Existence of Pure Quantum States

[msumm21]

Are systems ever in a pure quantum mechanical state? If they are, is it possible to know the precise pure QM state? The example I am thinking of is the spin of an electron. If we measure the spin about the "z-axis" and find the result to be "up" then we say the electron is in the pure state with spin up about the z-axis. However, given that the measurement device itself is described by QM I'm thinking we don't know exactly how it is oriented (or maybe more correctly it doesn't even have an exact orientation), all we really know is that it's measuring about an axis very close to the z-axis, with some statistical spread, right? So the resulting state of the electron is like a mixed state, except due to a fundamental/unavoidable uncertainty? I haven't thought this through yet in other situations. Any references that explain this further would be much appreciated.

 

[Fredrik]

Sounds about right. Pure states obviously exist in the theory, but they are idealizations of complete knowledge about the system, and for practical reasons, our knowledge is never complete.

 

[msumm21]

If this is true, we cannot claim it is impossible to measure spin down about the z-axis, and if that is true I'm wondering if something like the PBR theorem is not valid. The preliminary, simplified justification of the theorem they provide in the paper on arXiv (http://lanl.arxiv.org/abs/1111.3328) seems like it is not valid in this case. However, I don't yet understand the subsequent full proof they give which is evidentially more robust and may protect against this, not sure yet.

 

[Fredrik]

I'm not getting into a discussion about PBR again, but in general, practical problems are never relevant to something that deserves to be called a "theorem".

 

[atyy]

Maybe church of the larger Hilbert space? http://www.dagstuhl.de/Materials/Files/09/09311/09311.BennettCharles.Slides.pdf (p44)

 

[msumm21]

I'm not getting into a discussion about PBR again, but in general, practical problems are never relevant to something that deserves to be called a "theorem".

I was thinking this was not just a practical problem, but a fundamental limitation akin to and following from the uncertainty principle, in which case I think proofs that will fail when this assumption is removed are not as useful/meaningful. I'm not certain the full version of the PBR argument really needs that assumption, but I'm suspicious about that since the simplified (first) argument seems like it must fail when this is taken into account.

 

[Jano L.]

However, given that the measurement device itself is described by QM I'm thinking we don't know exactly how it is oriented (or maybe more correctly it doesn't even have an exact orientation), all we really know is that it's measuring about an axis very close to the z-axis, with some statistical spread, right?

It is not a good idea to describe the magnet quantum mechanically. The very idea of measuring spins requires we can set up and read the orientation of the magnet, and this is how experimenters studying spins use quantum formalism.

 

[Fredrik]

I was thinking this was not just a practical problem, but a fundamental limitation akin to and following from the uncertainty principle, in which case I think proofs that will fail when this assumption is removed are not as useful/meaningful. I'm not certain the full version of the PBR argument really needs that assumption, but I'm suspicious about that since the simplified (first) argument seems like it must fail when this is taken into account.

It's a practical problem that doesn't even have a lot to do with quantum mechanics. There are mixed states in classical mechanics too. If you flip a coin to determine how to prepare the system before the experiment, and you tell your experimentalist friend everything except the result of the flip, then he should consider the system as being in a mixed state (probability 1/2 of pure state A and probability 1/2 of pure state B) before the measurement. If we use a simultaneous measurement of position and momentum to maximize our knowledge about the system before the "actual" measurement, then the fact that measurements aren't perfect even in a classical world means that we won't be able to single out a pure state with certainty. There will always be a set of pure states with non-zero probabilities, and this means that the system can be described as being in a mixed state.

The reason why we still think of classical systems as always being in a pure state is just that this is a simpler interpretation of the mathematics. Our intuition about the real world is telling us that the universe is always in a pure state, and that we only need mixed states when we don't know what pure state we should be using. So we think of pure states as describing what the universe is actually doing, and mixed states as describing our knowledge of the system. But experiments do not really favor this interpretation over one that says that the mixed states describe what the universe is really doing. We reject this idea only because it's complicated and can't really be understood intuitively.

It is not a good idea to describe the magnet quantum mechanically. The very idea of measuring spins requires we can set up and read the orientation of the magnet, and this is how experimenters studying spins use quantum formalism.

It's sufficient that the indicator component of the measuring device behaves in a way that for all practical purposes is indistinguishable from classical. We don't have to (and shouldn't) assume that measuring devices are perfectly classical.

 

[VantagePoint72]

I was thinking this was not just a practical problem, but a fundamental limitation akin to and following from the uncertainty principle

The point is that mixed states due to experimental uncertainty are not the same as the Heisenberg uncertainty principle. The former is a limitation in practice; the latter is a limitation in principle. Maybe you're conflating the HUP with the observer effect?

 

[msumm21]

Thanks for the responses.

It's a practical problem that doesn't even have a lot to do with quantum mechanics.

Just to make sure we're talking about the same thing, by saying it seems fundamental and not like a "practical problem" I meant to say that it doesn't seem like a limitation due to current technology, limited observer knowledge, or something like that, it seems like an unavoidable truth that you can't put an electron's spin in a pure state, regardless of what technology/tools you have (given that your technology/tools themselves follow QM). Is this this same definition you were thinking of? If so, and if you still think it is a practical problem then do you know how the spin can be prepared in a pure state?

LastOneStanding: the reason I think it follows from HUP is because the "preparation devices" (the magnets in this case) obey HUP.

 

[vanhees71]

To prepare a quantum state is of course an art of its own, but it's not impossible as some suggest here. The most famous example, treated in every QM1 lecture is the Stern-Gerlach experiment to prepare spin states of silver atoms (performed in Frankfurt 1923). There with an in in principle arbitrarily high precision you can prepare pure spin-up and spin-down states (silver atoms have spin 1/2) are prepared.

Of course, nowadays these experiments got much refined and much better in the state representation. For neutrons you have a phantastic precision. For photons there is a whole industry. The quantum opticians prepare entangled two-photon states on demand and thus also single-photon states by absorbing one of the photons in the pair. The recent physics Nobel prize was given for work dealing with similar preparations in "cavity QED". There atomic Rydberg states were prepared with high precision, and also the quantum mechanical "state reduction", i.e., the transition from a mixed state to a pure state due to interactions with the "preparation apparatus" has been demonstrated, etc. etc.

So, nowadays, there are plenty examples for pure-state preparations!

 

[Fredrik]

Just to make sure we're talking about the same thing, by saying it seems fundamental and not like a "practical problem" I meant to say that it doesn't seem like a limitation due to current technology, limited observer knowledge, or something like that, it seems like an unavoidable truth that you can't put an electron's spin in a pure state, regardless of what technology/tools you have (given that your technology/tools themselves follow QM). Is this this same definition you were thinking of?

What I mean when I say that it's a practical problem is that even if QM is a perfect representation of reality, an actual state preparation procedure (and by that I mean an action performed in the real world) will never put the spin exactly in a pure state.

You could blame the uncertainty relations for this, but what I have in mind is a simpler idea: Every time we invent a new and improved measuring device or state preparation device, it will inherit some degree of imperfection from the devices involved in its construction and calibration. So even in a classical world, there wouldn't be any perfect state preparation devices.

This problem can be considered even more fundamental than the uncertainty relations, since it's not derived from any specific theory.

LastOneStanding: the reason I think it follows from HUP is because the "preparation devices" (the magnets in this case) obey HUP.

What I'm saying is that we would have essentially the same problem even without the uncertainty relations.

 

[Fredrik]

To prepare a quantum state is of course an art of its own, but it's not impossible as some suggest here. The most famous example, treated in every QM1 lecture is the Stern-Gerlach experiment to prepare spin states of silver atoms (performed in Frankfurt 1923). There with an in in principle arbitrarily high precision you can prepare pure spin-up and spin-down states (silver atoms have spin 1/2) are prepared.

I thought about Stern-Gerlach devices before I posted in this thread, and I came to the conclusion that even though a Stern-Gerlach device oriented in (for example) the z direction would give us a pure state, this doesn't mean that we can prepare pure states. The problem is that there's no way to ensure that the device is oriented in exactly the right direction.

If we assume that the device is oriented in some direction, and that we just don't know exactly what that direction is, we could perhaps say that the spin is really in a pure state, and that the only reason to use a mixed state is our lack of knowledge of which pure state the spin is in.

This assumption would however be naive. The magnet that we're trying to orient in the desired direction is "for all practical purposes classical", but not exactly classical. So it's not going to have an exactly well-defined orientation, only an approximate one.

The conclusion that the spin would "really" be in a pure state if the device had an exact orientation is somewhat naive too, because QM doesn't make claims like "pure states describe the properties of the system, and mixed states describe our knowledge of the system". This is the sort of claim that's made by interpretations of QM, not by QM itself.

 

[msumm21]

vanhees71: are you saying there is some trick with Stern-Gerlach and other preparation tools that can avoid the uncertainty in the preparation device from "bleeding over" into the prepared system? Stern-Gerlach & spin is the example I gave in the original post, but what I was saying is that, regardless of how well you make it, it seems like it should always be subject to QM uncertainty itself, and hence the axis it measures "spin up" about is not perfectly known, so a pure quantum state of the prepared system is not achieved. Right?