Albert Einstein's Theory of Quantum Mechanics
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 two fold. 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 wave function
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 wave function 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
wave function 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 xaxis 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
wave function 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 wave function, 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 waveparticle 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.
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