Uncertainty Relations

The position-momentum uncertainty relation (we discuss only this pair of conjugate variables here) is usually written in the form

ℏ-- △x △p ≥ . 2

(1)

and interpreted as follows: The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. Interpreted in a slightly more general way as an uncertainty principle (see Heisenberg [62], quoted here),

It has turned out that it is in principle impossible to know, to measure the position and velocity of a piece of matter with arbitrary accuracy.
Werner Heisenberg, (1901-1976).

Eq. (1) - interpreted in this sense will be referred to as “individual uncertainty principle” (IUP). The IUP represents the most important cornerstone of the standard (Copenhagen) interpretation of QT. As a principle used in physics it requires empirical justification. This justification may either come directly from experimental data or, more indirectly, from the theoretical formalism of QT which is an extremely well tested theory and is considered to be true. Thus, the central question of this section is: Why should we believe in the IUP ?

The answer to this question is, as will be shown below, that in fact (essentially) two variants of Eq. (1) exist, but in both of these the meaning of $△x$ , $△p$ is different from the meaning of $△x$ , $△p$ in the IUP. In other words, evidence for two different uncertainty principles may be found in nature but none of them agrees with the IUP.

2.1 Preparation and Measurement

The first variant of Eq. (1) may be justified by referring to several “Gedankenexperimente” formulated by Heisenberg [60]. In this case $△x$ and $△p$ are uncertainties in position and momentum of a single particle after an attempt to determine these values. In the most famous of these Gedankenexperiments the measuring device is a light microscope. If Eq. (1) refers to such an experimental situation we will write it in the form

s,a s,a ℏ △x △p ≥ ---, 2

(2)

to indicate that the uncertainties refer to a single particle (s) after (a) an attempt to assign definite numbers to position and momentum. The quantity $△x$ denotes the error in position measurement. The quantity $△p$ denotes the momentum disturbance induced by the measurement (the value of $p$ is assumed to be known at the time of measurement [60]). The estimate (2) is not an exact result of the quantum mechanical formalism, despite the fact that both the de Broglie wave length²² and the Compton recoil of the photon has been used in its derivation.

The second variant of Eq. (1) is an exact result of the quantum mechanical formalism. In this case $△x$ and $△p$ are statistical quantities. The standard deviation $△x$ is defined as the square-root of the variance of $x$ . The variance of $x$ is defined as the mean value of the second power of the deviation from the expectation value, say $x¯$ , of $x$ ,

∘ -------------- 2 △x = ( ¯x - x ) .

(3)

In Eq. (3) we followed the common confusing habit to denote uncertainties of different meaning by the same symbol. In what follows statistical uncertainties defined as in (3) will be denoted by $( △x )sep$ and $( △p )sep$ to indicate that they do not describe single measurements on individual particles but rather a large (infinite) number of experiments on identically prepared individual systems. In these experiments $x$ and $p$ are not measured simultaneously but separately (note also that the measured values are assumed to be exact, i.e. the measurement errors of $x$ and $p$ vanish). For these quantities, the exact inequality

sep sep ℏ (△x ) (△p ) ≥ --. 2

(4)

It turns out that neither Eq. (4) nor Eq. (2) can be used to establish the uncertainty principle as defined by Heisenberg. Eq. (4) cannot be used because it is a statistical relation and does not refer to individual events (on top of that there is no simultaneous measurement of $x$ and $p$ ). Eq. (2), which refers to individual events, cannot be used because it describes a preparation - i.e. it describes a relation between future uncertainties - and not a measurement. Both Eqs. (2) and (4) describe universal limitations concerning the preparation of states. On the other hand, Eq. (2) does not forbid simultaneous measurement of $x$ and $p$ with an accuracy higher than allowed by the r.h.s of (2).

Let us discuss this very important last point in more detail. A measurement is a procedure to obtain the values of variables ’at a particular time’; this means: immediately before the interaction with a measuring apparatus takes place. The values of the variable after the interaction will be generally different from the measured ones (as a consequence of the interaction). The data obtained before the interaction are the measurement results, the data obtained afterwards are the preparation results. This difference between measurement and preparation is negligible in macroscopic experiments - if ’microscopic’ measuring devices are used - but important in QT. More detailed descriptions of the measurement/preparation processes in QT may be found in works by Margenau [96], [97], Prugovecki [123] and others.

Measurements of a higher degree of precision than is permitted by the uncertainty principle are not, I shall try to show, incompatible with the system of formulae of quantum theory, or with its statistical interpretation.
Karl Popper (1902-1994), The Logic of Scientific Discovery, p. 210.

Consider as an example Heisenberg’s light microscope experiment. Here, the incoming electron is assumed to have sharp momentum. The light microscope is then used to measure the position of the electron. This is, according to (2), possible with an accuracy $s,a △x ~ △x$ at the cost of an induced uncertainty (disturbance) $△ps,a$ in momentum. However, the latter uncertainty does not affect the measurement of momentum, which refers to a time immediately before the perturbation and is assumed to be known with certainty in this Gedankenexperiment. Thus, the measured uncertainty product can be made arbitrarily small by increasing $s,a △p$ accordingly. Note that this is not a simultaneous measurement of $x$ and $p$ in the sense that both data have been measured at the same time. Only $x$ is measured by means of the microscope; the exact value of $p$ is assumed to be known from a previous measurement/preparation procedure. Nevertheless, the values for $x$ and $p$ at one and the same time may be obtained by means of this idealized arrangement - given that Heisenberg’s assumption of sharp momentum of the incoming electron is really true. Eq. (2) is frequently referred to as error-disturbance relation.

The above discussion may be found in the published literature, in particular in works by Ballentine [8], Prugovecki [123], Margenau [96], [97], Popper [119], and others. Several other experimental arrangements (Gedankenexperimente), where the uncertainty product is smaller than $ℏ ∕2$ , have been published; see Prugovecki [123] Park and Margenau [113] (p. 239-245) Ballentine [8], Popper [121], and Holze and Scott [67]. These counterexamples show that the abstract version of ’Heisenberg’s uncertainty principle’ (the IUP) is probably not realized in nature; no justification can be found for it - at least on the basis of ’Gedankenexperimente’.

The number of papers criticizing these counterexamples is smaller than one would expect considering the importance of the IUP for questions of interpretation. Poppers thought experiment has been criticized by Collet [93], see also [118] and by Sudberry [147]. The major point of this latter criticism has been shown to be unjustified [see p. 1853 of [81] (arXiv:quant-ph/9905039)]. I found critical comments on the Gedankenexperiment of Park and Margenau [113] in section 7 of a review article by Busch et. al [33] (arXiv:quant-ph/0609185). These authors claim that a refutation of Park and Margenau’s experiment may be found in a paper by Quadt [126]. But Quadt escapes the confrontation with Park and Margenau’s simple physical reasoning by using a very abstract formal language (of quantum proposition lattices) which seems tailor-made to express Bohr’s ideas. An earlier, slightly less formal criticism by Busch and Lahti [34] arrives at the conclusion that different interpretations may coexist insofar as they complement each other; the choice depends, according to these authors, essentially on philosophical positions. One can only agree on that. Frequently, attempts are made [33] to justify the philosophical positions of the CI by referring to the theorems of Bell and Kochen-Specker (see section 5). It is, however, clear that such claims are principally and fundamentally erroneous, because no philosophical conclusion will ever be obtained by means of mathematical reasoning. A mathematical derivation, leading to a theorem, can only transform interpreted input into interpreted output; the interpretation of the mentioned theorems changes dramatically (in favor of the SI) if the interpretation of the input is changed (in favor of the SI)²³ .

Heisenberg’s proposal was based on an error, the mixing up of measurement and preparation. Nobody was aware of the importance of this distinction at that time; the same error was ’committed’ by such physicists as Einstein and Wigner. Margenau, who discovered the distinction, wrote ([98], p.473):

"To find oneself at odds with Einstein and Wigner is not a pleasant situation, especially when the issue is largely verbal. But as the sequel will show, there are advantages in the view I advocate, and these force me to a position of apostasy which I am far from enjoying."
Henry Margenau (1901-1997).

Today the distinction is (more or less) accepted by the scientific community²⁴ , entering even the realm of good textbooks on QT [12], [116]. Modern definitions of preparation and measurement may be found e.g. in works by Jauch [74] and Beltrametti [22]. The works of Margenau, and his student Park, are very profound. While the dominant role of the CI could not be affected, at least one of the important points of his analysis, the distinction between measurement and preparation, is now - forty years later - generally accepted. Hopefully, the rest of his penetrating analysis will be rediscovered by the scientific community within the period of the next forty years.

Thus, the measurement-preparation error is - although still widely unknown - not the most serious problem anymore. But later, Heisenberg and many others used the statistical relation (4) to justify the IUP. This is a second erroneous mixing up - this time of individual events and statistical events. This second error has been realized by many physicists but is nevertheless still alive and waiting for its detection (elimination) by the large majority of physicists working on fundamental questions of QT. Its survival is a consequence of the dominating role of the CI and the fundamental (erroneous) assumption of the CI that the quantum mechanical wave function is able to describe the behavior of an individual particle. More on that in sections 5 and 6.

Note that a test of the uncertainty principle by means of real experiments was completely out of reach at that time. The present experimental situation will be discussed shortly. Let us first comment briefly on the further historical development after Heisenberg’s declaration of the universal uncertainty principle (called here IUP) in 1929. During the following years Heisenberg’s principle was repeated again and again and transferred as a truth from one generation to the next. I mention, as an example, Messiah’s two-volume textbook [105] , which was an authoritative source of QT for a whole generation of physicists (including the present author). In the first volume Heisenberg’s reasoning is reproduced in essentially unaltered form²⁵ . As a consequence, Heisenberg’s principle (as interpreted in the sense of the IUP) attained the status of a truth. It has been repeated so many times, in published papers and lecture halls, that it could not be wrong; the fact, that it never has been verified by observation, became of secondary importance²⁶ . In short, it was - although actually a philosophical principle²⁷ - considered as an essential part of (quantum) physics.

2.2 Uncertainty from ”unsharp observables”

"This objective 'description'. . . reveals itself as a kind of 'ideological superstructure', which has little to do with immediate reality..."
Heisenberg about Bohm's theory, 1955. In "Niels Bohr and the Development of Physics", p.18

The distinction between measurement and preparation could not longer be overlooked in the second part of the last century. Consequently, a number of physicists tried to derive more precisely defined uncertainty relations using the mathematical apparatus of quantum theory; an incomplete list of papers includes Arthurs and Kelly [7], Davies and Lewis [37], Prugovecki [124], Werner [159], Vorontsov [158], Ozawa [111, 110, 109], and Busch et. al. [33]. At least two of the three different meanings of the uncertainty relations, discussed above, have been clearly distinguished in these works [110, 109], [33] . However, the results obtained are, as they are derived from QT, necessarily statistical in nature and as such do not prove the validity of the IUP, which refers to individual events.

Since QT in its established form is unable to describe such things as simultaneous measurements of several observables, it must be supplemented by additional assumptions about the state of a particle after measurement of the first observable (more details on that in later sections). Such additional assumptions - which are not triggered by any unexplained experimental data but by wishful thinking concerning interpretation - must be postulated in order to establish a mathematical framework for the idea that a wave function can be associated with a single object. A coherent set of such assumptions is called a measurement theory. Measurement theories are mathematical superstructures built on top of QT which are introduced for philosophical reasons and cannot be verified experimentally²⁸ .

"Mathematics is the art of giving the same name to different things"
Jules Henri Poincaré (1854 - 1912), French mathematician, theoretical physicist, engineer, and philosopher of science

This approach [7, 37, 124, 159, 158, 111, 110, 109, 33] is intended to be an improvement of the simplest (von Neumann’s) measurement theory. It is sometimes referred to as theory of ”unsharp observables”; a detailed description will not be given here. The measurement apparatus is considered as part of the measurement process but a complete quantum-mechanical description is, of course, impossible. Therefore, the measurement apparatus is defined by mathematical axioms, as a structure in Hilbert-space, and the interaction between sample and measurement apparatus is described by a variety of mathematical models, which lead to different conclusions. A set of operators, called POVM’s, representing possible measurement results, is introduced. This set of is not unique and has to be determined in practice mostly by mathematical reasoning.

It does not seem that anyone of the problems associated with the individuality interpretation of QT can be solved in this framework²⁹ . Generally, the ad hoc association of physical meaning to abstract mathematical objects is far from unique and strongly interpretation-dependend. Consequently, most of the above works are characterized by a predominantly mathematical way of thinking; an exception may be found in Peres’ book [116]. The arbitrariness in fundamental assumptions of these works has been discussed in more detail by Hofmann [66] (arXiv:1205.0073).

"Everything is vague to a degree you do not realize till you have tried to make it precise"
Bertrand A. W. Russell, 3rd Earl Russell (1872 - 1970) British philosopher

The concept underlying these works is neither purely phenomenological (as in Heisenberg’s famous discussion of the light microscope) nor purely quantum-theoretical. Both the quantum formalism and additional assumptions (the projection postulate and others), expressing the fact that QT is a theory of single particles, are used. From a fundamental point of view of interpretation of QT, these works are logically circular, as far as they claim to derive the IUP.³⁰ : Heisenberg’s uncertainty relation, interpreted in the sense of the IUP, is (implicitly) used to infer the individuality interpretation of QT. But in attempts to derive the IUP from the formalism of QT, the interpretation inferred from the IUP - namely the individuality interpretation of QT - is used (in form of the projection postulate and other assumptions). An essential element of the assertion one wants to prove has been used as premise.

On the other hand, one cannot assert that all results obtained in such theories of unsharp observables are necessarily devoid of any physical meaning. Despite the fundamental (individualistic) erroneous starting point of such theories, the assumption of wave-function collapse may (partly) be meaningful as describing effectively a preparation procedure for the second measurement. A systematic investigation of such theories, from the point of view of the SI, has unfortunately not yet been undertaken.

"Now, what that means is that there is fundamental indeterminacy from quantum mechanics, but besides that there are other sources of effective indeterminacy"
Murray Gell-Mann (1929-), received the 1969 Nobel Prize in physics.

Although not directly related to the present question of the validity of the IUP, it is interesting to mention Ozawa’s derivation [109] (arXiv:quant-ph/0207121) of a statistical version of Heisenberg’s error - disturbance relation (2) for non-commuting variables. In the present context Ozawa’s result is interesting for the following reasons:

In contrast to other unsharp-observable-theories, which lead to statistical results in agreement with (statistical versions of) (2), Ozawa’s theory takes both sources of uncertainty (preparation and measurement) into account. As a consequence two new terms appear describing uncertainties in the two observables due to the preparation procedure. In fact, there is no reason why the fluctuation uncertainties found by Kennard [80] and Robertson [132], which lead to Eq. (4) and whose existence is beyond any doubt, should not contribute to the total uncertainty in a statistical experiment. Ozawa’s result has been verified experimentally, see subsection 2.4 for more details.

Ozawa claims that the new terms in his universally valid uncertainty relation allow effectively for a violation of Heisenberg’s error-disturbance relation (2). And this is also what experiments testing his relation are believed to show³¹ . The problem is that not Heisenberg’s error-disturbance relation (2) (which refers to single particles and not to ensembles) is investigated but a statistical version (describable in the mathematical framework of QT) of it. Such a statistical version of his own uncertainty relation was never postulated by Heisenberg³² . Thus, the statistical version of (2) whose validity was tested experimentally as well as theoretically, was Ozawas (and other’s) own creation. This ’Heisenberg relation’ has never been derived or verified; the precise physical and mathematical meaning of the quantities occuring in it is unknown. This situation has been recognized by Fujikawa [55] (arXiv:1205.1360). Combining Ozawa’s relation with Robertson’s relation he derived a new statistical uncertainty relation, which is also universally valid but differs from Ozawa’s universally valid uncertainty relation. Fujikawa’s universally valid uncertainty relation happens to agree with a statistical version of Heisenberg’s single particle error-disturbance relation (2). But this second statistical version of (2) is not the one created by Ozawa but by Fujikawa. Both statistical versions look ’reasonable’ from a physical point of view and both are of comparable simplicity. Both versions are in agreement with the experimental data mentioned in subsection 2.4. Many other inequivalent (statistical) uncertainty relations derived by different authors may be found in the literature. This confusion is not unexpected from the point of view of the SI, which claims that superstructures of all kinds, built on top of QT, are a waste of time.

"Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means"
Bertrand Russell (1872 - 1970), British philosopher and logician

Slightly more relevant for the validity of the IUP are derivations of the (statistical version of the) IUP³³ . Several variants of Eq. (4) have been derived [159] (arXiv:quant-ph/0405184), [33], where the non-commuting observables (here $x$ and $p$ ) were not measured separately but simultaneously. The resulting relation may be written in the form

sim sim ℏ (△x ) (△p ) ≥ ---. 2

(5)

It avoids the naive mixing up of preparation and measurement, as done by Heisenberg in the early days of QT, and is, in this respect, slighly closer to the IUP. Eq. (5) is of course not a proof of the IUP (it leaves room for individual violations of the bound) but is at least compatible with the IUP. However, preparation and measurement uncertainties have not been separately taken into account in these works [159, 33]. Ozawa, using the same line of reasoning as in his error-disturbance theory, derived an inequality different from Eq. (5), which contains again two new terms due to the imperfect preparation of the two non-commuting observables [110] (pdf). He suggests that under certain conditions the Heisenberg limit may be violated, but does not go into details. One expects similar problems as for the error-disturbance relation discussed above. Apparently, Ozawa’s (joint measurement) uncertainty relation [110] has not yet been tested experimentally. For this reason, this work will not be discussed here further.

Summarizing, there is no support at all for the validity of the IUP from unsharp variable theories. The conclusion of some of these theories, that not even a statistical version of the IUP holds true, could even be seen as support for the SI, were it not that, according to the SI, there is no sound basis for these theories.

2.3 Individual and statistical predictions

"The true logic of this world is the calculus of probabilities"
James Clerk Maxwell (1831 - 1879), Scottish physicist

Summarizing, we have four relations with different meanings - which all go under the name ’uncertainty relation’. Eq. (1) is the abstract relation, referred to as IUP, expressing Heisenberg’s uncertainty principle (as defined above) which is most important for the (Copenhagen) interpretation of QT. The remaining three relations (2), (4), (5) are much more ’concrete’ in the sense that they correspond to realistic (Gedanken) experiments or to theoretical results obtained in the mathematical framework of QT. We have to answer the following question: Can the IUP (1) be confirmed with the help of (at least) one of the relations (2), (4), (5) ?

We have already shown that Eq. (2) cannot be used for that purpose because it does not describe a relation between measurement errors. We also mentioned that the same is obviously true for (4), which describes statistical fluctuations of measurements performed separately and not simultaneously. The remaining question is wether or not Eq. (5) can be used to confirm the IUP³⁴ .

"Science is uncertain. Theories are subject to revision; observations are open to a variety of interpretations, and scientists quarrel amongst themselves. This is disillusioning for those untrained in the scientific method, who thus turn to the rigid certainty of the Bible instead. There is something comfortable about a view that allows for no deviation and that spares you the painful necessity of having to think"
Isaac Asimov (1920 - 1992), American science-fiction author and professor of biochemistry

The third uncertainty relation, Eq. (5), claims to describe simultaneous measurements but is still statistical in nature, as mentioned already. As a single relation, describing essentially the average of an enormous number of results on individual events, it contains much less information than the IUP, which represents an enormous number of relations, each one referring to an individual event. Let us explain this last point in more detail. In order to see the difference between the IUP, Eq. (1), and statistical relations like (5) (or (4)) we ask and answer the following question: How can these relations be verified experimentally ?

The above argument is similar in spirit to (a simple version of) Margenau’s and Park’s criticism [99] of an axiomatic framework constructed by Prugovecki [124].

"In all affairs it's a healthy thing now and then to hang a question mark on the things you have long taken for granted"
Bertrand Russell (1872 - 1970) , British philosopher

Thus, the theoretical results described in subsection 2.2 do not provide support for the validity of the IUP. This is contrary to common wisdom. The difference between individual measurements and statistical measurements has not (in contrast to the difference between measurement and preparation) yet been realized by the scientific community. This crucial difference is, e.g., not even mentioned in the recent works on uncertainty relations quoted in the last subsection 2.2. The deeper reason for this denial is that even today an overwhelming majority of physicists believes that the wave function describes single particles.³⁹ .

Summarizing, we found no support for the IUP from the formalism of QT. Let us now turn to the question whether experimental confirmation/refutation can be found.

2.4 Empirical support

For the present question of interpretation of QT, experimental tests related to the IUP (1) are of primary importance. Let us, nevertheless, first mention two recent tests of Heisenberg’s error-disturbance relation (2), which is not completely unrelated to our problem.

"Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin-measurement"
Hasegawa and coworkers, Vienna 2012

Two experimental tests of Ozawa’s generalized error-disturbance relation (briefly discussed in subsection 2.2) have recently been reported, namely by Erhart et. al. [48] (arXiv:1201.1833) and by Rozema et. al. [135] (arXiv:1208.0034). Using different techniques, the final conclusion of both works agree insofar as Ozawa’s relation is confirmed and a violation of Ozawa’s statistical version of Heisenberg’s error-disturbance relation (2) is found [see the discussion in subsection 2.2]. The significance of the difference between the original single-particle relation (2) and statistical versions of it, is not recognized in these works⁴⁰ .

"Facts do not cease to exist because they are ignored"
Aldous L. Huxley (1894 - 1963), English writer, Best known for his novel Brave New World.

In view of the fact, that Heisenberg’s error-disturbance relation is (erroneously) considered by many physicists as one of the most fundamental relations of QT, the excitement caused by this breakdown of one of the (supposed) fundaments of QT is amazingly small. One gets the impression that the ”uncertainty principle” (no matter which one) became an object of uncritical admiration, which cannot be affected any more by new insights or facts. ⁴¹ . This guarantees also the survival of erroneous conclusions drawn from it in the past.

"There are no facts, only interpretations"
Friedrich Nietzsche (1844 - 1900), German philosopher

Let us now come back to our central question, the validity of the IUP. The general method to test the validity of the IUP, described in the last subsection has never been realized. It seems impracticable, at least at the present time; experimental devices to measure the (random) position and (random) momentum of a particle at a particular time ’with arbitrary accuracy’ simply do not exist. Thus, an experimental determination of the individual numbers $Δxi$ , $Δpi$ ( $i = 1, ..., N$ ) occurring in the IUP, and required to test its validity, has not been possible so far⁴² .

Therefore, in real experiments (as well as in most theoretical considerations) statistical data, typically widths of slits and of peaks in momentum distributions, are taken as measures for properties of individual particles. This is a very rough estimate. The strange fact, that these statistical results are often considered as an experimental confirmation of a principle (the IUP) ruling the behavior of individual particles is closely related to the almost universal acceptance of the erroneous idea that the wave function is able to describe the behavior of a single particle.

Thus, a direct experimental test of the IUP has not been possible so far. Let us nevertheless consider the question whether or not the experimental statistical estimates obtained so far are in agreement with the IUP. We will ask and answer in consecutive order the two questions:

A positive answer to the first question does not present a final verification of the IUP, but would nevertheless support it. A positive answer to the second question presents evidence for the nonexistence of the IUP, since its violation in a statistical experiment necessarily implies its violation in individual observations⁴³ .

Turning now to the question of the empirical support [for the uncertainty principle], we unhesitatingly declare that rarely in the history of physics has there been a principle of such universal importance with so few credentials of experimental tests.
Max Jammer (1915-), Philosophy of Quantum Mechanics, p.81.

In the same paragraph (on p.81 of his profound work on the Philosophy of Quantum Mechanics [73]) Jammer adds:

”..no such measurement on individual particles has ever been performed with sufficient precision to be of any significance..”

This assessment was written in 1974. Today, the situation has not changed. In a recent review article by Busch et al [33] the experimental situation concerning joint measurements of position and momentum is summarized as follows ([33], p.170):

”To the best of our knowledge, and despite some claims to the contrary, there is presently no experimental realization of a joint measurement of position and momentum. Thus there can as yet be no question of an experimental test of the uncertainty relation for inaccuracies in joint measurements of these quantities”.

The uncertainty relations Busch et al. refer to⁴⁴ are of the statistical type (5). While no experimental support for position/momentum variables does exist, successful joint measurements of conjugate quadrature components of quantum optical fields (using multiport homodyme detection) have been reported. These field experiments cannot be used as a proof of the IUP, which is a principle ruling the behavior of single particles.

Thus, there seem to be essentially no experimental (statistical) results supporting the IUP⁴⁵ . The most natural explanation seems to be that the IUP is not realized in nature. We have already mentioned that thought experiments by Ballentine [8], Prugovecki [123], Park and Margenau [113], and Popper [121] as well as theoretical results by Uffink and Hilgevoord [154] and by Mensky [103] exist which show a violation of the IUP. The remaining important question is now whether or not such violations of the IUP can also be found in real experiments⁴⁶ .

Several violations of the IUP have been reported in the literature. We shall restrict ourselves to a single work, the verification of Popper’s thought experiment [121] reported by Kim and Shih [81] (arXiv:quant-ph/9905039).

Popper suggested an experimental arrangement - making essential use of two-particle entanglement - designed to test the validity of the IUP. According to his prediction, the measured uncertainty product $Δx Δp$ should be markedly smaller than $ℏ$ . The results (see Fig. 5 of [81]), obtained using entangled photons, agree clearly with Popper’s prediction.

Kim and Shih avoid a direct attack on the IUP-monument by noting that what is actually measured is an entangled two-photon state (referred to as ’biphoton’ state) instead of the state of a single, individual photon. On the other hand they claim that their result is in agreement with Popper’s prediction and with quantum mechanics. In essence, they seem to suggest that the IUP remains valid but should be modified - restricted to not-entangled particles. Most scientists, taking part in the following discussion, did not share this point of view; entanglement was not considered as a sufficient reason to dispense with the concept of individuality altogether.

Kim and Shih’s experiment (or rather its interpretation as a violation of the IUP) was criticized by Short [142] (arXiv:quant-ph/0005063), and others. Support for Kim and Shih’s results and/or Popper’s attack against IUP and CI came from Sancho [136] (pdf), Hunter [70] (arXiv:quant-ph/0507009), Rigolin [131], (arXiv:quant-ph/0008100), Unnikrishnan [155] and others. I mention also an interesting but somewhat cryptic paper by Qureshi [127] (arXiv:quant-ph/0505158).

This section is not the right place to go into the details of this controversial discussion, which involves, besides the uncertainty relations, also questions like completeness of QT (see section 5). I consider Kim and Shih’s experiment as convincing evidence for violation of the IUP in a particular case. This leads necessarily to the breakdown of this ’universal principle’ as a whole. On top of that the outcome of this experiment can be considered as a late confirmation of Popper’s ”metaphysical realism” and his pertaining criticism of the CI (more on that in section 6).

Summarizing, the central question of this section (Why should we believe in the IUP ?) can be answered as follows: Uncertainty relations of the statistical type, like (4), are of course valid, being a straight consequence of the formalism of QT. On the other hand, there is no reason to believe that uncertainty relations like the IUP (1) - ruling the measurement of conjugate observables of individual particles - will be realized in nature. No rational basis for this central postulate of the CI can be found.

2 Uncertainty Relations

2.1 Preparation and Measurement

2.2 Uncertainty from ”unsharp observables”

2.3 Individual and statistical predictions

2.4 Empirical support