On May 15, 1935, Albert Einstein co-authored a paper with his two postdoctoral research associates, Boris Podolsky and Nathan Rosen, at the Institute for Advanced Study. First published in the Physical Review, the article was entitled “Can Quantum Mechanical Description of Physical Reality Be Considered Complete?”, and generally referred to as “EPR” owing to the first initials of the authors’ last names, this paper quickly became a staple in debates, both current and old, over the correct interpretation of quantum theory. In fact, it is ranked among the top ten of all papers ever published in Physical Review journals, and EPR is still near the top of their list of most-cited articles due to its pivotal role in the development of quantum information theory. Within the paper itself and at the heart of the matter, two quantum systems are joined in such a way as to link both their spatial positions in a certain direction and also their linear momenta in their respective directions, even when the systems are nowhere near each other in space. As a result of this “entanglement”, determining either position or momentum for one system would fix, respectively, the position or the momentum of the other. On that basis, they argue that one cannot maintain both the accepted view of quantum mechanics and the completeness of the theory; in essence, only one of the two can be correct. This essay describes the central argument of that 1935 paper, explores its possible solutions, and probes the ongoing significance of the issues that the paper raises.By 1935, the conceptual understanding of the quantum theory was dominated by Niels Bohr’s ideas concerning complementarity as described by the Copenhagen Interpretation. Those ideas centered around the observations and measurements obtained within the quantum domain, as according to the theory, observing a quantum object involves an inherent physical interaction with a measuring device that affects both systems in an uncontrolled way. The best picture to think of would be a photon-observing apparatus trying to measure the position of an electron, where the photons inherently strike the electrons and move them some distance. The effect that this produces on the measuring instrument as the “result” can only be predicted statistically, leading to inherit error within the measuring system. In addition, the effect experienced by the quantum object limits what other quantities can be co-measured with the same level of precision, and according to complementarity through Heisenberg’s Uncertainty Principle, when the position of an object is observed, its momentum is affected in some unknown capacity. Thus, both the position and momentum of the particle cannot be known at precisely the same level. In fact, a similar situation arises for the simultaneous determination of energy and time. Thus, complementarity necessitates a doctrine of unknowable physical interactions that, according to Bohr, are also the source of the statistical nature of the quantum theory.Initially, Einstein was excited about the quantum theory and had even expressed ardent support for its general approval. By 1935, however, while recognizing the theory’s significant achievements, his excitement had morphed into something else: disappointment. His reservations were twofold. First, he felt that the theory had wholeheartedly abandoned the historical task of natural science, which was to provide knowledge of the fundamental laws of nature that were independent of observers or their observations. Instead, the theory’s prevailing understanding of the quantum wavefunction was that it only treated the outcomes of any measurements as probabilities, as outlined by the Born Rule. In fact, the theory in no way mentioned what, if anything, was likely to be true if no observation had ever occurred. That there could be laws for a system undergoing observation, but no laws of any sort dictating how the system behaves independently of observation, painted the quantum theory as unrealistic at best and false at worst. Second, the quantum theory as defined by the Copenhagen Interpretation was essentially statistical. The probabilities built into the wavefunction were fundamental and, unlike the case with classical mechanics, they were not understood as a simple case of moving the decimals to get a finer and finer precision in the readouts of instruments. In this sense, the theory was indeterministic, and Einstein began to probe just how strongly the quantum theory was tied to indeterminism and the concept of determinism in general.He wondered whether it was possible, at least in principle, to attribute certain properties to a quantum system in the absence of measurement. Is it possible, for instance, that the decay of an atom actually occurs at a definite moment in time, even though such a definite decay time is not implied by the quantum wavefunction? In trying to answer such questions, Einstein began to ask whether the quantum theory’s descriptions of quantum systems was, in fact, complete. In other words, can all physically relevant truths about systems be derived from quantum states? In response, Bohr and others sympathetic to his theory of complementarity made bold claims, not just for the descriptive adequacy of the quantum theory, but also for its “finality”, claims that enshrined the features of indeterminism that worried Einstein. Thus, complementarity became Einstein’s target for investigation. In particular, Einstein had reservations about the uncontrollable physical effects extolled by Bohr in the context of measurement interactions and about their role in fixing the interpretation of the wave function. Accordingly, EPR’s focus on completeness was intended to support those reservations in a particularly dramatic way. The EPR text is concerned, in the first instance, with the logical connections between two assertions. The first assertion is that quantum mechanics is incomplete, and the second assertion is that incompatible quantities, like the value of the x-coordinate of a particle’s position and the value of that same particle’s linear momentum in the x direction, cannot have simultaneous “reality”; in other words, they cannot have simultaneously real, discrete values. The authors declare the contradiction of these two assumptions as their first premise: one or the other must hold. It follows that if quantum mechanics were complete, indicating that the first assertion failed, then the second one would hold; i.e., incompatible quantities cannot have real values simultaneously. They further take as a second premise that if quantum mechanics were complete, then incompatible quantities, in particular coordinates of position and momentum, could indeed have simultaneous, real values. They then conclude that quantum mechanics is incomplete for the reasons stated above. This conclusion certainly follows from their logic since otherwise, if the theory were complete, one would have a contradiction over simultaneous values.To establish these two premises more fully and flesh them out so that no doubt remains, EPR begins with a discussion over the idea of a complete theory. Here, the authors offer only one necessary condition: that for a theory to be complete, “every element of the physical reality must have a counterpart in the physical theory.” Although they do not define an “element of physical reality” explicitly in the text, that expression is used when referring to the values of physical quantities, like positions, momenta, and spins, that are determined by an underlying “real physical state”. The picture that EPR builds in this section is that quantum systems have real states that assign values to certain quantities, and while the authors waffle between saying the quantities in question have “definite values” or whether “there exists an element of physical reality corresponding to the quantity”, suppose the simpler terminology is adopted. If this assumption is true, a system can therefore be defined as definite if that quantity has a definite value; that is to say, if the real state of the system assigns a value, or an “element of reality”, to the quantity. Further, without a change in the real state, there will be no change among the values assigned to those quantities. With that understanding now in place, in order to investigate the issue of completeness, the major question that EPR now has to answer is when, exactly, a quantity has a definite value. For that purpose, they offer a minimal sufficient condition: if, without in any way disturbing a system, the prediction with absolute certainty of the value of a physical quantity is possible, then there must exist at least one element of reality corresponding to that quantity. This condition for an “element of reality” is known as the EPR Criterion of Reality, and by way of illustration, EPR points to the specific case when the solution to the quantum wavefunction is an eigenstate, since in an eigenstate, the corresponding eigenvalue has a probability of one. Thus, it has a definite value that one can determine, and hence predict with absolute certainty, without disturbing the system.With this understanding in place, the mathematics of eigenstates show that if, for instance, the values of position and momentum for a quantum system were definite and, accordingly, elements of reality, then the description provided by the wave function of the system would be incomplete, since no wavefunction can contain eigenvalue counterparts of one for both elements due to the generally accepted postulates of Heisenberg. Hence, the authors verify the first premise: either quantum theory is incomplete, or there can be no simultaneously real, “definite” values for incompatible quantities. The next challenge is to show that if quantum mechanics were complete, then incompatible quantities could have simultaneous real values, which is the basis of the second premise. This statement, however, is not as easy to demonstrate. Admittedly, what EPR proceeds to do from this point onwards is rather odd. Instead of assuming completeness, and on that basis, deriving that incompatible quantities can actually have real values simultaneously, they simply set out to derive the latter assertion without assuming any completeness at all. This “derivation” turns out to be the heart, and most controversial, part of the paper.For the proof of this derivation, they sketch and then unpack an iconic thought experiment whose variations continue to be widely discussed to this day. The experiment discusses two quantum systems that, while spatially distant from one another and perhaps quite far apart, the total quantum wavefunction for the pair links both the positions of the systems as well as their linear momenta together. Within the paper, the total linear momentum is zero along the x-axis, so that if the linear momentum of one of the systems along the x-axis were found to be p, the momentum of the other system in the x direction would therefore have to be ?p. At the same time, their positions along the x-axis are also strictly defined so that determining the position of one system on the x-axis allows us to infer the position of the other system along the axis. The authors then proceed to construct an explicit wavefunction for the total, combined system that embodies these links, despite the fact that the systems are perhaps very widely separated in space. Although others have later questioned the legitimacy of this wavefunction, it does, at least for the moment, appear to guarantee the required relationships for any such spatially separated system. In this way, the second premise of the paper is established, proving that quantum mechanics has more issues that need to be worked out.The authors resolve this paradox by a radical claim: that quantum mechanics, despite all of its success in a tremendous range of experiments, is actually an incomplete theory. In other words, there exists some underlying and as-of-yet undiscovered theory of nature to which quantum mechanics is merely a kind of statistical approximation, similar to the Small-Angle Approximation that physicists use to make mathematical equations easier to work with. Furthermore, unlike quantum mechanics, the more complete theory contains every variable corresponding to all of the different “elements of reality”, and these variables are the missing ingredient to what must be added to quantum mechanics to explain this entanglement without resorting to such concepts like action at a distance, or as Einstein has previously stated, “spooky action at a distance”. This theory, also called the Hidden Variable Theory, can be visualized in a rather simple example of the double-slit experiment. That experiment, traditionally showing the wave-particle duality of electrons, might have something different, something unseen, actually at play. What if, instead of the electrons randomly distributing in the wave-inspired pattern, there was, in fact, a variable, a certain “element of reality”, at each entrance to the slit silently directing the travel of each electron? While this premise may seem preposterous at the moment, it does give pause for thought. Maybe the experiments were proving the wrong theory all along? It is important to keep in mind that this example is rather simplistic, and a more sophisticated example might clear up any confusion, as would a serious challenge to the Hidden Variable Theory that would come in the form of a scientific experiment.Following its publication, for the next fifteen years, the EPR paradox was center stage whenever the conceptual difficulties of quantum theory came under fire. With the paradox limited to just a thought experiment, all of this back-and-forth resulted in nothing; it was all smoke to the fire of quantum mechanics. Then, in 1951, a professor at Princeton University named David Bohm showed that one could simulate the same situation of the EPR paradox by observing the dissociation of a diatomic molecule whose total spin angular momentum at the time of dissociation is zero.