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In a comment on Mechanism of quantum entanglement; proof of quantumness, Emilio Pisanty stated,

There's many types of entanglement, and only a few can be traced to any sort of conservation law.

Replying to a request to point out some examples of types of entanglement, he suggested that the request probably deserves its own thread. So, my question:

What types of entanglement are not the result of conservation principles?

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  • $\begingroup$ Almost all multi-particle states are entangled (in the strong sense that the non-entangled states have positive codimension in the state space). $\endgroup$
    – WillO
    Jul 13, 2018 at 18:47
  • $\begingroup$ Understood-- in fact, it's probably correct to say that any set of particles that have interacted are entangled in one or more ways. But the question is about the role of conservation principles in entanglement: does any kind of entanglement not result from a conservation principle? $\endgroup$
    – S. McGrew
    Jul 13, 2018 at 19:04
  • $\begingroup$ This question is too broad. Basically you can derive all of physics from conservation laws. $\endgroup$ Jul 13, 2018 at 20:43
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    $\begingroup$ That's what I thought. But I'm sure @EmelioPisanty had a good reason for saying that only a few kinds of entanglement can be traced to conservation laws. Hoping he will answer! $\endgroup$
    – S. McGrew
    Jul 13, 2018 at 21:48
  • $\begingroup$ I agree with both statements by Norbert and @WillO , but I’d love to see an answer to this question either way! $\endgroup$ Jul 13, 2018 at 22:38

2 Answers 2

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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.

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  • $\begingroup$ Perfect answer! $\endgroup$
    – S. McGrew
    Aug 5, 2018 at 22:49
  • $\begingroup$ "And, indeed, there are such other methods" This answer would be vastly more helpful if just one such other concrete method were cited. Then when ever the conservation law needed argument comes up, we just say ... that. $\endgroup$ Feb 11, 2023 at 1:32
  • $\begingroup$ Hi Emilio -- long time, yes? In reading your answer, the point that surprised me most was blaming conservation laws for hidden variable thinking when, at least for me, it was always the opposite: Local quantum rules combined with conservation require entanglement. Born kind of did that by accident, not by forethought. Einstein was the one who pointed out to him that some form of superluminal concept was implied. Regarding Hamiltonians and interactions, I can't figure out how you think of those as separate from the conservation laws, e.g., of energy, that make those interactions happen. $\endgroup$ Feb 12, 2023 at 2:54
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If Entanglement is deemed as consequence of conservation laws, then we have a realist explanation of entanglement and it would not be mysterious any more. Conservation laws are valid independent of whether entanglement takes place or not.

However at least in one state of entanglement (Bell's state), perfect anti correlation (always opposite spin when measured along same axis), is presented as entanglement phenomena. This perfect anti correlation can be explained by the conservation laws as well. Therefore it is easy to get mixed up. I have seen documentaries by celebrity scientists and they do mention conservation laws while talking about entanglement.

General-audience only hears/grasps this particular state of entanglement.

This is the only (or one of very few, I am not a QM literate) state(s) where we can directly link entanglement with conservation laws at a single pair level. In other states, we can not verify conservation laws at the level of a single entangled pair. However, they can be verified at the level of averages by measuring numerous pairs.

Therefore, if the quantum predictions are met by a very large set of pairs, then that also verifies the conservation laws for that set as a whole.

It would be impossible to separate conservation from any phenomena, let alone entanglement.

We do not have means to measure/verify conservation in case of a single pair in many states of entanglement but that does not mean conservation laws are not obeyed. If conservation laws were not obeyed, quantum predictions at average level would not be experimentally proven.

Now the question - is entanglement due to conservation laws? I think there is a link.

Can we prove the link? - 1) For individual pair - Yes, at least for one state. 2) At statistical level - Yes, for all states.

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