All components of a quantum system are always entangled. It's not some mysterious property that arises under unusual circumstances. In fact, I would argue that quantum entanglement is the defining feature of quantum mechanics that distinguishes it from classical mechanics.
However, the way that most articles (and even physicists) use the term "entanglement" is in a loose way. Within a good approximation, a multi-component quantum state can be represented as a tensor product of single component quantum states. When this isn't a good approximation anymore, you can say that the entanglement of the system is much more apparent.
To give an example of what I mean, there's a technique in quantum chemistry known as density functional theory (DFT), which is used to compute ground state energies of various molecules. For some molecules, it works pretty well. The benefit of DFT is that it is a fast calculation technique (well, as far as quantum chemistry goes), but it's speed comes at a price. Rather than using the Coulombic interaction of every pair of electrons to compute the energy of a molecule (or more technically, using every Coulombic interaction as terms in the Hamiltonian), a probability cloud of electron positions is computed instead, and this cloud is used as the energetic term.
This works really well for a lot of systems, but in some cases (superconductors, metals, solid-state physics, van der Waals interactions), this approximation falls apart because the effect of electron-electron correlation significantly affects the energy of the system. In this case, the full, non-separable wavefunction is required. The fact that this full wavefunction cannot easily be broken into an approximation of simpler, single electron wavefunctions means that the entanglement of the system is very apparent.
To calculate the correct energy of the electrons in a molecule it's necessary to use entanglement. But these molecular examples may be explained using some weird classical models with local hidden variables, and with electrons that interact with each other to conspire and provide the right result of the measurement. The quantum mechanical explanation is actually more simple, but not "intuitive".
Almost all the discussions about entanglement discuss the strange case of two entangled particles that are far away. The distance between the particles is only a trick to:
* be sure that one of them can't communicate to the other and tell the result of the measurement
* be sure that there is no a local hidden variable theory that explain the result
For someone who is more of a mathematician than physicist, the following article might be of interest in this context: http://math.ucr.edu/home/baez/quantum/node4.html
The gist is that the state space of a multicomponent quantum mechanical system (think electrons in a metal) is given by the tensor product of the single component state spaces. As might be familiar from linear algebra, not every element in the tensor product of two vector spaces is simply a tensor product of two vectors. That's really all there is to entanglement, but unfortunately you can't explain that to a lay person.
I would argue that using the term "entanglement" to refer to the situation where "the entanglement of the system is much more apparent" or "tensor products are not a good approximation anymore" is completely reasonable. In contrast, when you say that "all components of a quantum system [that appears in nature] are always entangled" then this is trivially true in a technical sense, but highly misleading. To produce and stabilize non-trivial amounts of entanglement in a way that it can be harnessed for quantum information processing is certainly an interesting and highly non-trivial task which people are spending lots of effort on.
I don't think it's trivially true in a technical sense at all. It certainly didn't have to be the case that all quantum systems exhibit entanglement. Physics could have been different, such that entanglement is a property only some systems can have. And that would lead to drastically different conclusions. In fact, treating entanglement as something special is, at least for me personally, the highly misleading concept.
When I was learning quantum mechanics, there were many concepts that confused and bothered me because people explained things in a loose and inaccurate way. Other incorrect statements that confused me for a long time include:
"Fermions cannot occupy the same quantum state. Bosons can."
"It's not possible to measure the position and momentum of a particle simultaneously."
And while not QM: "Mass can be converted to energy."
Essentially, schools teach the approximations first, which leads you to believe that reality works a certain way, and then you think of all sorts of situations where these rules lead to paradoxes and contradictions, and then you have to relearn everything the more accurate way, while trying to expunge the old, wrong stuff from your brain that likes to stick there.
In my undergrad, I thought of plenty of ways that "entanglement", based upon the way it was explained in class could be used to transfer information faster than the speed of light. The professor didn't have a rebuke against my argument (granted this was a chemistry professor, not a physics professor), and essentially it was because everyone was repeating wrong, catchy phrases to each other. I don't think "Entanglement is the fact that a multiparticle wavefunction cannot be decomposed" is a particularly complex idea to understand. Everything else kind of falls out of that statement.
In my opinion, the most accurate theory should be taught first, and then the less accurate approximations can be derived as limiting cases of the more accurate formulation. And if the math is too complex for the more accurate version, then clearly specify what fallacies or assumptions are being taken. One of my statistical mechanics books does this really, really well (the one by McQuarrie). Disclaimers are all throughout his book about how the equations apply to limiting cases and approximations, what those approximations are, and how using those approximations causes the result to differ from a more accurate theory.
I guess I was being generous when interpreting your statement that "all components of a quantum system are always entangled". For pure states, such a statement is trivially true in the sense any kind of interaction in the Hamiltonian will generically create entanglement.
For mixed states, your statement is of course false. It is unfortunately often the case in the real world that two subsystems are in a non-entangled quantum state.
Regarding your general theme: I understand that confusion can arise if seemingly informal language is taken verbatim as a formal statement. I do not think that the solution is to abolish the former, which can be extremely efficient to reason in and communicate with (you gave some examples in your post), but rather to educate on the interpretation. It is unfortunate that this is not always done, as your experiences suggest.
You have good points. However I'm going to be a little nitpicky here. Mixed states aren't exactly "real" in the sense that what is a pure state for Alice can be a mixed state for Bob (I can provide an example if you don't know what I mean). So in that case the "lack" of entanglement is simply due to ignorance of which pure state the system is actually in.
The point is that entanglement always refers to an a priori choice of subsystems (say, Alice and Bob). This is the part that makes the phenomenon non-trivial. If there are other systems around (say, Eve the environment) then the joint state of Alice and Bob will be usually be mixed as a consequence of for the "trivial" reason that we discussed in the previous posts (to adapt a famous saying, almost all components of a quantum system are always mixed ;-). There is nothing "unreal" about mixed states, and not all mixed states lack entanglement. However, for mixed states, being entangled is no longer the generic behavior. The unavoidable interactions with the environment are the reason why it is hard to maintain entanglement between subsystems.
To say that we should do better and bring the environment back into the picture is missing the point if we are interested in the correlations between Alice and Bob. These do exclusively depend on their joint state (mixed or not).
(Sorry for continuing the rather long discussion, but I'd like to achieve some resolution here.)
By "real", I mean that the concept of a density matrix is derived completely on top of the postulates of quantum mechanics in combination with the Born rule. There's nothing fundamental about mixed states; you can also have classical mixed states (e.g. deriving thermodynamics from classical statistical mechanics). It's essentially just taking the postulates for pure states and applying a layer of statistics on top of it. I believe it was von Neumann that originally did this? In other words, the density matrix formulation does not add any additional predictive capability to physics that the original QM from the 1920s did not already provide. It's just a more convenient tool for connecting QM to experimentally realizable systems. Do you disagree with this?
When you lose entanglement due to decoherence (specifically, the off-diagonal terms of the density matrix approaching zero), these correlations are lost because you're essentially performing a measurement. But they still exist in the whole Alice + Bob + you and your measurement device system! But then, this starts treading into the discussion of the whole unsolved measurement problem which I kind of wanted to lurk around, since no one ever gets anywhere with those discussions.
Indeed your last point ("To say that we should do better and bring the environment back into the picture is missing the point") is essentially the whole picture I'm focusing on. Perhaps my background with quantum chemistry has slanted the way I explain things on here, because you're never collapsing these systems when you perform simulations of them to calculate their properties.
The density matrix is as real as the wave function when it comes to describing the corresponding subsystem. In the situation I was sketching, there is no measurement, no collapse, and the "measurement problem" does not play a role. Here is a concrete example: Suppose that you have three spins that are in a superposition of |000> and |111>. Alice has one the spins, Bob the other spin, and the third one belongs to the environment. The reduced state of any two of the three spins is NOT entangled. Therefore, Alice and Bob which will not be able violate any Bell inequality, win a CHSH game, distill Bell pairs, etc. if they only control two of the three spins. It is irrelevant that Alice is entangled with the joint system of Bob and Eve.
Again, the basic point is that the notion of entanglement refers to a choice of subsystems. Your statement that "All components of a quantum system are always entangled" is either trivializing the discussion or demonstrably false.
The correct statement for the first one is that the wavefunction for a system of fermions must be antisymmetric under the exchange of any two particles (the wavefunction flips sign). For a system of bosons, it stays the same.
The correct statement for the third one is that energy is not a substance; it's a number that is conserved as a result of the invariance of physics under time translations (see Noether's theorem). Thus, mass has an energy associated with it (see stress-energy tensor), and a particle that has mass can be converted into other particles that do not have mass, but energy is just a property that remains constant (except under GR, but then that's a whole other can of worms).
DFT is a method. It has been used by physicists for the last 40 years to model the solid state. What you put forward in your third paragraph is not quite right - things are more subtle.
What you seem to be suggesting is that we should be free of the Kohn-Sham framework[1,2] or find the exact XC functional. Not surprisingly, the results of calculations depend on how the functionals were constructed. But that doesn't mean "DFT" is bad. DFT's just the method. There is no approximation in DFT. It is in itself exact for the description of the ground state.
Yeah, I know. My rant further down the page on inaccuracies in teaching is coming back to haunt me now. Ironic, isn't it? I was glossing over specifics and loosely conflating concepts to form a point.
> All components of a quantum system are always entangled.
No, that's not true. If it were then there could never be direct observation of quantum interference because entangled particles do not self-interfere.
It is true that entanglement is very common, and an unentangled state is the unusual case. But it's not true that unentangled states (or, to be precise, unentangled degrees of freedom) don't exist.
I think either you're confused on an issue or I'm confused on what you're trying to say. Quantum interference is observable with even a single particle.
> because entangled particles do not self-interfere
Entangled particles certainly can, and do interfere (as mentioned, electrons in an atom are entangled and definitely interfere; there's a whole field devoted to calculating this effect), so I'm not sure exactly what you mean here. Can you give me, in ket notation, an example of an unentangled state? (And if you're referring to a direct tensor product of two single particle states, I'm still considering that an entangled state since it can unitarily evolve out of that configuration into one that can't be written as a tensor product).
> (And if you're referring to a direct tensor product of two single particle states, I'm still considering that an entangled state since it can unitarily evolve out of that configuration into one that can't be written as a tensor product)
This reasoning does not make sense. The notion of entanglement is not invariant under "global" operations, nor should it be.
> Quantum interference is observable with even a single particle.
Yes, but only if that particle is in an unentangled state. If that particle is entangled then the only way to observer the interference is to make measurements on both the particle whose interference you wish to observe and the particle with which it is entangled.
This is simply the old canard about measurement "destroying" interference (because measurement and entanglement are the same thing).
> Entangled particles certainly can, and do interfere
That's true, but the way in which they interfere is different, and the procedure you need to perform to observe the interference is likewise different.
> Can you give me, in ket notation, an example of an unentangled state?
|U> + |D>
Or if you want two unentangled particles:
(|U1> + |D1>)(|U2> + |D2>)
as opposed to the entangled state:
|U1>|D2> + |D1>|U2>
(Where U and D are intended to designate something like spin Up or spin Down, but that doesn't really matter.)
> if you're referring to a direct tensor product of two single particle states, I'm still considering that an entangled state since it can unitarily evolve out of that configuration into one that can't be written as a tensor product
This debate has really been bothering me and making me anxious the last couple of days. I haven't been able to sleep. I need to know something, are you self-taught in quantum mechanics or do you have a PhD in theoretical physics?
I'm not doubting your argument, (for all I know you're a well-known physics professor), I would just really like an answer to this question, as I think it would put my mind at ease.
Self-taught. Why does that matter? (I do have a Ph.D. but it's in CS.)
P.S. It turns out you were right about this:
"it can unitarily evolve out of that configuration into one that can't be written as a tensor product"
That turns out to be true (and it retrospect it's obviously true because otherwise it would not be possible to create entangled pairs.) But you are wrong in your claim that therefore the antecedent state is also entangled. A state that can be factored is not entangled by definition.
It doesn't. All that matters is that the science is correct (see Ron Maimon for instance; self-taught but far more knowledgable than most physicists). However, I get in far too many discussions involving QM online (mostly Reddit) where I learn that I wasted my time arguing with a 14 year old that watched "Through the wormhole" or something.
> P.S. It turns out you were right about this
You sound surprised. My graduate research is in quantum chemistry (and related areas), so I should have at least a few concepts correct. As someone with more of a QC background, perhaps my definition of what theoretical physicists consider "entangled" is inaccurate, so I am reading up on the matter. But I'm finding their specification of "entangled" to be an unintuitive and unobvious choice of wording.
You really shouldn't be nonplussed about it (which, I'm not sure if you intended it this way, but calling it a "basic" concept seems a little passive aggressive. I could equally say that someone self-taught in QM should know the "basic" fact that product states would unitarily evolve out of that configuration). It just depends on what area you focus on and what your background is. Scott Aaronson says on his blog that he has explained Bell's Theorem to QFT professors that had never heard of it. Just wasn't relevant to their work. Does that make them bad physicists for not knowing such a well-known part of QM?
Heck, it's better than a professor I had that did DFT work who thought the single-particle wavefunction was the most fundamental aspect of quantum mechanics. Nevermind the fact that DFT is trashed for its lack of ability to account for electron correlation.
(And I do mean trashed. I was at the Gordon Research Conference in quantum chemistry this past summer where Peter Gill gave a lecture titled "the obituary of DFT". And if anyone has a right to rag on it, it's certainly him.)
> I could equally say that someone self-taught in QM should know the "basic" fact that product states would unitarily evolve out of that configuration.
Well, that would depend on how far my studies have gotten me, no? My general assumption is that anyone who actually does QM for a living knows at least as much as I do, and almost certainly a great deal more. (Case in point: I presume that QFT is quantum field theory, but what is DFT?) I prefer to view this as humility rather than passive aggression. In any case, I don't think either one of us should be losing any sleep over it :-)
But can we agree that there are states that can be factored, and states that can't, and that this is a useful distinction to make? And that this distinction is physical, i.e. that it produces observable effects? (And not just observable, but interesting and useful?)
> But can we agree that there are states that can be factored, and states that can't, and that this is a useful distinction to make? And that this distinction is physical, i.e. that it produces observable effects? (And not just observable, but interesting and useful?)
If you say so. "All components of a quantum system are always entangled" sounds like a contrary claim to me, but I don't want to quibble over it.
It does make an interesting puzzle, though, how it can be physically possible to prepare a state that is both unentangled and pure. I don't actually know the answer to that off the top of my head.
However, the way that most articles (and even physicists) use the term "entanglement" is in a loose way. Within a good approximation, a multi-component quantum state can be represented as a tensor product of single component quantum states. When this isn't a good approximation anymore, you can say that the entanglement of the system is much more apparent.
To give an example of what I mean, there's a technique in quantum chemistry known as density functional theory (DFT), which is used to compute ground state energies of various molecules. For some molecules, it works pretty well. The benefit of DFT is that it is a fast calculation technique (well, as far as quantum chemistry goes), but it's speed comes at a price. Rather than using the Coulombic interaction of every pair of electrons to compute the energy of a molecule (or more technically, using every Coulombic interaction as terms in the Hamiltonian), a probability cloud of electron positions is computed instead, and this cloud is used as the energetic term.
This works really well for a lot of systems, but in some cases (superconductors, metals, solid-state physics, van der Waals interactions), this approximation falls apart because the effect of electron-electron correlation significantly affects the energy of the system. In this case, the full, non-separable wavefunction is required. The fact that this full wavefunction cannot easily be broken into an approximation of simpler, single electron wavefunctions means that the entanglement of the system is very apparent.