This page gives plain-language summaries of physics papers. We draw no link to Dost or to any other text; reading side by side is the reader's own work.
Experiment: measured. Theory: calculated or proved. Interpretation: its meaning is disputed. The labels are our own classification; a paper may have more than one nature.
In 1935 Einstein, Podolsky and Rosen asked this question: does the description that quantum mechanics gives with the wave function describe physical reality completely? The paper defines what is necessary for a theory to be counted as complete and when a quantity can be called an 'element of reality', then with a two-particle example reaches the conclusion that quantum mechanics is incomplete.
On this page6. The EPR Paradox · Einstein 1935 — Can quantum-mechanical d · 7. Bell's Theorem …
Einstein, Podolsky and Rosen argued, on the basis of the correlation of two particles that have moved apart, that the wave function description is incomplete (Phys. Rev. 47, 777).
The paper begins by setting two criteria. The first is completeness: a theory is complete if every element of physical reality has a counterpart in the theory. The second is reality: if, without disturbing a system in any way, we can predict with certainty the value of a physical quantity, then there is an element of reality corresponding to this quantity.
Then an example is set up. Two particles interact for a while and then move apart; their initial states are such that the difference of their positions and the sum of their momenta are definite. If we measure the position of the first particle, we predict the position of the second with certainty; if we measure the momentum of the first, we predict the momentum of the second with certainty. Since nothing physical is done to the second particle by these measurements (the particles are now far apart), by the reality criterion both the position and the momentum of the second particle are real elements.
But in quantum mechanics position and momentum cannot carry definite values at the same time; the wave function does not give both. Therefore the wave function description does not contain every element of reality; that is, it is incomplete.
In the paper the reasoning based on the reality criterion is tied not only to the position and momentum example but also to a general rule: a measurement made on one of two systems that do not influence each other cannot change the real state of the other. This locality assumption became the focus of the later debate.
The paper does not say whether the incompleteness can be remedied in some way; it only argues that the theory is not complete. Its limit: the conclusion rests on the assumption that the two particles do not influence each other when far apart. After the first statement in this paper, Einstein set out his own view in other writings in other forms.
Afterwards: Bohr replied the same year (bohr-1935); in 1964 Bell showed that EPR's assumptions give consequences that can be tested by experiment.
7. Bell's Theorem
Main question: Can locality and quantum mechanics both be true together?
The EPR paper had suggested that a completion of quantum mechanics might be sought in which particles carry predetermined values and distant events do not influence each other instantaneously. Bell showed that any theory holding these two assumptions together will fail to agree with some quantum predictions. That is, the question was turned from a philosophical debate into a problem that can be decided by experiment.
Bell derived an inequality showing that no theory built with local hidden variables can give all the predictions of quantum mechanics (Physics 1, 195).
Bell begins his paper from the EPR debate. According to EPR, the results of measurements made without influencing a distant system must be explained by predetermined values (hidden variables). Bell sets up this idea in precise form: each particle pair has a common initial state; the result of a measurement on one side is determined only by this state and by the measurement setting on that side, and does not depend on the setting far away. This is called the locality assumption.
Bell shows that this assumption brings a mathematical bound on the correlations between measurement results on the two sides; this bound, known today as the Bell inequality, is independent of what the hidden variables are. Then, in the singlet case (the state of two spin-1/2 particles with total spin zero), he computes the correlation predicted by quantum mechanics and shows that for certain settings this correlation exceeds the bound.
Conclusion: no theory built with local hidden variables can give all the predictions of quantum mechanics. The paper also notes that the assumptions of von Neumann's proof that there can be no hidden variables were too strict.
The assumptions of the theorem were later analyzed in more detail: besides locality, 'silent' assumptions are also needed, such as that measurement settings are chosen independently of the hidden variables. Each of these assumptions is discussed as a way of escaping the conclusion.
The theorem does not directly say that nature is nonlocal. The theorem says that quantum predictions and the locality assumption cannot both be true together; there is also separate debate over which is to be given up and over which assumptions besides locality are used. Bell's original derivation also assumes perfect correlation; this condition was later relaxed.
Afterwards: Because Bell's original inequality required perfect correlation, it was not suited to experiment; in 1969 Clauser, Horne, Shimony and Holt derived a form suited to experiment.
8. Bell Experiments
Main question: Does nature really violate Bell inequalities?
Bell's theorem is a mathematical result; only experiment can say which side nature is on. These two papers describe the experiment-ready form of the theorem (CHSH) and one of the first strong experiments (Aspect). The question is whether the measured correlations exceed the bound allowed by local hidden variables, and, if they do, under which open doors left the result holds.
Clauser, Horne, Shimony and Holt derived an inequality that made Bell's result suitable for real experiments and proposed an experiment with photon polarizations.
Bell's 1964 inequality required the measured correlations to be perfect; in real experiments, however, detectors are imperfect and correlations are never complete. The CHSH paper overcomes this obstacle.
The authors define a quantity combining the correlations for four setting pairs of two distant observers, each of whom can measure at two different settings. If local hidden variables exist, the absolute value of this quantity can be at most 2. The authors' first derivation also assumed that the results were predetermined; Bell showed in 1971 that this assumption is not needed.
Quantum mechanics, on the other hand, predicts that at suitable settings this quantity can reach up to 2√2 (about 2.83) instead of 2; this bound was later called the Tsirelson bound. The paper describes how we can test this difference with photon polarization measurements in which each observer uses two settings, for example with two photons emitted in succession from an atom.
The inequality is also important for this reason: even if no signal passes between the two observers, quantum correlations can exceed this bound. The paper also deals with for which measurement angles this difference will be largest and how it is to be read in measurements made with imperfect detectors; thus testing Bell's result in the laboratory became a realistic goal. The proposed arrangement was designed so that it could be carried out with the techniques possible in those years.
Its limit: the paper is an experimental proposal, not an experimental result. Moreover, interpreting the experiment requires additional assumptions under conditions such as detectors being imperfect; these assumptions are called loopholes and were closed one by one in later experiments.
Afterwards: The CHSH form became the common language of all later Bell experiments; in 2015 three groups (Hensen, Giustina, Shalm) published experiments that together closed the main loopholes that had remained open until then.
Aspect, Dalibard and Roger carried out a Bell experiment that narrowed the 'communication loophole' of local hidden variables by rapidly changing the measurement settings while the photons were in flight (Phys. Rev. Lett. 49, 1804).
This experiment by Aspect and colleagues comes after the first experiments testing the Bell inequality and takes up a more specific objection. In earlier experiments the measurement settings were fixed at the start of the experiment and held constant. In that case, the information of the setting on one side reaching the other side by a route slower than the speed of light could influence the results. This objection is called the communication loophole.
In the experiment the settings were changed while the photon pair was leaving the source and traveling to the detectors. On each side there were two polarizers, and a fast switch directed the photon to one of them. The switching time was shorter than the light travel time between the two sides. Therefore the setting decision on one side could not influence the result of the measurement on the other side by a transfer of information slower than light.
The measured correlations came out in agreement with the prediction of quantum mechanics and exceeded the bound of local hidden variables. However, the switching was not a fully random choice.
The experiment produced the photon pairs with two lights emitted in succession by calcium atoms. The measured correlation came out in agreement with the prediction of quantum mechanics and exceeded the bound of local hidden variables. The experimental results were read not as a single experiment but together with the earlier measurements of the same team. The distance between the two sides in the arrangement was kept, relative to the switching time, so large that light could not reach from one side to the other; the aim of this design was to narrow the communication loophole.
Its limit: the experiment also worked with the low efficiency of photon detectors, that is, another loophole, the 'detection loophole', remained open; and the problem of choosing the settings completely at random was also not solved. For this reason the result was interpreted with some additional assumptions.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: These two loopholes were closed separately in later years; three experiments published in 2015 addressed all of them together. The 2022 Nobel Prize in Physics was given to Aspect, Clauser and Zeilinger for Bell experiments.
9. GHZ States
Main question: Is quantum strangeness stronger than EPR?
Bell's theorem is a statistical result: local hidden variables cannot exceed a bound on correlations, but in individual measurements there is no contradiction. Greenberger, Horne and Zeilinger showed that in an entangled state of three or more particles the same contradiction appears without any inequality, with a single measurement result.
M. Kafatos (ed.), Bell's Theorem, Quantum Theory and Conceptions of the Universe, Kluwer, s. 69-72 theory
Greenberger, Horne and Zeilinger showed that in a four-particle quantum state the assumptions of local reality contradict definite results ('Going Beyond Bell's Theorem').
The paper's starting point is a difference between Bell's theorem and EPR's reality criterion. EPR counted cases in which we can predict the value of a quantity with certainty as 'elements of reality'. Bell's theorem, on the other hand, does not contain this case of certain prediction; it is a statistical inequality. The authors show that if a more complex model is used, locally determinate models will not work even in cases where certain prediction is possible.
The proposed state is a special entangled form of multiple particles (today called the GHZ state). The original paper is set up for four spin-1/2 particles; later Mermin and other authors gave a simpler form for three particles. In this state the result of some combinations of measurements is certain according to quantum mechanics. If we assume that the measurement result of each particle is predetermined and independent of choices far away, a contradiction arises among the certain results: the same value is forced to be both plus and minus.
In Bell's theorem the contradiction becomes visible only in the statistics of many measurements. In the GHZ state, on the other hand, the contradiction appears in the result of a single combination, that is, in an 'all or nothing' form.
The authors explicitly separate the structure of the contradiction: quantum mechanics predicts with certainty the result of certain measurements. The local reality assumption says that each particle's own value is predetermined. These two claims give opposite results for the same combination of measurements.
Its limit: the theory was presented as a thought experiment; experimental realizations were made in later years and are interpreted with additional assumptions because of the imperfections of real measurements.
Afterwards: Three-photon GHZ states were produced experimentally in 2000 (Pan and colleagues); these states are among the basic resources also used in quantum information.
10. Quantum Nonlocality
Main question: Why does nature not use the maximum nonlocality allowed?
Bell experiments show that quantum mechanics gives stronger correlations than local theories allow. But these correlations too have an upper bound (the Tsirelson bound). Popescu and Rohrlich noticed that the rule of relativity forbidding sending messages faster than light allows stronger correlations than quantum mechanics allows. What is asked is why nature does not stop at the very edge of what is possible.
Popescu and Rohrlich showed a hypothetical model that is compatible with relativity but builds stronger correlations than quantum mechanics ('Quantum nonlocality as an axiom', Found. Phys. 24, 379).
The starting point of this paper is the question from which principles the nonlocal correlations of quantum mechanics can be derived. Two principles come to mind: the correlations cannot be explained by a local theory (Bell), but these correlations cannot be used to send a message (relativity).
The authors asked this question: if we set down only these two principles, do we arrive at the correlations that quantum mechanics predicts? The answer is no. They defined a hypothetical box in which each of two distant observers chooses an input and gets one bit of output; this box does not change the statistics of the other observer's output whatever one observer's input is, that is, it carries no message. But the CHSH quantity reaches 4 in this box; the upper bound of quantum mechanics is about 2.83, and the bound of local theories is 2.
Thus correlations that are compatible with relativity yet more 'nonlocal' than quantum mechanics are possible. The fact that quantum mechanics exceeds Bell's bound is not a consequence of the rule of relativity; the correlations of quantum mechanics are more restricted than these two principles.
From this situation the paper concludes that deriving quantum mechanics axiomatically from nonlocal correlations and relativity falls short; it says that an additional principle is needed.
The paper is cited for showing that compatibility with relativity alone does not determine quantum correlations; the result prepared the ground for later work seeking the principles of quantum theory. It should be noted that this box is defined by a rule linking one observer's input and output to the other's; the box is not proposed as a physical device.
Its limit: there is no proof that the box could actually exist; the box is only a theoretical tool. The additional principle that would explain the limit of quantum mechanics is still being researched.
This summary was written without looking at the paper's own abstract or text, relying on a known account; it should be checked against the source.
Afterwards: Boxes of this kind were later called 'PR boxes', and the question of why quantum correlations stop at this point continues as an open research subject in quantum information theory.