilemma:

1. The wavefunction specifies completely the physical properties of a physical system
2. The wavefunction evolves linearly according to Schrodinger's equation
3. Experiments have definite outcomes

My point is - as said in my last post - that sentences like yours do not answer anything in this context. They are nothing else but rephrasing "shut-up-and-calculate".

(excuse my, this us not an insult but just a quote; hope that you don't get it wrong)

 

[Simon Phoenix]

I can't say in which state something is after a measurement if I don't know the measurement apparatus

OK perhaps I should have made it clear. Supposing I perform a von Neumann measurement of type I on a system (if it is possible to do such a thing). Let's suppose we're measuring an observable ${\mathbf A}$ with eigenstates $| a_k \rangle$ and associated eigenvalues $a_k$. Then, from what I gather, you would disagree with the following statement :

If I obtain the eigenvalue $a_k$ as a result of my (ideal) measurement of ${\mathbf A}$ I must describe the state of the system after this measurement as $| a_k \rangle$

In what sense do you view this as 'wrong'? Doesn't it always allow us to correctly calculate the probabilities of subsequent measurement results?

Even the POVM formalism which allows one to calculate the density operator after a measurement (I hesitate to call this a 'state' even though I know many do) contains this as a special case.

I honestly don't know how you can view this process to be entirely described using quantities that evolve only unitarily. Of course we can construct FAPP models for this (decoherence) - is this what you had in mind?

by entangling position and spin component

I don't know what is meant by this - it would be like talking about entangling the energy and spin of a single electron. I have no idea what that means, or how to write down such a state. One can entangle 2 electrons, but I don't know what it means to entangle two properties of a single electron. Are you talking about something like taking the Hilbert space for an electron and splitting it into 2 subspaces and then considering states which cannot be written as product states from these subspaces to be entangled?

So you need an ensemble to verify that your preparation procedure leads to the state you claim to prepare

Of course. But, as far as I know, there isn't a single experiment that has falsified the 'projection postulate' - in other words the notion that, after one of these ideal measurements, we must describe the state of the system as being the eigenstate associated with the observed eigenvalue. I'm struggling to see in which sense this is actually 'wrong'. Of course there's the issue with spacelike measurements on entangled systems for which viewing the 'state' as some objective real thing in space time has one or two problems vis a vis the projection postulate - but using the 'updating of knowledge' kind of interpretations this isn't an issue.

 

[bhobba]

Any interpretation that does not or cannot define the terms used and cannot explain the relations between the mathematical formulation and the concept of "reality" addressed in the interpretation is in trouble.

Some do eg MW and CH and some leave it up in the air so to speak - eg Copenhagen. But Vanheees is correct - its like point and line in Euclidean Geometry - they are undefined in the theory but when you apply it its rather obvious. Is Euclidean Geometry up the creek as well ? (with all due respect to Hilbert and his axioms)

I don't like 'reality' in physics because there are so many different ideas about it eg read Penrose for a wacky way out view - with all due respect to his eminence and the fact I actually agreed with him once - what can I say - it was before I started posting here and realized the error of my ways.

Thanks
Bill

 

[tom.stoer]

But Vanheees is correct - its like point and line in Euclidean Geometry - they are undefined in the theory but when you apply it its rather obvious.

Whoops ... measurement in quantum mechanics is as obvious as line in geometry? That was not clear to me ...

I don't like 'reality' in physics

But physics is about reality.

A question does not vanish by declaring its irrelevance; doing this does not answer the question but simply narrows down the (own) area of interest. Questions outside will persist, even when being ignored.

I do neither claim that a minimalistic interpretation does not make sense from a pragmatic point of view, nor do I claim that everybody shall be interested in or work on these philosophical questions.

All I am saying is that some so-called interpretations are nothing else but "shut-up-and-calculate" in a new guise, and therefore do not answer questions that have been declared to be irrelevant when putting forward these interpretations.

 

[vanhees71]

The notion of "reality" is loaden with so much confusion by philosophers that it's pracatically useless to be discussed within the natural sciences. For a natural scientist reality is what can be objectively observed.

Why there should be a problem with the empirical fact that I find my bathroom in more or less the same state in the morning as I left it in the evening is not a problem of QT, but it is the only consistent description of this observed stability of matter, given the empirical fact that matter around us consists of atomic nuclei and electrons, bound to atoms, molecules and condensed-matter many-body state.

Further concerning the collapse: As I said, you cannot say, in which state the measured system is after the measurement without giving a sufficiently precise description of the measurement apparatus.

The Stern-Gerlach experiment is the paradigmatic example of a very good approximation for a von Neumann filter measurement, which is rather a preparation process than a measurement. You let a beam of uncharged particles (orginally silver atoms, nowadays with much more precision e.g., realized with neutrons) with magnetic moment run through an appropriately taylored inhomogeneous magnetic field (with a strong homogeneous component in a given direction, say the $z$ direction). As can be pretty easily shown by solving the time-dependent Schrödinger equation, the beam splits (for spin 1/2 particles) into two partial beams, where one beam consists to practically 100% of particles prepared in the spin-up state $|+1/2 \rangle$ and the other in the spin-down state $|-1/2 \rangle$, i.e., position and spin-$z$ component are (almost) perfectly entangled, and thus by blocking one beam (then the particles are absorbed in the material of the blocker and for sure not adequately described as a certain eigenstates of $\sigma_z$), you have left a beam of particles with (almost exactly) determined spin-$z$ component. In QT there's nothing needed to explain this preparation of $\sigma_z$ eigenstates than quantum-theoretical dynamics. There's no need for some mysterious "collapse" dynamics outside of QT, let alone the assumption of an instantaneous action in the entire space (which is, of course, a problem within relativistic physics and thus should not be used to base QT on).

 

[bhobba]

Whoops ... measurement in quantum mechanics is as obvious as line in geometry? That was not clear to me ...

What is a measurement is clear in practice like what a point and line is. Measurements that do not destroy what is being measured are simply a new preparation procedure. You can't always predict the result of a preparation procedure - my view is - big deal. We will have to disagree on this being an big issue - a discussion is likely not to really go anywhere.

Thanks
Bill

 

[PeroK]

But physics is about reality.

Physics tries to explain reality, usually in terms of mathematical models and the results of experiments. For example, you might model a baseball bat as a continuous mass distribution; whereas, "in reality" it is no such thing. We continue to model gravity as a force; whereas, "in reality" it is not a force. And so on.

That's where I agree with the sentiment that I think @bhobba was expressing. Physics is, in a sense, a mapping of "reality" into a scheme of models and experiments. I don't believe we have anything in physics that is reality in an absolute sense.

 

[RockyMarciano]

- its like point and line in Euclidean Geometry - they are undefined in the theory but when you apply it its rather obvious. Is Euclidean Geometry up the creek as well ? (with all due respect to Hilbert and his axioms)

Euclidean geometry is perhaps the wrong name to use here(it would be more appropriate maybe just geometry or differential geometry) as non-euclidean geomtries are actually what's at the root of relativity and QFT. But if something is undefined, instead of ignoring it, wouldn't redefining it in more clear or physical terms work better?

I don't like 'reality' in physics because there are so many different ideas about it.

Thanks
Bill

I think here the word "reality" is actually a substitute for "measurement", which is the true "elephant in the room" of modern science. But again, instead of ignoring it or despising it "because there are so many different ideas about it", why not try to define it mor erigorously in physical and mathematical terms. Just saying that this is unnecessary(even if the critique is precisely that there are many ideas and confusion about it, which should prompt everyone to correct this state of affairs) when it is so obviously it is not, and insisting that everybody doing measurements knows very well what is doing FAPP(even if no one can define it), and that SUAC is the only possible answer only leads to more confusion and disparate pseudo-interpretations(as Tom stoer correctly points out).

 

[Grinkle]

is doing FAPP

What is FAPP?

edit:

Never mind - For All Practical Purposes

 

[mikeyork]

The notion that Schrodinger's eqn tells us the time evolution of an isolated quantum system is based on the idea that a quantum state depends on some external variable called time. However, in reality, our concept of time comes from observing change -- i.e. quantum transitions -- and what we know is that the transition between observations is governed by a unitary transformation. What we see as a "direction" in time is the result of probability rules increasing disorder in the universe -- it's easier to break an egg than to put it back together. Thus time is a measure of that increasing disorder.

To then describe the unitary transformation as $e^{-i\hat{H}t/\hbar}$ where $\hat{H}$ is Hermitian then has a certain circularity and should be seen as a definition of $\hat{H}$. Modeling $\hat{H}$ is then a purely phenomenological process, however successful it is in predicting transitions.

By contrast, S-matrix theory says let's model the unitary operator in terms of its elements ( time-independent scattering amplitudes) directly.

 

[PeroK]

What we see as a "direction" in time is the result of probability rules increasing disorder in the universe -- it's easier to break an egg than to put it back together. Thus time is a measure of that increasing disorder.

It's strange, therefore, that the egg forms from relative disorder in the first place!

And, how is the motion of the Earth not a measure of time? Where is the increasing disorder there?

 

[PeroK]

It's strange, therefore, that the egg forms from relative disorder in the first place!

And, how is the motion of the Earth not a measure of time? Where is the increasing disorder there?

In fact, none of the clocks I can think of measure increasing disorder: sundial, hourglass, pendulum, quartz crystal, caesium atom.

 

[mikeyork]

The motion of the Earth and all the other clocks you mention are measuring change. Anything that changes can be a clock and its interpretation as time is merely a matter of calibration.

The creation of the egg is a low probability event and does not happen in isolation (second law of thermodynamics) but requires energy/interaction with other parts of the universe (just like a refrigerator).

 

[bhobba]

In fact, none of the clocks I can think of measure increasing disorder: sundial, hourglass, pendulum, quartz crystal, caesium atom.

Feynman wrote an interesting essay on that - if I recall its in the Character of Physical Law - the chapter on the distinction of past as future. Again, if I recall correctly, it would seem it's subtly involved with any way to measure time.

But really a new thread is required for it - its way off topic.

Thanks
Bill

 

[Blue Scallop]

About how to understand unitarity in QM.

They say decoherence occurs while the system is still unitarity meaning prior to any outcome.

So if you can somehow interface with any system and no outcome chosen. Is it still unitarity?

 

[Simon Phoenix]

As I said, you cannot say, in which state the measured system is after the measurement without giving a sufficiently precise description of the measurement apparatus.

Hmm. That's a bit of a cop out. For any given observable a sufficiently ingenious experimenter may be able to devise several technically different ways of realizing one of these ideal measurements. Granted it might take a considerable degree of ingenuity. Is your position that QM can not answer the question at all? So, on the assumption that I can construct an experiment to perform one of these idealized measurements, what does QM say about the state after the measurement?

I disagree that one actually needs the specific experimental details to even discuss this. I think QM tells us very nicely what the state is given a particular result - on the assumption that we can actually do one of these kinds of experiments.

On the other hand, is it even a reasonable assumption that these kinds of measurements even exist? I'm not sure - it's certainly beyond my ingenuity to think of a really clean-cut example (I don't consider the Stern-Gerlach set-up to be such a 'clean-cut' example).

position and spin-z component are (almost) perfectly entangled

Ah OK, I see what you mean by this now - the word 'component' threw me a bit and I thought you were talking about entangling different properties of a single system, which I couldn't make sense of. The magnetic field is acting like a beamsplitter. There's an entanglement between the path and the spin - in the same way we have an entanglement of the spatial mode (path) and polarization in a polarizing beamsplitter - which isn't a measurement either but a process by which we can filter different polarization states.

The real measurement is done when there is a subsequent measurement of the particles in one of the beam arms. So in the Stern-Gerlach case the absorption is acting rather like the technique for the production of heralded photons using parametric downconversion.

In the photon polarizing BS case you'd have a mixed state along the lines of $| 0 \rangle \langle 0| + | 1 \rangle \langle 1|$ in each timeslot in one of the arms before making a measurement on the other arm (an absorptive photodetection). After this 'heralding' measurement you'd have pure states in each timeslot (either a polarized photon or the vacuum).

 

[Blue Scallop]

Hmm. That's a bit of a cop out. For any given observable a sufficiently ingenious experimenter may be able to devise several technically different ways of realizing one of these ideal measurements. Granted it might take a considerable degree of ingenuity. Is your position that QM can not answer the question at all? So, on the assumption that I can construct an experiment to perform one of these idealized measurements, what does QM say about the state after the measurement?

I disagree that one actually needs the specific experimental details to even discuss this. I think QM tells us very nicely what the state is given a particular result - on the assumption that we can actually do one of these kinds of experiments.

On the other hand, is it even a reasonable assumption that these kinds of measurements even exist? I'm not sure - it's certainly beyond my ingenuity to think of a really clean-cut example (I don't consider the Stern-Gerlach set-up to be such a 'clean-cut' example).
Ah OK, I see what you mean by this now - the word 'component' threw me a bit and I thought you were talking about entangling different properties of a single system, which I couldn't make sense of. The magnetic field is acting like a beamsplitter. There's an entanglement between the path and the spin - in the same way we have an entanglement of the spatial mode (path) and polarization in a polarizing beamsplitter - which isn't a measurement either but a process by which we can filter different polarization states.

The real measurement is done when there is a subsequent measurement of the particles in one of the beam arms. So in the Stern-Gerlach case the absorption is acting rather like the technique for the production of heralded photons using parametric downconversion.

In the photon polarizing BS case you'd have a mixed state along the lines of $| 0 \rangle \langle 0| + | 1 \rangle \langle 1|$ in each timeslot in one of the arms before making a measurement on the other arm (an absorptive photodetection). After this 'heralding' measurement you'd have pure states in each timeslot (either a polarized photon or the vacuum).

Simon Phoenix. As an expert in practical quantum mechanics and setups. Can you please help think of a practical experimental setup to test whether it is possible to "scan" a unitary evolution without measurements? It is commonly believed that you need measurements to even tell when a system is unitary or entangled.. I want to convince myself this is an ironclad rule. So maybe you can encode some data in a superposition in an experiment setup and without measurements.. one will test whether it can be decoded while in unitary evolution. Any idea how to do the simplest experimental setup for this test?

 

[Simon Phoenix]

As an expert in practical quantum mechanics and setups

Lol. Vanhees and Bill are much more expert than I - I have a rather 'old fashioned' view of QM and I struggle a bit with the more modern (and logically sound) approaches.

Can you please help think of a practical experimental setup to test whether it is possible to "scan" a unitary evolution without measurements? It is commonly believed that you need measurements to even tell when a system is unitary or entangled..

I'm sorry - I don't really understand what you're asking here. Is there another way of phrasing your question? By 'unitary' here, I think you mean separable. To extract any real information you need to do a measurement of some kind.

 

[Blue Scallop]

Lol. Vanhees and Bill are much more expert than I - I have a rather 'old fashioned' view of QM and I struggle a bit with the more modern (and logically sound) approaches.
I'm sorry - I don't really understand what you're asking here. Is there another way of phrasing your question? By 'unitary' here, I think you mean separable. To extract any real information you need to do a measurement of some kind.

I read this in a paper about quantum information:
"The crucial distinction between quantum and classical information appears when one attempts
to use these other states as alternatives for transmitting information. Suppose we attempt
to encode ten bits of information onto a spin- 1/2 particle, by preparing it so that its spin points in
one of 2^10 possible directions. Then we send the particle to you. Can you read out the ten bits?
Of course not. Quantum theory forbids any measurement to distinguish all 1024 possibilities."

How do you do this actual physical experimental setup where you can encode ten bits of information onto a spin 1/2 particle?