Nature wrote an article with the list of top 15 "sleeping beauty" papers that were initially almost ignored but many decades later, they exploded and began to attract lots of followups.
Almost all of them are about the physics of surfaces and closely related issues in solid state physics. One exception, ranking as the #14 sleeping beauty, is the Einstein-Podolsky-Rosen 1935 paper
This "sleeping beauty" woke up in 1994, i.e. 59 years after it was written. Obviously, some people would like to say that people were shallow, uncritical, and uncurious for those 59 years before some clever brave folks began to appreciate the wisdom in the paper. As you know, my summary is the opposite one. The physicists have been competent and understood that the EPR arguments weren't really right for those 59 years before the scientific community started to be flooded by folks who understood quantum mechanics at least as incorrectly as Einstein and his collaborators (without having found any results that could compare to Einstein's).
The initial letters of the authors – EPR – are currently used as a de facto synonym of "quantum entanglement" even though this paper (and Einstein's phrase "spooky action at a distance") is an alternative, fringe attack against the quantum entanglement. Einstein's and his collaborators' formulations are fresh and clear – unlike their followers' – but the paper is deeply flawed.
So let's begin with the paper in Physical Review. It starts with a question in the title:
The abstract starts as follows:
Note that EPR's definition of "reality", when translated to the quantum mechanical language, actually depends on the state of the system (it must be an "eigenstate" for "reality" to hold). And the state of the physical system depends on the previous measurements. So measurable quantities aren't ever "universally real" if we agree both with EPR's definition of reality and the quantum mechanical predictions. The same tension will be described in various ways below.
The key EPR claim is that if (1) is false, then (2) is false as well. However, this claim is incorrect, as we will discuss later.
The first paragraph reads:
EPR effectively want to place "something" or "some ideas" above all theories. That's the self-evident purpose of the otherwise useless "distinction". What they want to place above all theories is clearly the basic framework of classical physics. But this framework has simply turned out to be wrong. Their attempt to "distinguish" this "objective reality" from the theory is nothing else than the desire to protect these ideas from falsification, and that's an utterly unscientific approach.
Particular features of reality, such as the question whether it exists objectively and before the measurement, are not independent of the theory. Different theories have different answers to these general questions. In this sense, one cannot "distinguish" or "separate" reality from the theories. And it just turns out that the general views about reality advocated by EPR – the basic framework of classical physics – is a wrong, falsified assumption about Nature.
They repeat the comment in the abstract that "reality" is something that may be predicted with certainty and without disturbing the system. As I said, this set turns out to be basically empty, quantum mechanics implies. While EPR admit that "elements of reality must be found by experiments", not by philosophical dogmas, they were clearly assuming that the character of reality and the template for the predictions should be determined by philosophical dogmas, not by science.
A particular definition of a set ("reality") invented by a human may have any number of elements. For their definition of "reality", the number is zero. That means nothing wrong about quantum mechanics.
A paragraph is ending on the 2nd page with a weird claim:
So EPR's ideas about reality clearly contradict the quantum-mechanical ones.
(A measurement only "mechanically disturbs" the observables that don't commute with the measured one. But thanks to the generic entanglement or correlations between all observables, as incorporated in the state vector, a measurement generally changes the state vector describing – our knowledge of – the mutually commuting observables, too. Whether one uses the verb "disturb" for the latter change of the wave function may be up to taste but it's important that it's no "mechanical disturbance" that could be used to transfer real information at a distance.)
They make this conflict "disappear" by the funny insertion that they want to regard their definition of reality (which is incompatible with quantum mechanics as it stands) as a sufficient condition for reality. This is exactly just like saying that "regarded as a sufficient, not necessary condition for the emergence of diverse animal species, the creation of animals by God in 7 days is in agreement both with the Bible and with Darwin's theory of evolution". LOL.
Yes, no. What are you saying here, Albi? If you allow the creation to be just "a sufficient condition", i.e. if it can be omitted, then it may be in agreement with anything. But in that form, the statement is 100% vacuous and any interpretation that makes it even infinitesimally non-vacuous is 100% wrong. Creation of species is simply not in agreement with Darwin's theory and the classical notion of reality is analogously not in agreement with quantum mechanics.
On the rest of Page 2, EPR describe the textbook situation of one particle with the operators \(x,p\) that don't commute. They don't have any simultaneous eigenstates which means, in EPR's jargon, that \(x\) and \(p\) cannot be simultaneously predictable. Only probabilities may be calculated. In particular, a well-defined \(p\) wave function – a plane wave – has the same probability for all values of \(x\). All this basic review seems totally OK.
They also correctly say that if the predicted probabilities are strictly between 0% and 100%, the measurement determines the result but also unavoidably distorts the physical system. Using their jargon (that they say to be common at that time), "when the momentum of a particle is known, its coordinate has no physical reality". They state and justify the two possibilities (1) and (2) in the abstract.
At the end of Page 2, they correctly say that quantum mechanicians normally assume that the description is complete. And they propose a contradiction between non(1) and (2) in the second half of the paper.
The EPR contradiction
They study the combination of subsystems I and II that interact for \(0\leq t \leq T\). At the final moment, the interaction stops and the combined system has evolved into a generic entangled state they write as\[
\Psi(x_1,x_2) = \sum_{n=1}^\infty \psi_n(x_2) u_n(x_1)
\] If you measure the observable \(A\) of subsystem I whose eigenstates are mutually orthogonal (mutually exclusive) wave functions \(u_n(x_1)\) and the measured eigenstate is \(u_k\), then you know that the second system is in (or "has collapsed into") the corresponding \(\psi_k\). This decomposition of the two-system pure state according to the eigenvalue of \(A\) may always be written down – you must just appreciate that if the decomposition is done in this way, the wave functions \(\psi_n(x_2)\) for the second subsystem are neither mutually orthogonal nor having the same norm in general (but they will happen to be orthonormal in the EPR example).
If you measure a different, noncommuting quantity \(B\) on the subsystem I, you need to use its different eigenfunctions \(v_n(x_1)\). There will be a different decomposition of \(\Psi\) with \(v,\phi\) replacing \(u,\psi\).
Now, you probably know what their claimed contradiction is. If we measure \(A\) on subsystem I, then we collapse the subsystem II into one of the states \(\psi_n\) while if we measure \(B\) on subsystem I, we collapse the subsystem II into one of the states \(\phi_n\) which are different than \(\psi_n\).
So the wave function after the measurement of subsystem I and before the measurement of subsystem II depends on what observable, \(A\) or \(B\) or another one, we measured on the subsystem I. The measurement of subsystem I couldn't have influenced the subsystem II so EPR decide that "it is thus possible to assign two different wave functions to the same reality". The meaning of "reality" is always ambiguous but I would say that this conclusion is correct. The reality of the subsystem II hasn't changed by the measurement done on the subsystem I, so it's the "same reality", according to my common understanding of the words "same reality". And yes, the observer who measured subsystem I has assigned different wave functions, either \(\psi_m\) or \(\phi_n\), to the subsystem II after the measurement.
But I add, there is absolutely no contradiction here because the values of the wave function are not "reality" by themselves. In particular, the wave function for a particular value of \(x_2\) cannot be measured in one particular physical situation, not even in principle; mathematically speaking, the wave function is not an observable because it is not a Hermitian operator on the Hilbert space (it is an element of this space). The wave function is the quantum raw material to compute all the probabilistic distributions describing results of measurements that actually can be made – measurements of observables i.e. linear Hermitian operators. It encodes all the observer's knowledge about both physical systems. It "collapses" during the measurement because the observer has learned something from the measurement.
The subsystem II "collapses" either to \(\psi_m\) or \(\phi_n\) if the observer I measures the operator \(A\) or \(B\) simply because from this \(A\) or \(B\) measurement, he has learned different things about the subsystem I i.e. about the whole system I+II which means, thanks to the correlations that developed for \(0\leq t\leq T\) between I and II, that the observer I has learned different things about the subsystem II, too.
The measurement almost always influences the measured system; it is "creating" the new reality. And that's the reason behind the measurement-on-I-dependent possible post-measurement state vectors \(\psi_m\) or \(\phi_n\) for the subsystem II, too. While all sorts of correlations between the two subsystems exist thanks to the interactions for \(0\leq t \leq T\), if we study the results of measurements on the subsystem II only, we may easily prove that those (probabilities of all outcomes) are calculable from the (reduced) density matrix for the subsystem II and nothing else, and therefore completely independent of all events and decisions done in the region I. That's why there's physically no action at a distance, no physical influence of the region I on region II.
These days, people like to use finite Hilbert spaces and "qubits" as their simple examples of entanglement and pretend that EPR have almost pioneered the research of "discrete quantum information". This is not true at all. They always prefer to talk about the infinite-dimensional Hilbert spaces similar to the Hilbert space of a particle on a line (including operators with continuous spectra). If someone should be praised as an early developer of the "discrete quantum information", it's probably Wolfgang Pauli.
And the difference between the two decompositions is what they discuss on Page 4, too. They describe an entangled state of 2 particles on a line. In the \(x\)-\(x\) representation, their entangled wave function is basically \(\delta(x_1-x_2)\) which means that the two particles are sitting at an unknown point, all points are equally likely, but we know that the position of both particles is exactly the same.
(They don't care about the normalization factor of the wave function, and neither will I. It may be made finite if the space is compactified or replaced by a box.)
It is straightforward to doubly Fourier transform this two-particle wave function to the momentum, \(p\)-\(p\) representation. The wave function on the two-particle Hilbert space is "similar" to the identity operator. But it's not really an operator – the other Hilbert space isn't complex conjugated. If it were an identity operator, the double Fourier transform would be the identity operator \(\delta(p_1-p_2)\) again. But thanks to the (missing) complex conjugation, the wave function of the entangled state is \(\delta(p_1+p_2)\), with the relative plus sign.
It implies that the momentum of each particle is totally uncertain, all values are equally likely, but we know that \(p_1=-p_2\). This is of course nothing else than the conservation of the total momentum which is obeyed because the two-particle wave function in the \(x\)-\(x\) language may be seen to be translationally invariant (under translations of both particles' positions by the same distance).
So much like in the finite-dimensional Hilbert spaces of photons' or electrons' polarizations that became popular later, EPR conclude that \(x_1\) and \(x_2\) are perfectly correlated if you decide to measure both positions, much like \(p_1\) and \(p_2\) if you happen to measure the momenta.
OK. How do they complete their wrong proof of the contradiction between the complementarity and completeness? It's simple.
As I say, they say that an observable is an "element of reality" if it can be predicted with certainty. Once you measure \(x_1\), the other particle collapses to the corresponding eigenstate of \(x_2\) with \(x_2=x_1\), so \(x_2\) is seen to be an "element of reality". But if you decide to measure \(p_1\) instead, the second particle collapses to a momentum eigenstate with \(p_2=-p_1\) so the prediction of \(p_2\) becomes "certain" so \(p_2\) is an "element of reality".
EPR combine these two possible measurements to conclude that both \(x_2\) and \(p_2\) must be elements of reality because there exist situations and measurements – that only differ by the action on the first particle which can no longer affect the second particle – in which \(x_2\) is known with certainty before it's measured; or \(p_2\) is known with certainty before it's measured.
Their interpretation is that \(x_2\) and \(p_2\) have "simultaneous reality", after all.
This interpretation is clearly flawed: they miss the whole point of complementarity. The procedures by which they decided that \(x_2\) was real or \(p_2\) was real couldn't have been carried out "simultaneously", in the same universe. Indeed, they describe two possible histories of the Universe – two histories that differ by decisions what property of the first particle is measured. These two histories or measurements of the first particle can't be done or exist "simultaneously" which is why the final interpretation of the procedures, in terms of the "reality" of \(x_2\) and/or \(p_2\), can't be said to be "simultaneous", either.
You may also say that their mistake is a widespread (among "interpreters") confusion of the words "and" and "or". By considering two different histories – with different measurements of the first particle – they could have determined that "either" \(x_2\) "or" \(p_2\) was real. But they could not correctly determine that "both" \(x_2\) "and" \(p_2\) were real "simultaneously" because their two thought experiments couldn't have taken place simultaneously, either.
It is actually the same confusion as the widespread interpretation of the superposition states as "dead and alive cat". In reality, the cat is still "dead or alive", just like you would expect. If you just replace the cat by the physicist who measures the first particle, this analogy may be seen to be an isomorphism. The observer's brain may be in a superposition of states "I want to measure the position" and "I want to measure the momentum". But the superposition means that one or the other takes places. It does not mean that they occur simultaneously! As any pair of two orthogonal states, they are mutually exclusive.
Unlike most of their followers, EPR were sort of aware of this obvious criticism. In the last two paragraphs, they admit that one may object because their reality "was not sufficiently restrictive". Instead, the critic of EPR could say that \(x_2\) and \(p_2\) may only be said to be "simultaneously real" if they can actually be "simultaneously measured" which is not the case thanks to the nonzero commutator. If one is more careful, EPR correctly articulate the EPR critics' comments, the reality of observables generally depends on the measurements that are being made on the first system.
After they described this correct resolution almost flawlessly, they wrote:
It's funny because the 100% unjustified and self-evidently incorrect assertion "no reasonable definition of reality could be expected to permit this" is the central point that decides about the validity or, in this case, invalidity of the whole paper. This is the point of the paper saying "here a miracle occurs". Quantum mechanics changes our notions of reality in such a way that exactly the "forbidden" insight is true and fundamental in the whole theory: the reality always depends on the observables we can make, and realities of noncommuting observables are always mutually exclusive.
Bohr has called it the complementarity principle. The sentence about "permissions" is 100% equivalent to the statement "no reasonable physics ideas could permit Bohr's complementarity principle". Bohr has tried to explain those things to Einstein since 1927. The debates took many years. The men must have enjoyed each other, it must have been fun for them.
But when it comes to learning or settling questions in physics, just imagine how complete waste of time the 8 years of the "Bohr-Einstein debates" must have been if at the end, Einstein and his collaborators dismiss Bohr's complementarity with one sentence "no reasonable definition of reality admits Bohr's celebrated principle" and they don't even find it appropriate to elaborate on this point – a point that is nothing else than the straight denial of one of the most important findings of 20th century physics. I have only spent less than 2 hours by debunking this EPR paper – Bohr has spent a portion of 8 years. ;-)
It was pretty bad but when the post-Einstein anti-quantum zeal became a mass movement a few decades ago, it got much worse...
In 1935, most papers – like the EPR paper – didn't contain any list of references at the end (at most a few references written as footnotes at particular pages). These days, such an omission could have make it much harder for the readers to follow the work. But I actually have doubts whether the incorporation of the mandatory lists of references were such a great advance. Before that, the paper probably were more self-sufficient.
Off-topic link: Unusually enough, Sean Carroll wrote a sensible text, one against warp drives and EM drives
Almost all of them are about the physics of surfaces and closely related issues in solid state physics. One exception, ranking as the #14 sleeping beauty, is the Einstein-Podolsky-Rosen 1935 paper
Can quantum mechanical description of physical reality be considered complete? (4 pages, full text)that I will critically review below. The main author came to the U.S. 2 years earlier, he was probably the primary source of the "ideas", and the two collaborators were postdocs at the Institute for Advanced Studies in Princeton (this description of affiliations sounds just like today!).
This "sleeping beauty" woke up in 1994, i.e. 59 years after it was written. Obviously, some people would like to say that people were shallow, uncritical, and uncurious for those 59 years before some clever brave folks began to appreciate the wisdom in the paper. As you know, my summary is the opposite one. The physicists have been competent and understood that the EPR arguments weren't really right for those 59 years before the scientific community started to be flooded by folks who understood quantum mechanics at least as incorrectly as Einstein and his collaborators (without having found any results that could compare to Einstein's).
The initial letters of the authors – EPR – are currently used as a de facto synonym of "quantum entanglement" even though this paper (and Einstein's phrase "spooky action at a distance") is an alternative, fringe attack against the quantum entanglement. Einstein's and his collaborators' formulations are fresh and clear – unlike their followers' – but the paper is deeply flawed.
So let's begin with the paper in Physical Review. It starts with a question in the title:
Can quantum mechanical description of physical reality be considered complete?Yes, you bet, Albert. It not only "can" be considered complete. It may be shown to be complete and for this reason, it actually "is" considered complete. By this statement, I mean that there can't exist any valid yet inequivalent, more complete "completion". The wave function – a pure state, a ket vector – describes a maximum information we may have about a given state; we may be sure that the system is in a pure state by measuring a maximal set of commuting observables. If the information is non-maximal, we describe the system with the density matrix. But the laws involving the pure states or density matrices are so closely related that they should be considered the same laws and these laws are complete.
The abstract starts as follows:
In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system.In quantum mechanics, there's also an element corresponding to each element of reality. But the set is empty if we adopt EPR's definition of reality. Indeed, the theory teaches us that in a generic state, almost no physical quantity may be predicted with certainty and almost no physical quantity may be determined without disturbing the system. Indeed, these two conditions happen to be equivalent. We may predict operators \(L\) with certainty if the state vector is an eigenstate of \(L\), and in that case, the measurement of \(L\) doesn't modify the state at all. But it's very rare for the state vector to be an eigenstate of a random operator so almost all the things that are physically meaningful – that may be measured – are not reality according to EPR's definition.
Note that EPR's definition of "reality", when translated to the quantum mechanical language, actually depends on the state of the system (it must be an "eigenstate" for "reality" to hold). And the state of the physical system depends on the previous measurements. So measurable quantities aren't ever "universally real" if we agree both with EPR's definition of reality and the quantum mechanical predictions. The same tension will be described in various ways below.
In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.The proposition (1) is false because the quantum mechanical description is complete. But (2) is true: non-commuting quantities cannot have simultaneous reality (as defined by EPR above). If \([L,M]\neq 0\), then the state vector can't be an eigenstate of both \(L\) and \(M\) – well, unless it's an eigenstate of the operator \([L,M]\) with the zero eigenvalue, but that's even rarer and not "generic".
The key EPR claim is that if (1) is false, then (2) is false as well. However, this claim is incorrect, as we will discuss later.
The first paragraph reads:
Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves.This is a highly problematic assertion because it implicitly involves the assumption that the "foundations" of physics cannot change – they must be those of classical physics. One may perhaps distinguish objective reality from the theories or pictures describing it but one must always be aware of the fact that our knowledge or opinions about the reality is always only through some theories or pictures, and those may be right or wrong.
EPR effectively want to place "something" or "some ideas" above all theories. That's the self-evident purpose of the otherwise useless "distinction". What they want to place above all theories is clearly the basic framework of classical physics. But this framework has simply turned out to be wrong. Their attempt to "distinguish" this "objective reality" from the theory is nothing else than the desire to protect these ideas from falsification, and that's an utterly unscientific approach.
Particular features of reality, such as the question whether it exists objectively and before the measurement, are not independent of the theory. Different theories have different answers to these general questions. In this sense, one cannot "distinguish" or "separate" reality from the theories. And it just turns out that the general views about reality advocated by EPR – the basic framework of classical physics – is a wrong, falsified assumption about Nature.
In attempting to judge the succes [sic] of a physical theory, we may ask ourselves two questions: (1) "Is the theory correct?" and (2) "Is the description given by the theory complete?"EPR admit that quantum mechanics is "correct" because it agrees with the observations but want to argue that the quantum mechanical description is "incomplete". They declare the main necessary condition for a completeness:
every element of the physical reality must have a counterpart in the physical theory.Great. Literally, this is clearly true in quantum mechanics. All elements of the reality may be said to be measurements or their results. And each of them has a counterpart in the physical theory, a Hermitian linear operator on the Hilbert space. An unstated problem for EPR's views is that quantum mechanics has different rules how to use these operators to make predictions and what the character of the predictions is – and what are the assumptions for the predictions to be physically meaningful.
They repeat the comment in the abstract that "reality" is something that may be predicted with certainty and without disturbing the system. As I said, this set turns out to be basically empty, quantum mechanics implies. While EPR admit that "elements of reality must be found by experiments", not by philosophical dogmas, they were clearly assuming that the character of reality and the template for the predictions should be determined by philosophical dogmas, not by science.
A particular definition of a set ("reality") invented by a human may have any number of elements. For their definition of "reality", the number is zero. That means nothing wrong about quantum mechanics.
A paragraph is ending on the 2nd page with a weird claim:
Regarded not as a necessary, but merely as a sufficient, condition of reality, this criterion is in agreement with classical as well as quantum-mechanical ideas of reality.What? This comment makes it rather clear that in 1935, Einstein was already completely decoupled from top physics research. Three years earlier, in 1932, Heisenberg gave a Nobel lecture where he clearly explained why the exact opposite is true. Reality cannot be determined without distorting the system. Measurements always involve a modification of the system and they cannot be determined with certainty.
So EPR's ideas about reality clearly contradict the quantum-mechanical ones.
(A measurement only "mechanically disturbs" the observables that don't commute with the measured one. But thanks to the generic entanglement or correlations between all observables, as incorporated in the state vector, a measurement generally changes the state vector describing – our knowledge of – the mutually commuting observables, too. Whether one uses the verb "disturb" for the latter change of the wave function may be up to taste but it's important that it's no "mechanical disturbance" that could be used to transfer real information at a distance.)
They make this conflict "disappear" by the funny insertion that they want to regard their definition of reality (which is incompatible with quantum mechanics as it stands) as a sufficient condition for reality. This is exactly just like saying that "regarded as a sufficient, not necessary condition for the emergence of diverse animal species, the creation of animals by God in 7 days is in agreement both with the Bible and with Darwin's theory of evolution". LOL.
Yes, no. What are you saying here, Albi? If you allow the creation to be just "a sufficient condition", i.e. if it can be omitted, then it may be in agreement with anything. But in that form, the statement is 100% vacuous and any interpretation that makes it even infinitesimally non-vacuous is 100% wrong. Creation of species is simply not in agreement with Darwin's theory and the classical notion of reality is analogously not in agreement with quantum mechanics.
On the rest of Page 2, EPR describe the textbook situation of one particle with the operators \(x,p\) that don't commute. They don't have any simultaneous eigenstates which means, in EPR's jargon, that \(x\) and \(p\) cannot be simultaneously predictable. Only probabilities may be calculated. In particular, a well-defined \(p\) wave function – a plane wave – has the same probability for all values of \(x\). All this basic review seems totally OK.
They also correctly say that if the predicted probabilities are strictly between 0% and 100%, the measurement determines the result but also unavoidably distorts the physical system. Using their jargon (that they say to be common at that time), "when the momentum of a particle is known, its coordinate has no physical reality". They state and justify the two possibilities (1) and (2) in the abstract.
At the end of Page 2, they correctly say that quantum mechanicians normally assume that the description is complete. And they propose a contradiction between non(1) and (2) in the second half of the paper.
The EPR contradiction
They study the combination of subsystems I and II that interact for \(0\leq t \leq T\). At the final moment, the interaction stops and the combined system has evolved into a generic entangled state they write as\[
\Psi(x_1,x_2) = \sum_{n=1}^\infty \psi_n(x_2) u_n(x_1)
\] If you measure the observable \(A\) of subsystem I whose eigenstates are mutually orthogonal (mutually exclusive) wave functions \(u_n(x_1)\) and the measured eigenstate is \(u_k\), then you know that the second system is in (or "has collapsed into") the corresponding \(\psi_k\). This decomposition of the two-system pure state according to the eigenvalue of \(A\) may always be written down – you must just appreciate that if the decomposition is done in this way, the wave functions \(\psi_n(x_2)\) for the second subsystem are neither mutually orthogonal nor having the same norm in general (but they will happen to be orthonormal in the EPR example).
If you measure a different, noncommuting quantity \(B\) on the subsystem I, you need to use its different eigenfunctions \(v_n(x_1)\). There will be a different decomposition of \(\Psi\) with \(v,\phi\) replacing \(u,\psi\).
Now, you probably know what their claimed contradiction is. If we measure \(A\) on subsystem I, then we collapse the subsystem II into one of the states \(\psi_n\) while if we measure \(B\) on subsystem I, we collapse the subsystem II into one of the states \(\phi_n\) which are different than \(\psi_n\).
So the wave function after the measurement of subsystem I and before the measurement of subsystem II depends on what observable, \(A\) or \(B\) or another one, we measured on the subsystem I. The measurement of subsystem I couldn't have influenced the subsystem II so EPR decide that "it is thus possible to assign two different wave functions to the same reality". The meaning of "reality" is always ambiguous but I would say that this conclusion is correct. The reality of the subsystem II hasn't changed by the measurement done on the subsystem I, so it's the "same reality", according to my common understanding of the words "same reality". And yes, the observer who measured subsystem I has assigned different wave functions, either \(\psi_m\) or \(\phi_n\), to the subsystem II after the measurement.
But I add, there is absolutely no contradiction here because the values of the wave function are not "reality" by themselves. In particular, the wave function for a particular value of \(x_2\) cannot be measured in one particular physical situation, not even in principle; mathematically speaking, the wave function is not an observable because it is not a Hermitian operator on the Hilbert space (it is an element of this space). The wave function is the quantum raw material to compute all the probabilistic distributions describing results of measurements that actually can be made – measurements of observables i.e. linear Hermitian operators. It encodes all the observer's knowledge about both physical systems. It "collapses" during the measurement because the observer has learned something from the measurement.
The subsystem II "collapses" either to \(\psi_m\) or \(\phi_n\) if the observer I measures the operator \(A\) or \(B\) simply because from this \(A\) or \(B\) measurement, he has learned different things about the subsystem I i.e. about the whole system I+II which means, thanks to the correlations that developed for \(0\leq t\leq T\) between I and II, that the observer I has learned different things about the subsystem II, too.
The measurement almost always influences the measured system; it is "creating" the new reality. And that's the reason behind the measurement-on-I-dependent possible post-measurement state vectors \(\psi_m\) or \(\phi_n\) for the subsystem II, too. While all sorts of correlations between the two subsystems exist thanks to the interactions for \(0\leq t \leq T\), if we study the results of measurements on the subsystem II only, we may easily prove that those (probabilities of all outcomes) are calculable from the (reduced) density matrix for the subsystem II and nothing else, and therefore completely independent of all events and decisions done in the region I. That's why there's physically no action at a distance, no physical influence of the region I on region II.
These days, people like to use finite Hilbert spaces and "qubits" as their simple examples of entanglement and pretend that EPR have almost pioneered the research of "discrete quantum information". This is not true at all. They always prefer to talk about the infinite-dimensional Hilbert spaces similar to the Hilbert space of a particle on a line (including operators with continuous spectra). If someone should be praised as an early developer of the "discrete quantum information", it's probably Wolfgang Pauli.
And the difference between the two decompositions is what they discuss on Page 4, too. They describe an entangled state of 2 particles on a line. In the \(x\)-\(x\) representation, their entangled wave function is basically \(\delta(x_1-x_2)\) which means that the two particles are sitting at an unknown point, all points are equally likely, but we know that the position of both particles is exactly the same.
(They don't care about the normalization factor of the wave function, and neither will I. It may be made finite if the space is compactified or replaced by a box.)
It is straightforward to doubly Fourier transform this two-particle wave function to the momentum, \(p\)-\(p\) representation. The wave function on the two-particle Hilbert space is "similar" to the identity operator. But it's not really an operator – the other Hilbert space isn't complex conjugated. If it were an identity operator, the double Fourier transform would be the identity operator \(\delta(p_1-p_2)\) again. But thanks to the (missing) complex conjugation, the wave function of the entangled state is \(\delta(p_1+p_2)\), with the relative plus sign.
It implies that the momentum of each particle is totally uncertain, all values are equally likely, but we know that \(p_1=-p_2\). This is of course nothing else than the conservation of the total momentum which is obeyed because the two-particle wave function in the \(x\)-\(x\) language may be seen to be translationally invariant (under translations of both particles' positions by the same distance).
So much like in the finite-dimensional Hilbert spaces of photons' or electrons' polarizations that became popular later, EPR conclude that \(x_1\) and \(x_2\) are perfectly correlated if you decide to measure both positions, much like \(p_1\) and \(p_2\) if you happen to measure the momenta.
OK. How do they complete their wrong proof of the contradiction between the complementarity and completeness? It's simple.
As I say, they say that an observable is an "element of reality" if it can be predicted with certainty. Once you measure \(x_1\), the other particle collapses to the corresponding eigenstate of \(x_2\) with \(x_2=x_1\), so \(x_2\) is seen to be an "element of reality". But if you decide to measure \(p_1\) instead, the second particle collapses to a momentum eigenstate with \(p_2=-p_1\) so the prediction of \(p_2\) becomes "certain" so \(p_2\) is an "element of reality".
EPR combine these two possible measurements to conclude that both \(x_2\) and \(p_2\) must be elements of reality because there exist situations and measurements – that only differ by the action on the first particle which can no longer affect the second particle – in which \(x_2\) is known with certainty before it's measured; or \(p_2\) is known with certainty before it's measured.
Their interpretation is that \(x_2\) and \(p_2\) have "simultaneous reality", after all.
This interpretation is clearly flawed: they miss the whole point of complementarity. The procedures by which they decided that \(x_2\) was real or \(p_2\) was real couldn't have been carried out "simultaneously", in the same universe. Indeed, they describe two possible histories of the Universe – two histories that differ by decisions what property of the first particle is measured. These two histories or measurements of the first particle can't be done or exist "simultaneously" which is why the final interpretation of the procedures, in terms of the "reality" of \(x_2\) and/or \(p_2\), can't be said to be "simultaneous", either.
You may also say that their mistake is a widespread (among "interpreters") confusion of the words "and" and "or". By considering two different histories – with different measurements of the first particle – they could have determined that "either" \(x_2\) "or" \(p_2\) was real. But they could not correctly determine that "both" \(x_2\) "and" \(p_2\) were real "simultaneously" because their two thought experiments couldn't have taken place simultaneously, either.
It is actually the same confusion as the widespread interpretation of the superposition states as "dead and alive cat". In reality, the cat is still "dead or alive", just like you would expect. If you just replace the cat by the physicist who measures the first particle, this analogy may be seen to be an isomorphism. The observer's brain may be in a superposition of states "I want to measure the position" and "I want to measure the momentum". But the superposition means that one or the other takes places. It does not mean that they occur simultaneously! As any pair of two orthogonal states, they are mutually exclusive.
Unlike most of their followers, EPR were sort of aware of this obvious criticism. In the last two paragraphs, they admit that one may object because their reality "was not sufficiently restrictive". Instead, the critic of EPR could say that \(x_2\) and \(p_2\) may only be said to be "simultaneously real" if they can actually be "simultaneously measured" which is not the case thanks to the nonzero commutator. If one is more careful, EPR correctly articulate the EPR critics' comments, the reality of observables generally depends on the measurements that are being made on the first system.
After they described this correct resolution almost flawlessly, they wrote:
No reasonable definition of reality could be expected to permit this.LOL. They don't elaborate; the final 5-line paragraph only conveys EPR's belief that a "complete" (=classical) description of reality should exist.
It's funny because the 100% unjustified and self-evidently incorrect assertion "no reasonable definition of reality could be expected to permit this" is the central point that decides about the validity or, in this case, invalidity of the whole paper. This is the point of the paper saying "here a miracle occurs". Quantum mechanics changes our notions of reality in such a way that exactly the "forbidden" insight is true and fundamental in the whole theory: the reality always depends on the observables we can make, and realities of noncommuting observables are always mutually exclusive.
Bohr has called it the complementarity principle. The sentence about "permissions" is 100% equivalent to the statement "no reasonable physics ideas could permit Bohr's complementarity principle". Bohr has tried to explain those things to Einstein since 1927. The debates took many years. The men must have enjoyed each other, it must have been fun for them.
But when it comes to learning or settling questions in physics, just imagine how complete waste of time the 8 years of the "Bohr-Einstein debates" must have been if at the end, Einstein and his collaborators dismiss Bohr's complementarity with one sentence "no reasonable definition of reality admits Bohr's celebrated principle" and they don't even find it appropriate to elaborate on this point – a point that is nothing else than the straight denial of one of the most important findings of 20th century physics. I have only spent less than 2 hours by debunking this EPR paper – Bohr has spent a portion of 8 years. ;-)
It was pretty bad but when the post-Einstein anti-quantum zeal became a mass movement a few decades ago, it got much worse...
In 1935, most papers – like the EPR paper – didn't contain any list of references at the end (at most a few references written as footnotes at particular pages). These days, such an omission could have make it much harder for the readers to follow the work. But I actually have doubts whether the incorporation of the mandatory lists of references were such a great advance. Before that, the paper probably were more self-sufficient.
Off-topic link: Unusually enough, Sean Carroll wrote a sensible text, one against warp drives and EM drives
Reviewing EPR, a "sleeping beauty" paper
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Reviewed by DAL
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May 26, 2015
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