I When does quantum entanglement occur?

[Spathi]

TL;DR Summary

The question arises whether it is possible to consider molecules of water in a glass as entangled: if one knows the momenta of all molecules except one, he can recalculate the momentum of this individual molecule.

If I understand the idea of EPR correctly, the quantum entanglement occurs between two particles for which the total momentum is known (therefore, knowing the momentum of one particle, one can recalculate the other, and this contradicts the uncertainty principle). Then the question arises whether it is possible to consider, on the same basis, molecules of water in a glass as entangled: if one knows the momenta of all molecules except one, he can recalculate the momentum of this individual molecule. I apologize if I ask naive questions, but I do not understand how to determine where quantum entanglement arises and where it does not.

 

[PeroK]

Perhaps it's not a good idea to try to learn QM from a paper (EPR) whose aim is to undermine it?

 

[PeterDonis]

If I understand the idea of EPR correctly, the quantum entanglement occurs between two particles for which the total momentum is known

This is just one particular case of entanglement that was used in the original EPR paper as an example (and one that is not easily realizable in an actual experiment--that is why later papers, such as those of Bell, used entanglement of spins instead).

In general, quantum systems become entangled when they interact; any kind of interaction will do, and any kind of quantum degrees of freedom can be involved.

the question arises whether it is possible to consider, on the same basis, molecules of water in a glass as entangled

Since the molecules in a glass of water are certainly interacting with each other, yes, they will be entangled. But not in a way that is very useful; see below.

if one knows the momenta of all molecules except one

Which is never going to actually happen, since that would require carefully preparing the entire glass of water in a very precisely controlled state for all the molecules, which is astronomically unachievable in practice. In an actual glass of water, while the molecules will be entangled, they won't be entangled in a way that fixes a value for any useful conserved quantity, which is what would be required for an EPR-type inference to be made.

I do not understand how to determine where quantum entanglement arises and where it does not.

See my statement about interaction above. That is the general rule.

 

[DrChinese]

If I understand the idea of EPR correctly, the quantum entanglement occurs between two particles for which the total momentum is known (therefore, knowing the momentum of one particle, one can recalculate the other, and this contradicts the uncertainty principle).

There is no violation of the uncertainty principle when the momentum of 2 entangled particles are measured precisely. The uncertainty principle applies primarily when an attempt is made to deduce conjugate observables, such as momentum and position, on a SINGLE particle with high precision (i.e. low standard deviation). Further:

a) You can know the total momentum of 2 particles precisely, and that does NOT mean they are entangled.

b) They ARE entangled if you know the total momentum of 2 particles precisely, but do not know their individual momenta. Usually you might start with situation a), and then allow the 2 particles to interact so that their individual momenta change.

 

[vanhees71]

Perhaps it's not a good idea to try to learn QM from a paper (EPR) whose aim is to undermine it?

One should learn from sources, which are understandable. The EPR paper is far from that!

If you want to learn from the original papers, I'd recommend to read Dirac and Pauli rather than anything close to the Copenhagen gang (particularly Bohr and Heisenberg; an exception are the early papers written by Born, Jordan and by Born, Jordan, and Heisenberg). Very good are also the early textbooks by Dirac and Pauli (on wave mechanics).

I'd, however, also not recommend this as a starting, because more modern textbooks are usually better to read, and if it comes to entanglement, Bell, and all that of course you can find this stuff only in more modern textbooks. My favorites are

J. J. Sakurai, Modern Quantum Mechanics (the revised edition co-authored by Tuan; the more recent editions co-authored by Napolitano are almost the same but include a new chapter on "relativistic wave mechanics", which usually does more harm than good, because relativistic QT is more easy to understand using relativistic QFT to formulated).

L. Ballentine, Quantum mechanics (uses the rigged-Hilbert space approach on a level suited for physicists and the minimal statistical interpretation).

 

[vanhees71]

There is no violation of the uncertainty principle when the momentum of 2 entangled particles are measured precisely. The uncertainty principle applies primarily when an attempt is made to deduce conjugate observables, such as momentum and position, on a SINGLE particle with high precision (i.e. low standard deviation). Further:

a) You can know the total momentum of 2 particles precisely, and that does NOT mean they are entangled.

b) They ARE entangled if you know the total momentum of 2 particles precisely, but do not know their individual momenta. Usually you might start with situation a), and then allow the 2 particles to interact so that their individual momenta change.

Indeed, in the original EPR example of a decaying particle the two resulting asymptotic free particles are represented by a wave function, where both the total momentum is quite well defined (in the center-of-mass frame it's 0) and the relative position is quite well determined to be at a distance $L$ large compared to the range of the interaction between the particles. The wave function is thus described by something like
$$\psi(\vec{x}_1,\vec{x}_2)=\phi_{\epsilon}(\vec{x}_1-\vec{x}_2-\vec{L}) \int_{\mathbb{R}^3} \mathrm{d}^3 P \tilde{\psi}_{\epsilon'} (\vec{P}) \exp[\mathrm{i} \vec{P} \cdot \vec{R}/hbar]$$
with $\vec{R}=(m_1 \vec{x}_1+m_2 \vec{x}_2)/(m_1+m_2)$.

Here $\phi_{\epsilon}$ is a position wave function sharply peaked with some small width $\epsilon$ around 0, i.e., $\vec{r}=\vec{x}_1-\vec{x}_2$ is very likely around $\vec{L}$, and $\tilde{\psi}_{\epsilon'}$ is a momentum-space wave function sharply peaked around 0 too, such that the total momentum is very likely around 0. Of course the center-center-of-mass position $\vec{R}$ is very indetermined (uncertainty relation between $\vec{P}$ and $\vec{R}$ components: $\Delta R_j \Delta P_j \geq \hbar/2$) and the momentum of the relative motion $\mu \dot{\vec{r}}$ is also very indetermined. That $\vec{P}$ and $\vec{r}$ are both very well determined is no contradiction to the uncertainty relation of course, because $\hat{\vec{P}}=\hat{\vec{p}}_1 + \hat{\vec{p}}_2$ and $\hat{\vec{r}}=\hat{\vec{x}_1}-\hat{\vec{x}}_2$ compute, i.e., are compatible observables.

The entanglement is such that although each $\vec{x}_1$ and $\vec{x}_2$ (the positions of each of the particles) is also pretty indetermined, but if you measure the position of one particle pretty precisely, you also know the position of the other particle pretty precisely due to the fact that $\vec{r}$ is known pretty precisely in the EPR state. Then of course the momenta of either particle are still pretty indetermined.

The same holds for the individual particle momenta, which are also both pretty indetermined, but when measuring one of the particles momenta very accurately you also know the other particle's momentum pretty accurately to be just the opposite momentum, because the total momentum is prepared to be well determined in this EPR state. Then of course also the position of either particle is still pretty indetermined.

There is no paradox, as soon as you accept the statistical meaning of the quantum state. The choice of which single-particle observable you measure accurately decides which observable of the other particle is then also well determined (if you measure momentum of particle 1 accurately, you know the momentum of particle 2 also accurately, but neither particle's positions and vice versa). The correlations in case of the one or the other measurement are due to the preparation in this entangled state and there's no need of an action at a distance influencing one particle by measuring the other at a far distant place.