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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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!