Types of entanglement not due to conservation principles

The notion that entanglement is due to conservation laws is an unfortunate misconception caused by a popular way to explain the concept at the general-audience level.

That argument goes something like the following:

You take two particles, which are known to have total spin $S_z=0$, and then you separate them. Then if you measure particle A and it has spin $S_z=+1/2$, then you know that particle B must have spin $S_z=-1/2$, and the particles are entangled.

As general-audience presentations go, it's not that bad, particularly if it is embedded in a larger piece that must juggle other concepts as well and cannot devote that much space to a detailed explanation of entanglement. There's a few things to say about this argument:

• The good: this argument does represent (though not in enough detail to fully specify it) one possible way to get entanglement. More specifically, if you know that particles A and B share a spin singlet state $|\psi⟩ = \tfrac{1}{\sqrt{2}}\big(|{↑↓}⟩-|{↓↑}⟩\big)$, and you spatially separate them, then yes, you do get an entangled pair.

• The bad:

  • That said, however, the real argument relies on the spin-singlet state, and not on its $S_z=0$ aspect. In particular, you could have an identical argument if you started in the spin triplet state, $|\psi_+⟩ = \tfrac{1}{\sqrt{2}}\big(|{↑↓}⟩+|{↓↑}⟩\big)$, which has an opposite sign in the superposition, but it is absolutely critical that you know what that phase is. If all you have is a box that produces $|\psi⟩$ and $|\psi_+⟩$ with 50% probability but without telling you which, then $S_z=0$ is still true, but you have completely destroyed the entanglement.
  • Moreover, that argument ignores the fact that there are perfectly valid pure states, such as the bare $|↑↓⟩=|↑⟩\otimes|↓⟩$, that are consistent with the $S_z=0$ property but don't have any entanglment at all.

  • Even worse, basing the argument on the conservation law sets the reader up for one of the biggest misconceptions of all when it comes to quantum entanglement $-$ that entanglement is like JS Bell's description of Bertlmann's socks, or, in other words, that entanglement can be explained by a hidden-variable theory, where each particle has a well-determined spin before we look, and the observation merely reveals that value. This is provably wrong! We know from Bell's theorem that such a description of entangled states (generally known as 'local and realist') is incompatible with the predictions of quantum mechanics, and we know from experiments that nature follows the QM predictions and it is not bound by the constraints imposed by local realism.

  • In addition to that, the presentation structure of that argument is of the form

    what is entanglement? well, here is one way to produce entangled states

    and, if not handled properly, leads the reader directly into a faulty generalization. No matter how solidly the case for the method has been established, the argument says nothing about whether there are other ways to produce entangled states.

And, indeed, there are such other methods. More to the point:

Entanglement is generic

Any time you have two quantum systems A and B interacting with a nontrivial hamiltonian $\hat H_\mathrm{AB}$, the generic outcome is that they will come out entangled with each other, i.e. something of the form $$ U(|\psi\rangle\otimes|\varphi\rangle)\longrightarrow |\Psi\rangle. $$ (Contrary to what was stated in the comments, this does not need to involve any conservation laws at all.) Entanglement is simply a product of interactions, and it does not require particularly symmetric conditions, or configurations that are amenable to a simple conservation-law analysis, to appear.

On the other hand, there is an important difference to be drawn between controlled entanglement, i.e. the entangled states which are technologically useful and experimentally verifiable, and entangled states where we don't have enough of a controllable handle on the state to make it do anything useful. When we don't have such a handle, entanglement morphs into the other side of the coin - decoherence, which is nothing less than an uncontrolled entanglement with parts of the environment that we cannot address. As the top answer to the linked question makes clear, this type of entanglement is something we'd very happily have less of, in a ton of different circumstances.

Controlled entanglement, on the other hand, is relatively fragile, because there are all sorts of factors that can mess it up (such as dephasing on the superpositions degrading the quality, or entanglement with other degrees of freedom that we don't want to include). This is why (controlled) entanglement is often considered a valuable resource - but if you drop the qualifiers, it's not that hard to get.