One answer to this question — probably a common one — is that entanglement was discovered by Einstein in the Einstein-Podolsky-Rosen paper of the 15th of May, 1935: `Can Quantum-Mechanical Description of Physical Reality be Considered Complete?’. It was discovered only to be instantly repudiated! The non-locality implicit in a change in a wave function at place A leading to a change in a wave function at place B seemed to Einstein to be inconsistent with relativity and the assumption that A was separable from B was one of his assumptions of reality.

That definition of separability was:

Separability: Whatever we regard as existing (real) should be localised in time and space (space-time). Or, more weakly, if two dynamical systems are space-like separated then each system can be characterised by its own properties, independently of the properties of the other system.

Since this separability was the denial of entanglement and so EPR discovered entanglement — or so the story might go — by denying its possibility.

However, even though Einstein et al were the first to publish on this the better answer as to who discovered entanglement was Schrödinger. Schrödinger published the paper `Discussion of probability relations between separated systems’ in the Proceedings of the Cambridge Philosophical Society for 1935, in fact in August of that year, only a few months after the Einstein et al paper, and he and Einstein had been in correspondence on this for several years, both with the idea that the measurement process suggested something deeply problematic about quantum theory. But Schrödinger is better at drawing out the consequences and indicating that it arose from the nature of the tensor product itself. What he says on this is far clearer than what is in the Einstein et al paper — and Einstein would probably have agreed, as he was known to have disliked how Rosen and Podolsky (mostly the latter) wrote up the idea after their talks together.

Whereas, by 1935 Einstein was well-settled in Princeton, Schrödinger was unsettled in Oxford. He had just had a baby daughter by his mistress, Hilde March, and he had just won the Nobel Prize (in 1933), but he had no permanent position and he was unhappy with the home he had been given by ICI who were paying his stipend — a stipend that he also didn’t feel was enough. Nor did Oxford take entirely to Schrödinger: Frederick Lindemann, the head of physics, strongly objected to Schrödinger’s ménage à trois and wanted to get rid of `this bounder’. And yet out of this chaos Schrödinger managed to write, in lucid prose, an account of the characteristic trait of quantum mechanics.

Here is Schrödinger’s conclusion:

When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or $\psi$-functions) have become entangled. To disentangle them we must gather further information by experiment, although we know as much as anyone could know about all that happened. Of either system, taken separately, all previous knowledge may be entirely lost, leaving us but one privilege: to restrict experiments to one only of the two systems. After reestablishing one representative by observation, the other one can be inferred simultaneously. In what follows the whole of this procedure will be called \textit{disentanglement}. Its sinister importance is due to its being involved in every measuring process and therefore forming the basis of the quantum theory of measurement, threatening us thereby with at least a regressus in infinitum, since it will be noticed that the procedure itself involves measurement.

We might note that Schrödinger distinguishes between the state and its representative; why does he do this? The reason is that the state is not Lorentz invariant, so it changes as we change Lorentz frames. So Schrödinger is careful not to speak of the state, as though it were something absolute — which would be as bad a mistake as speaking of the velocity of an object in the light of Galilean relativity. For Schrödinger the quantum state is something rather mysterious, and he thought that others were far too free with the notion, and too incurious as to what lay behind it. But though we cannot speak of the state we can speak of its \textit{representative} in a frame. It should be noted that this frame relativity does not mean that entanglement is also relative: it is frame invariant (because the fact that the state for the joint system is a pure state is an invariant).

The entanglement that we have here was called, by Schrödinger, in his German publications, Verschränkung — cross-linking.

He continues:

Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separated and therefore virtually capable of being “best possibly known”, i.e. of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself.

Attention has recently been called to the obvious but disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter’s mercy in spite of his having no access to it. — Schrödinger 1935, p. 556.

 

It might seem as though everyone would pay attention to this startling prediction. But as it happened, it was almost universally ignored. For decades it lay dormant.