The Action That Really Spooked Einstein

“The reason I cannot seriously believe in it is that the theory cannot be reconciled with the principle that physics has to describe a reality in space and time without spooky action at a distance.”

Albert Einstein, 1947 [1]

“For it seems hard to-imagine a complete separation, whilst the systems are still so close to each other, that, from the classical point of view, their interaction could still be described as an unretarded actio in distans. And ordinary quantum mechanics, on account of its thoroughly unrelativistic character, really only deals with the actio in distans case.”

Erwin Schrödinger, 1936 [2]

“Spooky Action at a Distance: Albert Einstein famously referred to entanglement as “spooky action at a distance” because of how one particle’s state seems to instantaneously affect the other, no matter the distance between them. This appears to violate classical physics’ principle that nothing can travel faster than light, but according to quantum mechanics, no information is actually being transmitted faster than light.”

Grok 2, 2024

As you can see from Grok’s quote, Einstein’s words “spooky action at a distance” are nowadays most commonly associated with the most general and simplified scenario of quantum entanglement. Grok’s definition is somewhat blurred but the purpose of this post is not to correct Grok’s statements but to clarify which action at a distance in particular spooked Einstein. The mathematical formalism of quantum mechanics, according to Erwin Schrödinger, dictated that the free will of an experimentator measuring one part of an entangled system could instantaneously affect the state of its another part no matter how far away from each other were those entangled parts. That scenario of quantum entanglement makes it really spooky.

In 1936 Schrödinger published in Oxford a paper Probability Relations Between Separated Systems, that followed his previous paper The Present Status of Quantum Mechanics from 1935 that became famous due to the cat thought experiment and in which he introduced the term quantum entanglement. In this follow on paper he expanded on just one scenario of entanglement that concerned him more than others because it included really spooky action at a distance. He summarised his findings as follows:

“An earlier paper dealt with the following fact. If for a system which consists of two entirely separated systems the representative (or wave function) is known, then current interpretation of quantum mechanics obliges us to admit not only that by suitable measurements, taken on one of the two parts only, the state (or representative or wave function) of the other part can be determined without interfering with it, but also that, in spite of this non-interference, the state arrived at depends quite decidedly on what measurements one chooses to take — not only on the results they yield. So the experimenter, even with this indirect method, which avoids touching the system itself, controls its future state in very much the same way as is well known in the case of a direct measurement. In this paper it will be shown that the control, with the indirect measurement, is in general not only as complete but even more complete. For it will be shown that in general a sophisticated experimenter can, by a suitable device which does not involve measuring non-commuting variables, produce a non-vanishing probability of driving the system into any state he chooses; whereas with the ordinary direct method at least the states orthogonal to the original one are excluded.

The statement is hardly more than a corollary to a theorem about “ mixtures “(+) for which I claim no priority but the permission of deducing it in the following section, for it is certainly not well known.

(+) The valuable conception of a mixture and the appropriate way of handling a mixture by the Statistical Operator is due to Johann von Neumann; see his Mathematische Grundlagen der Quantenmechanik, Berlin, Springer, 1932; especially pp. 225 ” [2]

In the earlier paper this scenario was described in § 11. The Lifting of Entanglement. The Result Depends on the Free Will of the Experimenter. I think it is important to quote that section in full:

“We return to the general case of entanglement, not just the special case of a measurement process. Suppose that the catalogues of expectations of two bodies A and B have been entangled as a result of interaction in the past. Then I can take one, say B, and complete my knowledge about it, which has become submaximal, by successive measurements. I claim that as soon as I have succeeded in doing this, and no sooner, the entanglement with A has been rescinded and, moreover, the measurements on B combined with the conditional statements allow us to obtain maximal knowledge of A as well.

To see this, note first of all that the knowledge about the total system will remain maximal, as it will not be spoiled by good and accurate measurements. Secondly, conditional statements of the form “when for A … then for B …” no longer exist as soon as we have a maximal catalogue for B. For such a catalogue is not conditional, and nothing concerning B can be added to it. Thirdly, conditional statements in the reverse direction (“when for B … then for A …”) can be converted into statements about A alone since all probabilities for B are already known unconditionally. The entanglement has therefore been completely removed. And since the knowledge about the total system has remained maximal, the only possibility is that the maximal catalogue for B is complemented by one for A. Nor can it happen that A has become maximal through measurements on B, while B is not yet maximal. Because then all the above reasoning can be reversed, and it follows that B is also maximal. Both systems will be maximal at the same time, as claimed. Notice, by the way, that this would also be true if the measurements are not restricted to one of the two systems. Most interesting is, however, that one can restrict it to one of the two; that this already has the desired effect.

Which measurements on B are used is entirely up to the experimenter. He does not need to use particular variables to be able to make use of the conditional statements. He can simply make a plan to arrive at a maximal knowledge of B, even when he knew nothing about B. It is also no harm if he carries out this plan to the end. To save himself superfluous work, he might consider if he has already achieved his aim, however.

Which A-catalogue is thus obtained depends of course on the measured values for B (before the entanglement has vanished completely; not on those of the superfluous measurements). Now suppose that in a certain case I have found a catalogue for A in this way. Then I can think and wonder if I might have found a different catalogue if I had used a different plan for measuring B. Since I did not touch A in the actual experiment nor in the imaginary experiment, the statements in the other catalogue must also be correct, whatever they are. They must also be entirely contained in the first, and vice versa. It must therefore be identical to the first.

Strangely enough, the mathematical structure of the theory does not satisfy this condition automatically. Even stronger, one can construct examples in which this condition is necessarily violated. It is true that in each experiment, only one set of measurements can actually be performed (on B) after which the entanglement has vanished and one does not learn any more about A with further measurements on B. But there are types of entanglement for which there exist two specific measurement programmes for B such that 1. the entanglement is lifted, 2. the one leads to an A catalogue to which the other cannot possibly lead, no matter what values the measurements result in. The fact is that the two classes of A catalogues that can turn up in the one respectively the other programme, are purely separated and cannot have any element in common.” [3]

Schrödinger’s quote below explains why I am, after him, referring to two entangled parts of one system rather than to two entangled systems:

“The strange theory of measurement, the apparent jumping of the ψ-function, and finally the counter intuitive entanglement, all arise from the simple way in which the calculational apparatus of quantum mechanics allows two separate systems to be mentally combined into one, and for which it seems to be predestined.” [3]

References:

1. Zeilinger, Anton, Light for the quantum. Entangled photons and their applications: a very personal perspective. Published 15 June 2017 • © 2017 The Royal Swedish Academy of Sciences
2. Schrödinger Erwin. Probability relations between separated systems. Mathematical Proceedings of the Cambridge Philosophical Society. 1936;32(3):446–452. doi:10.1017/S0305004100019137
3. Schrödinger, Erwin, The Present Status of Quantum Mechanics, Die Naturwissenschaften 1935. Volume 23, Issue 48.