## 1. Introduction



The uncertainty principle is certainly one of the most famous aspects
of quantum mechanics. It has often been regarded as the most
distinctive feature in which quantum mechanics differs from classical
theories of the physical world. Roughly speaking, the uncertainty
principle (for position and momentum) states that one cannot assign
exact simultaneous values to the position and momentum of a physical
system. Rather, these quantities can only be determined with some
characteristic "uncertainties" that cannot become
arbitrarily small simultaneously. But what is the exact meaning of
this principle, and indeed, is it really a principle of quantum
mechanics? (In his original work, Heisenberg only speaks of
uncertainty relations.) And, in particular, what does it mean to say
that a quantity is determined only up to some uncertainty? These are
the main questions we will explore in the following, focusing on the
views of Heisenberg and Bohr.



The notion of "uncertainty" occurs in several different
meanings in the physical literature. It may refer to a lack of
knowledge of a quantity by an observer, or to the experimental
inaccuracy with which a quantity is measured, or to some ambiguity in
the definition of a quantity, or to a statistical spread in an
ensemble of similarly prepared systems. Also, several different names
are used for such uncertainties: inaccuracy, spread, imprecision,
indefiniteness, indeterminateness, indeterminacy, latitude, etc. As we
shall see, even Heisenberg and Bohr did not decide on a single
terminology for quantum mechanical uncertainties. Forestalling a
discussion about which name is the most appropriate one in quantum
mechanics, we use the name "uncertainty principle" simply
because it is the most common one in the literature.


## 2. Heisenberg


### 2.1 Heisenberg's road to the uncertainty relations



Heisenberg introduced his famous relations in an article of 1927,
entitled *Ueber den anschaulichen Inhalt der quantentheoretischen
Kinematik und Mechanik*. A (partial) translation of this title is:
"On the *anschaulich* content of quantum theoretical
kinematics and mechanics". Here, the term *anschaulich*
is particularly notable. Apparently, it is one of those German words
that defy an unambiguous translation into other languages.
Heisenberg's title is translated as "*On the physical
content* ..." by Wheeler and Zurek (1983). His
collected works (Heisenberg 1984) translate it as "*On the
perceptible content* ...", while Cassidy's
biography of Heisenberg (Cassidy 1992), refers to the paper as
"*On the perceptual content* ...". Literally,
the closest translation of the term *anschaulich* is
"visualizable". But, as in most languages, words that make
reference to vision are not always intended literally. Seeing is
widely used as a metaphor for understanding, especially for immediate
understanding. Hence, *anschaulich* also means
"intelligible" or
 "intuitive".[1]




Why was this issue of the *Anschaulichkeit* of quantum
mechanics such a prominent concern to Heisenberg? This question has
already been considered by a number of commentators (Jammer 1974;
Miller 1982; de Regt 1997; Beller 1999). For the answer, it turns out,
we must go back a little in time. In 1925 Heisenberg had developed the
first coherent mathematical formalism for quantum theory (Heisenberg
1925). His leading idea was that only those quantities that are in
principle observable should play a role in the theory, and that all
attempts to form a picture of what goes on inside the atom should be
avoided. In atomic physics the observational data were obtained from
spectroscopy and associated with atomic transitions. Thus, Heisenberg
was led to consider the "transition quantities" as the
basic ingredients of the theory. Max Born, later that year, realized
that the transition quantities obeyed the rules of matrix calculus, a
branch of mathematics that was not so well-known then as it is now. In
a famous series of papers Heisenberg, Born and Jordan developed this
idea into the matrix mechanics version of quantum theory.



Formally, matrix mechanics remains close to classical mechanics. The
central idea is that all physical quantities must be represented by
infinite self-adjoint matrices (later identified with operators on a
Hilbert space). It is postulated that the matrices \(\bQ\)
and \(\bP\) representing the canonical position and
momentum variables of a particle satisfy the so-called canonical
commutation rule



\[\tag{1} 
\bQ\bP - \bP\bQ = i\hslash\]


 where \(\hslash = h/2\pi\), \(h\) denotes
Planck's constant, and boldface type is used to represent
matrices (or operators). The new theory scored spectacular empirical
success by encompassing nearly all spectroscopic data known at the
time, especially after the concept of the electron spin was included
in the theoretical framework.



It came as a big surprise, therefore, when one year later, Erwin
Schrodinger presented an alternative theory, that became known as
wave mechanics. Schrodinger assumed that an electron in an atom
could be represented as an oscillating charge cloud, evolving
continuously in space and time according to a wave equation. The
discrete frequencies in the atomic spectra were not due to
discontinuous transitions (quantum jumps) as in matrix mechanics, but
to a resonance phenomenon. Schrodinger also showed that the two
theories were equivalent.[2]




Even so, the two approaches differed greatly in interpretation and
spirit. Whereas Heisenberg eschewed the use of visualizable pictures,
and accepted discontinuous transitions as a primitive notion,
Schrodinger claimed as an advantage of his theory that it was
*anschaulich*. In Schrodinger's vocabulary, this
meant that the theory represented the observational data by means of
continuously evolving causal processes in space and time. He
considered this condition of *Anschaulichkeit* to be an
essential requirement on any acceptable physical theory.
Schrodinger was not alone in appreciating this aspect of his
theory. Many other leading physicists were attracted to wave mechanics
for the same reason. For a while, in 1926, before it emerged that wave
mechanics had serious problems of its own, Schrodinger's
approach seemed to gather more support in the physics community than
matrix mechanics.



Understandably, Heisenberg was unhappy about this development. In a
letter of 8 June 1926 to Pauli he confessed that "The more I
think about the physical part of Schrodinger's theory, the
more disgusting I find it", and: "What Schrodinger
writes about the *Anschaulichkeit* of his theory, ... I
consider *Mist*" (Pauli 1979: 328). Again, this last
German term is translated differently by various commentators: as
"junk" (Miller 1982) "rubbish" (Beller 1999)
"crap" (Cassidy 1992), "poppycock"
(Bacciagaluppi & Valentini 2009) and perhaps more literally, as
"bullshit" (Moore 1989; de Regt 1997). Nevertheless, in
published writings, Heisenberg voiced a more balanced opinion. In a
paper in *Die Naturwissenschaften* (1926) he summarized the
peculiar situation that the simultaneous development of two competing
theories had brought about. Although he argued that
Schrodinger's interpretation was untenable, he admitted
that matrix mechanics did not provide the *Anschaulichkeit*
which made wave mechanics so attractive. He concluded: 



> 
> 
> to obtain a contradiction-free *anschaulich* interpretation, we
> still lack some essential feature in our image of the structure of
> matter. 
> 
> 
> 



The purpose of his 1927 paper was to provide exactly this lacking
feature.


### 2.2 Heisenberg's argument



Let us now look at the argument that led Heisenberg to his uncertainty
relations. He started by redefining the notion of
*Anschaulichkeit*. Whereas Schrodinger associated this
term with the provision of a causal space-time picture of the
phenomena, Heisenberg, by contrast, declared:



> 
> 
> We believe we have gained *anschaulich* understanding of a
> physical theory, if in all simple cases, we can grasp the experimental
> consequences qualitatively and see that the theory does not lead to
> any contradictions. Heisenberg 1927: 172)
> 
> 
> 



His goal was, of course, to show that, in this new sense of the word,
matrix mechanics could lay the same claim to *Anschaulichkeit*
as wave mechanics.



To do this, he adopted an operational assumption: terms like
"the position of a particle" have meaning only if one
specifies a suitable experiment by which "the position of a
particle" can be measured. We will call this assumption the
"measurement=meaning principle". In general, there is no
lack of such experiments, even in the domain of atomic physics.
However, experiments are never completely accurate. We should be
prepared to accept, therefore, that in general the meaning of these
quantities is also determined only up to some characteristic
inaccuracy.



As an example, he considered the measurement of the position of an
electron by a microscope. The accuracy of such a measurement is
limited by the wave length of the light illuminating the electron.
Thus, it is possible, in principle, to make such a position
measurement as accurate as one wishes, by using light of a very short
wave length, e.g., \(\gamma\)-rays. But for \(\gamma\)-rays, the
Compton effect cannot be ignored: the interaction of the electron and
the illuminating light should then be considered as a collision of at
least one photon with the electron. In such a collision, the electron
suffers a recoil which disturbs its momentum. Moreover, the shorter
the wave length, the larger is this change in momentum. Thus, at the
moment when the position of the particle is accurately known,
Heisenberg argued, its momentum cannot be accurately known:



> 
> 
> At the instant of time when the position is determined, that is, at
> the instant when the photon is scattered by the electron, the electron
> undergoes a discontinuous change in momentum. This change is the
> greater the smaller the wavelength of the light employed, i.e., the
> more exact the determination of the position. At the instant at which
> the position of the electron is known, its momentum therefore can be
> known only up to magnitudes which correspond to that discontinuous
> change; thus, the more precisely the position is determined, the less
> precisely the momentum is known, and conversely. (Heisenberg 1927:
> 174-5)
> 
> 
> 



This is the first formulation of the uncertainty principle. In its
present form it is an epistemological principle, since it limits what
we can *know* about the electron. From "elementary
formulae of the Compton effect" Heisenberg estimated the
"imprecisions" to be of the order 



\[\tag{2} \delta p\delta q \sim h\]


He continued: "In this circumstance we see the direct
*anschaulich* content of the relation \(\boldsymbol{QP} -
\boldsymbol{PQ} = i\hslash\)."



He went on to consider other experiments, designed to measure other
physical quantities and obtained analogous relations for time and
energy: 



\[\tag{3} \delta t \delta E \sim h\]

 and action \(J\) and angle \(w\) 



\[\tag{4} \delta w \delta J \sim h\]

 which
he saw as corresponding to the "well-known" relations



\[\tag{5} 
\boldsymbol{tE} - \boldsymbol{Et} = 
 i\hslash \text{ or } \boldsymbol{wJ} - \boldsymbol{Jw} = i\hslash\]


However, these generalisations are not as straightforward as
Heisenberg suggested. In particular, the status of the time variable
in his several illustrations of relation
 (3)
 is not at all clear (Hilgevoord 2005; see also
 Section 2.5).



Heisenberg summarized his findings in a general conclusion: all
concepts used in classical mechanics are also well-defined in the
realm of atomic processes. But, as a pure fact of experience (*rein
erfahrungsgemass*), experiments that serve to provide
such a definition for one quantity are subject to particular
indeterminacies, obeying relations
 (2)-(4)
 which prohibit them from providing a simultaneous definition of two
canonically conjugate quantities. Note that in this formulation the
emphasis has slightly shifted: he now speaks of a limit on the
definition of concepts, i.e., not merely on what we can *know*,
but what we can meaningfully *say* about a particle. Of course,
this stronger formulation follows by application of the above
measurement=meaning principle: if there are, as Heisenberg claims, no
experiments that allow a simultaneous precise measurement of two
conjugate quantities, then these quantities are also not
simultaneously well-defined.



Heisenberg's paper has an interesting "Addition in
proof" mentioning critical remarks by Bohr, who saw the paper
only after it had been sent to the publisher. Among other things, Bohr
pointed out that in the microscope experiment it is not the change of
the momentum of the electron that is important, but rather the
circumstance that this change cannot be precisely determined in the
*same* experiment. An improved version of the argument,
responding to this objection, is given in Heisenberg's Chicago
lectures of 1930.



Here (Heisenberg 1930: 16), it is assumed that the electron is
illuminated by light of wavelength \(\lambda\) and that the scattered
light enters a microscope with aperture angle \(\varepsilon\).
According to the laws of classical optics, the accuracy of the
microscope depends on both the wave length and the aperture angle;
Abbe's criterium for its "resolving power", i.e.,
the size of the smallest discernable details, gives 




\[\tag{6} 
\delta q \sim \frac{\lambda}{\sin \varepsilon}.\]


On the other hand, the direction of a scattered photon, when it enters
the microscope, is unknown within the angle \(\varepsilon\), rendering
the momentum change of the electron uncertain by an amount 



\[\tag{7} 
\delta p \sim \frac{h \sin \varepsilon}{\lambda}\]

leading again to the result
 (2).



Let us now analyse Heisenberg's argument in more detail. Note
that, even in this improved version, Heisenberg's argument is
incomplete. According to Heisenberg's "measurement=meaning
principle", one must also specify, in the given context, what
the meaning is of the phrase "momentum of the electron",
in order to make sense of the claim that this momentum is changed by
the position measurement. A solution to this problem can again be
found in the Chicago lectures (Heisenberg 1930: 15). Here, he assumes
that initially the momentum of the electron is precisely known, e.g.,
it has been measured in a previous experiment with an inaccuracy
\(\delta p\_{i}\), which may be arbitrarily small. Then, its position
is measured with inaccuracy \(\delta q\), and after this, its final
momentum is measured with an inaccuracy \(\delta p\_{f}\). All three
measurements can be performed with arbitrary precision. Thus, the
three quantities \(\delta p\_{i}, \delta q\), and \(\delta p\_{f}\) can
be made as small as one wishes. If we assume further that the initial
momentum has not changed until the position measurement, we can speak
of a definite momentum until the time of the position measurement.
Moreover we can give operational meaning to the idea that the momentum
is changed during the position measurement: the outcome of the second
momentum measurement (say \(p\_{f}\) will generally differ from the
initial value \(p\_{i}\). In fact, one can also show that this change
is discontinuous, by varying the time between the three
measurements.



Let us try to see, adopting this more elaborate set-up, if we can
complete Heisenberg's argument. We have now been able to give
empirical meaning to the "change of momentum" of the
electron, \(p\_{f} - p\_{i}\). Heisenberg's argument claims that
the order of magnitude of this change is at least inversely
proportional to the inaccuracy of the position measurement:




\[\tag{8} \abs{p\_{f} - p\_{i}} \delta q \sim h\]


However, can we now draw the conclusion that the momentum is only
imprecisely defined? Certainly not. Before the position measurement,
its value was \(p\_{i}\), after the measurement it is \(p\_{f}\). One
might, perhaps, claim that the value at the very instant of the
position measurement is not yet defined, but we could simply settle
this by a convention, e.g., we might assign the mean value \((p\_{i} +
p\_{f})/2\) to the momentum at this instant. But then, the momentum is
precisely determined at all instants, and Heisenberg's
formulation of the uncertainty principle no longer follows. The above
attempt of completing Heisenberg's argument thus overshoots its
mark.



A solution to this problem can again be found in the Chicago Lectures.
Heisenberg admits that position and momentum can be known exactly. He
writes:



> 
> 
> If the velocity of the electron is at first known, and the position
> then exactly measured, the position of the electron for times previous
> to the position measurement may be calculated. For these past times,
> \(\delta p\delta q\) is smaller than the usual bound. (Heisenberg
> 1930: 15)
> 
> 
> 



Indeed, Heisenberg says: "the uncertainty relation does not hold
for the past".



Apparently, when Heisenberg refers to the uncertainty or imprecision
of a quantity, he means that the value of this quantity cannot be
given *beforehand*. In the sequence of measurements we have
considered above, the uncertainty in the momentum after the
measurement of position has occurred, refers to the idea that the
value of the momentum is not fixed just *before* the final
momentum measurement takes place. Once this measurement is performed,
and reveals a value \(p\_{f}\), the uncertainty relation no longer
holds; these values then belong to the past. Clearly, then, Heisenberg
is concerned with *unpredictability*: the point is not that the
momentum of a particle changes, due to a position measurement, but
rather that it changes by an unpredictable amount. It is, however
always possible to measure, and hence define, the size of this change
in a subsequent measurement of the final momentum with arbitrary
precision.



Although Heisenberg admits that we can consistently attribute values
of momentum and position to an electron in the past, he sees little
merit in such talk. He points out that these values can never be used
as initial conditions in a prediction about the future behavior of the
electron, or subjected to experimental verification. Whether or not we
grant them physical reality is, as he puts it, a matter of personal
taste. Heisenberg's own taste is, of course, to deny their
physical reality. For example, he writes, 



> 
> 
> I believe that one can formulate the emergence of the classical
> "path" of a particle succinctly as follows: *the
> "path" comes into being only because we observe it*.
> (Heisenberg 1927: 185) 
> 
> 
> 



Apparently, in his view, a measurement does not only serve to give
meaning to a quantity, it *creates* a particular value for this
quantity. This may be called the "measurement=creation"
principle. It is an ontological principle, for it states what is
physically real.



This then leads to the following picture. First we measure the
momentum of the electron very accurately. By "measurement=
meaning", this entails that the term "the momentum of the
particle" is now well-defined. Moreover, by the
"measurement=creation" principle, we may say that this
momentum is physically real. Next, the position is measured with
inaccuracy \(\delta q\). At this instant, the position of the particle
becomes well-defined and, again, one can regard this as a physically
real attribute of the particle. However, the momentum has now changed
by an amount that is unpredictable by an order of magnitude 
\(\abs{p\_{f} - p\_{i}} \sim h/\delta q\). The meaning and validity of
this claim can be verified by a subsequent momentum measurement.



The question is then what status we shall assign to the momentum of
the electron just before its final measurement. Is it real? According
to Heisenberg it is not. Before the final measurement, the best we can
attribute to the electron is some unsharp, or fuzzy momentum. These
terms are meant here in an ontological sense, characterizing a real
attribute of the electron.


### 2.3 The interpretation of Heisenberg's uncertainty relations



Heisenberg's relations were soon considered to be a cornerstone
of the Copenhagen interpretation of quantum mechanics. Just a few
months later, Kennard (1927) already called them the "essential
core" of the new theory. Taken together with Heisenberg's
contention that they provide the intuitive content of the theory and
their prominent role in later discussions on the Copenhagen
interpretation, a dominant view emerged in which the uncertainty
relations were regarded as a fundamental principle of the theory.



The interpretation of these relations has often been debated. Do
Heisenberg's relations express restrictions on the experiments
we can perform on quantum systems, and, therefore, restrictions on the
information we can gather about such systems; or do they express
restrictions on the meaning of the concepts we use to describe quantum
systems? Or else, are they restrictions of an ontological nature,
i.e., do they assert that a quantum system simply does not possess a
definite value for its position and momentum at the same time? The
difference between these interpretations is partly reflected in the
various names by which the relations are known, e.g., as
"inaccuracy relations", or: "uncertainty",
"indeterminacy" or "unsharpness relations".
The debate between these views has been addressed by many authors, but
it has never been settled completely. Let it suffice here to make only
two general observations.



First, it is clear that in Heisenberg's own view all the above
questions stand or fall together. Indeed, we have seen that he adopted
an operational "measurement=meaning" principle according
to which the meaningfulness of a physical quantity was equivalent to
the existence of an experiment purporting to measure that quantity.
Similarly, his "measurement=creation" principle allowed
him to attribute physical reality to such quantities. Hence,
Heisenberg's discussions moved rather freely and quickly from
talk about experimental inaccuracies to epistemological or ontological
issues and back again.



However, ontological questions seemed to be of somewhat less interest
to him. For example, there is a passage (Heisenberg 1927: 197), where
he discusses the idea that, behind our observational data, there might
still exist a hidden reality in which quantum systems have definite
values for position and momentum, unaffected by the uncertainty
relations. He emphatically dismisses this conception as an unfruitful
and meaningless speculation, because, as he says, the aim of physics
is only to describe observable data. Similarly, in the Chicago
Lectures, he warns against the fact that the human language permits
the utterance of statements which have no empirical content, but
nevertheless produce a picture in our imagination. He notes, 



> 
> 
> One should be especially careful in using the words
> "reality", "actually", etc., since these words
> very often lead to statements of the type just mentioned. (Heisenberg
> 1930: 11) 
> 
> 
> 



So, Heisenberg also endorsed an interpretation of his relations as
rejecting a reality in which particles have simultaneous definite
values for position and momentum.



The second observation is that although for Heisenberg experimental,
informational, epistemological and ontological formulations of his
relations were, so to say, just different sides of the same coin, this
is not so for those who do not share his operational principles or his
view on the task of physics. Alternative points of view, in which
e.g., the ontological reading of the uncertainty relations is denied,
are therefore still viable. The statement, often found in the
literature of the thirties, that Heisenberg had *proved* the
impossibility of associating a definite position and momentum to a
particle is certainly wrong. But the precise meaning one can
coherently attach to Heisenberg's relations depends rather
heavily on the interpretation one favors for quantum mechanics as a
whole. And because no agreement has been reached on this latter issue,
one cannot expect agreement on the meaning of the uncertainty
relations either.


### 2.4 Uncertainty relations or uncertainty principle?



Let us now move to another question about Heisenberg's
relations: do they express a *principle* of quantum theory?
Probably the first influential author to call these relations a
"principle" was Eddington, who, in his Gifford Lectures of
1928 referred to them as the "Principle of Indeterminacy".
In the English literature the name uncertainty principle became most
common. It is used both by Condon and Robertson in 1929, and also in
the English version of Heisenberg's Chicago Lectures (Heisenberg
1930), although, remarkably, nowhere in the original German version of
the same book (see also Cassidy 1998). Indeed, Heisenberg never seems
to have endorsed the name "principle" for his relations.
His favourite terminology was "inaccuracy relations"
(*Ungenauigkeitsrelationen*) or "indeterminacy
relations" (*Unbestimmtheitsrelationen*). We know only
one passage, in Heisenberg's own Gifford lectures, delivered in
1955-56 (Heisenberg 1958: 43), where he mentioned that his
relations "are usually called relations of uncertainty or
principle of indeterminacy". But this can well be read as his
yielding to common practice rather than his own preference.



But does the relation
 (2)
 qualify as a principle of quantum mechanics? Several authors,
foremost Karl Popper (1967), have contested this view. Popper argued
that the uncertainty relations cannot be granted the status of a
principle on the grounds that they are derivable from the theory,
whereas one cannot obtain the theory from the uncertainty relations.
(The argument being that one can never derive any equation, say, the
Schrodinger equation, or the commutation relation
 (1),
 from an inequality.)



Popper's argument is, of course, correct but we think it misses
the point. There are many statements in physical theories which are
called principles even though they are in fact derivable from other
statements in the theory in question. A more appropriate departing
point for this issue is not the question of logical priority but
rather Einstein's distinction between "constructive
theories" and "principle theories".



Einstein proposed this famous classification in Einstein 1919.
Constructive theories are theories which postulate the existence of
simple entities behind the phenomena. They endeavour to reconstruct
the phenomena by framing hypotheses about these entities. Principle
theories, on the other hand, start from empirical principles, i.e.,
general statements of empirical regularities, employing no or only a
bare minimum of theoretical terms. The purpose is to build up the
theory from such principles. That is, one aims to show how these
empirical principles provide sufficient conditions for the
introduction of further theoretical concepts and structure.



The prime example of a theory of principle is thermodynamics. Here the
role of the empirical principles is played by the statements of the
impossibility of various kinds of perpetual motion machines. These are
regarded as expressions of brute empirical fact, providing the
appropriate conditions for the introduction of the concepts of energy
and entropy and their properties. (There is a lot to be said about the
tenability of this view, but that is not our topic here.)



Now obviously, once the formal thermodynamic theory is built, one can
also *derive* the impossibility of the various kinds of
perpetual motion. (They would violate the laws of energy conservation
and entropy increase.) But this derivation should not misguide one
into thinking that they were no principles of the theory after all.
The point is just that empirical principles are statements that do not
rely on the theoretical concepts (in this case entropy and energy) for
their meaning. They are interpretable independently of these concepts
and, further, their validity on the empirical level still provides the
physical content of the theory.



A similar example is provided by special relativity, another theory of
principle, which Einstein deliberately designed after the ideal of
thermodynamics. Here, the empirical principles are the light postulate
and the relativity principle. Again, once we have built up the modern
theoretical formalism of the theory (Minkowski space-time), it is
straightforward to prove the validity of these principles. But again
this does not count as an argument for claiming that they were no
principles after all. So the question whether the term
"principle" is justified for Heisenberg's relations,
should, in our view, be understood as the question whether they are
conceived of as empirical principles.



One can easily show that this idea was never far from
Heisenberg's intentions. We have already seen that Heisenberg
presented the relations as the result of a "pure fact of
experience". A few months after his 1927 paper, he wrote a
popular paper "*Uber die Grundprincipien der
Quantenmechanik*" ("On the fundamental principles of
quantum mechanics") where he made the point even more clearly.
Here Heisenberg described his recent break-through in the
interpretation of the theory as follows: "It seems to be a
general law of nature that we cannot determine position and velocity
simultaneously with arbitrary accuracy". Now actually, and in
spite of its title, the paper does not identify or discuss any
"fundamental principle" of quantum mechanics. So, it must
have seemed obvious to his readers that he intended to claim that the
uncertainty relation was a fundamental principle, forced upon us as an
empirical law of nature, rather than a result derived from the
formalism of the theory.



This reading of Heisenberg's intentions is corroborated by the
fact that, even in his 1927 paper, applications of his relation
frequently present the conclusion as a matter of principle. For
example, he says "In a stationary state of an atom its phase is
*in principle* indeterminate" (Heisenberg 1927: 177,
[emphasis added]). Similarly, in a paper of 1928, he described the
content of his relations as: 



> 
> 
> 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. (Heisenberg 1984: 26, [emphasis added])
> 
> 
> 



So, although Heisenberg did not originate the tradition of calling his
relations a principle, it is not implausible to attribute the view to
him that the uncertainty relations represent an empirical principle
that could serve as a foundation of quantum mechanics. In fact, his
1927 paper expressed this desire explicitly: 



> 
> 
> Surely, one would like to be able to deduce the quantitative laws of
> quantum mechanics directly from their *anschaulich*
> foundations, that is, essentially, relation
>  [(2)].
>  (*ibid*: 196) 
> 
> 
> 



This is not to say that Heisenberg was successful in reaching this
goal, or that he did not express other opinions on other
occasions.



Let us conclude this section with three remarks. First, if the
uncertainty relation is to serve as an empirical principle, one might
well ask what its direct empirical support is. In Heisenberg's
analysis, no such support is mentioned. His arguments concerned
thought experiments in which the validity of the theory, at least at a
rudimentary level, is implicitly taken for granted. Jammer (1974: 82)
conducted a literature search for high precision experiments that
could seriously test the uncertainty relations and concluded they were
still scarce in 1974. Real experimental support for the uncertainty
relations in experiments in which the inaccuracies are close to the
quantum limit have come about only more recently (see Kaiser, Werner,
and George 1983; Uffink 1985; Nairz, Andt, and Zeilinger 2002).



A second point is the question whether the theoretical structure or
the quantitative laws of quantum theory can indeed be derived on the
basis of the uncertainty principle, as Heisenberg wished. Serious
attempts to build up quantum theory as a full-fledged Theory of
Principle on the basis of the uncertainty principle have never been
carried out. Indeed, the most Heisenberg could and did claim in this
respect was that the uncertainty relations created "room"
(Heisenberg 1927: 180) or "freedom" (Heisenberg 1931: 43)
for the introduction of some non-classical mode of description of
experimental data, not that they uniquely lead to the formalism of
quantum mechanics. A serious proposal to approach quantum mechanics as
a theory of principle was provided more recently by Bub (2000) and
Chiribella & Spekkens (2016). But, remarkably, this proposal does
not use the uncertainty relation as one of its fundamental principles.
Third, it is remarkable that in his later years Heisenberg put a
somewhat different gloss on his relations. In his autobiography
*Der Teil und das Ganze* of 1969 he described how he had found
his relations inspired by a remark by Einstein that "it is the
theory which decides what one can observe"--thus giving
precedence to theory above experience, rather than the other way
around. Some years later he even admitted that his famous discussions
of thought experiments were actually trivial since 



> 
> 
> ... if the process of observation itself is subject to the laws
> of quantum theory, it must be possible to represent its result in the
> mathematical scheme of this theory. (Heisenberg 1975: 6)
> 
> 
> 


### 2.5 Mathematical elaboration



When Heisenberg introduced his relation, his argument was based only
on qualitative examples. He did not provide a general, exact
derivation of his
 relations.[3]
 Indeed, he did not even give a definition of the uncertainties
\(\delta q\), etc., occurring in these relations. Of course, this was
consistent with the announced goal of that paper, i.e., to provide
some qualitative understanding of quantum mechanics for simple
experiments.



The first mathematically exact formulation of the uncertainty
relations is due to Kennard. He proved in 1927 the theorem that for
all normalized state vectors \(\ket{\psi}\) the following
inequality holds:


 
\[\tag{9} 
\Delta\_{\psi}\bP \Delta\_{\psi}\bQ \ge \hslash/2 \]


Here, \(\Delta\_{\psi}\bP\) and
\(\Delta\_{\psi}\bQ\) are standard deviations of position
and momentum in the state vector \(\ket{\psi}\), i.e.,



\[\tag{10}
\begin{align\*}
(\Delta\_{\psi}\bP)^2 &= 
 \expval{\bP^2}\_{\psi} - \expval{\bP}\_{\psi}^2 \\
(\Delta\_{\psi}\bQ)^2 &= 
 \expval{\bQ^2}\_{\psi} - \expval{\bQ}\_{\psi}^2
\end{align\*}\]


 where \(\expval{\cdot}\_{\psi} = \expvalexp{\cdot}{\psi}\)
denotes the expectation value
in state \(\ket{\psi}\). Equivalently we can use the wave
function \(\psi(q)\) and its Fourier transform: 



\[\begin{align\*} 
\tag{11} \psi(q) &= \braket{q}{\psi} \\
\notag \tilde{\psi}(p) & = \braket{p}{\psi} 
 =\frac{1}{\sqrt{2\pi \hbar} }\int \! \! dq\, e^{-ipq/\hbar} \psi(q) 
\end{align\*}\]

 to write



\[\begin{align\*} 
(\Delta\_\psi {\bQ})^2 & = \! \int\!\! dq\, \abs{\psi(q)}^2 q^2 - 
 \left(\int \!\!dq \, \abs{\psi(q)}^2 q \right)^2 \\ 
(\Delta\_\psi {\bP})^2 & = \! \int \!\!dp \, \abs{\tilde{\psi}(p)}^2 p^2 - 
 \left(\int\!\!dp \, \abs{\tilde{\psi}(p)}^2 p \right)^2 
\end{align\*}\]

 The inequality
 (9)
 was generalized by Robertson (1929) who proved that for all
observables (self-adjoint operators) \(\bA\) and
\(\bB\): 



\[\tag{12} 
\Delta \_{\psi}\bA \Delta\_{\psi}\bB \ge 
 \frac{1}{2} \abs{\expval{[\bA,\bB]}\_{\psi}}
\]

 where \([\bA,\bB] := \bA\bB - \bB\bA\) denotes the commutator.



Since the above inequalities
 (9)
 and
 (12)
 have the virtue of being exact, in contrast to Heisenberg's
original semi-quantitative formulation, it is tempting to regard them
as the exact counterpart of Heisenberg's relations
 (2)-(4).
 Indeed, such was Heisenberg's own view. In his Chicago Lectures
(Heisenberg 1930: 15-19), he presented Kennard's
derivation of relation
 (9)
 and claimed that "this proof does not differ at all in
mathematical content" from his semi-quantitative argument, the
only difference being that now "the proof is carried through
exactly".



But it may be useful to point out that both in status and intended
role there is a difference between Kennard's inequality and
Heisenberg's previous formulation
 (2).
 The inequalities discussed here are not statements of empirical fact,
but theorems of the quantum mechanical formalism. As such, they
presuppose the validity of this formalism, and in particular the
commutation relation
 (1),
 rather than elucidating its intuitive content or to create
"room" or "freedom" for the validity of this
formalism. At best, one should see the above inequalities as showing
that the formalism is consistent with Heisenberg's empirical
principle.



This situation is similar to that arising in other theories of
principle where, as noted in
 Section 2.4,
 one often finds that, next to an empirical principle, the formalism
also provides a corresponding theorem. And similarly, this situation
should not, by itself, cast doubt on the question whether
Heisenberg's relation can be regarded as a principle of quantum
mechanics.



There is a second notable difference between
 (2)
 and
 (9).
 Heisenberg did not give a general definition for the
"uncertainties" \(\delta p\) and \(\delta q\). The most
definite remark he made about them was that they could be taken as
"something like the mean error". In the discussions of
thought experiments, he and Bohr would always quantify uncertainties
on a case-to-case basis by choosing some parameters which happened to
be relevant to the experiment at hand. By contrast, the inequalities
 (9)
 and
 (12)
 employ a single specific expression as a measure for
"uncertainty": the standard deviation. At the time, this
choice was not unnatural, given that this expression is well-known and
widely used in error theory and the description of statistical
fluctuations. However, there was very little or no discussion of
whether this choice was appropriate for a general formulation of the
uncertainty relations. A standard deviation reflects the spread or
expected fluctuations in a series of measurements of an observable in
a given state. It is not at all easy to connect this idea with the
concept of the "inaccuracy" of a measurement, such as the
resolving power of a microscope. In fact, even though Heisenberg had
taken Kennard's inequality as the precise formulation of the
uncertainty relation, he and Bohr never relied on standard deviations
in their many discussions of thought experiments, and indeed, it has
been shown (Uffink and Hilgevoord 1985; Hilgevoord and Uffink 1988)
that these discussions cannot be framed in terms of standard
deviations.



Another problem with the above elaboration is that the
"well-known" relations
 (5)
 are actually false if energy \(\boldsymbol{E}\) and action
\(\boldsymbol{J}\) are to be positive operators (Jordan 1927). In that
case, self-adjoint operators \(\boldsymbol{t}\) and \(\boldsymbol{w}\)
do not exist and inequalities analogous to
 (9)
 cannot be derived. Also, these inequalities do not hold for angle and
angular momentum (Uffink 1990). These obstacles have led to a quite
extensive literature on time-energy and angle-action uncertainty
relations (Busch 1990; Hilgevoord 1996, 1998, 2005; Muga et al. 2002;
Hilgevoord and Atkinson 2011; Pashby 2015).


## 3. Bohr



In spite of the fact that Heisenberg's and Bohr's views on
quantum mechanics are often lumped together as (part of) "the
Copenhagen interpretation", there is considerable difference
between their views on the uncertainty relations.


### 3.1 From wave-particle duality to complementarity



Long before the development of modern quantum mechanics, Bohr had been
particularly concerned with the problem of particle-wave duality,
i.e., the problem that experimental evidence on the behaviour of both
light and matter seemed to demand a wave picture in some cases, and a
particle picture in others. Yet these pictures are mutually exclusive.
Whereas a particle is always localized, the very definition of the
notions of wavelength and frequency requires an extension in space and
in time. Moreover, the classical particle picture is incompatible with
the characteristic phenomenon of interference.



His long struggle with wave-particle duality had prepared him for a
radical step when the dispute between matrix and wave mechanics broke
out in 1926-27. For the main contestants, Heisenberg and
Schrodinger, the issue at stake was which view could claim to
provide a single coherent and universal framework for the description
of the observational data. The choice was, essentially between a
description in terms of continuously evolving waves, or else one of
particles undergoing discontinuous quantum jumps. By contrast, Bohr
insisted that elements from both views were equally valid and equally
needed for an exhaustive description of the data. His way out of the
contradiction was to renounce the idea that the pictures refer, in a
literal one-to-one correspondence, to physical reality. Instead, the
applicability of these pictures was to become dependent on the
experimental context. This is the gist of the viewpoint he called
"complementarity".



Bohr first conceived the general outline of his complementarity
argument in early 1927, during a skiing holiday in Norway, at the same
time when Heisenberg wrote his uncertainty paper. When he returned to
Copenhagen and found Heisenberg's manuscript, they got into an
intense discussion. On the one hand, Bohr was quite enthusiastic about
Heisenberg's ideas which seemed to fit wonderfully with his own
thinking. Indeed, in his subsequent work, Bohr always presented the
uncertainty relations as the symbolic expression of his
complementarity viewpoint. On the other hand, he criticized Heisenberg
severely for his suggestion that these relations were due to
discontinuous changes occurring during a measurement process. Rather,
Bohr argued, their proper derivation should start from the
indispensability of both particle and wave concepts. He pointed out
that the uncertainties in the experiment did not exclusively arise
from the discontinuities but also from the fact that in the experiment
we need to take into account both the particle theory and the wave
theory. It is not so much the unknown disturbance which renders the
momentum of the electron uncertain but rather the fact that the
position and the momentum of the electron cannot be simultaneously
defined in this experiment (see the "Addition in Proof" to
Heisenberg's paper).



We shall not go too deeply into the matter of Bohr's
interpretation of quantum mechanics since we are mostly interested in
Bohr's view on the uncertainty principle. For a more detailed
discussion of the former we refer to Scheibe (1973), Folse (1985),
Honner (1987) and Murdoch (1987). It may be useful, however, to sketch
some of the main points. Central in Bohr's considerations is
the *language* we use in physics. No matter how abstract and
subtle the concepts of modern physics may be, they are essentially an
extension of our ordinary language and a means to communicate the
results of our experiments. These results, obtained under
well-defined experimental circumstances, are what Bohr calls the
"phenomena". A phenomenon is "the comprehension of
the effects observed under given experimental conditions" (Bohr
1939: 24), it is the resultant of a physical object, a measuring
apparatus and the interaction between them in a concrete experimental
situation. The essential difference between classical and quantum
physics is that in quantum physics the interaction between the object
and the apparatus cannot be made arbitrarily small; the interaction
must at least comprise one quantum. This is expressed by Bohr's
quantum postulate:



> 
> 
> [... the] essence [of the formulation of the quantum theory] may
> be expressed in the so-called quantum postulate, which attributes to
> any atomic process an essential discontinuity or rather individuality,
> completely foreign to classical theories and symbolized by
> Planck's quantum of action. (Bohr 1928: 580)
> 
> 
> 



A phenomenon, therefore, is an indivisible whole and the result of a
measurement cannot be considered as an autonomous manifestation of the
object itself independently of the measurement context. The quantum
postulate forces upon us a new way of describing physical
phenomena:



> 
> 
> In this situation, we are faced with the necessity of a radical
> revision of the foundation for the description and explanation of
> physical phenomena. Here, it must above all be recognized that,
> however far quantum effects transcend the scope of classical physical
> analysis, the account of the experimental arrangement and the record
> of the observations must always be expressed in common language
> supplemented with the terminology of classical physics. (Bohr 1948:
> 313)
> 
> 
> 



This is what Scheibe (1973) has called the "buffer
postulate" because it prevents the quantum from penetrating into
the classical description: A phenomenon must always be described in
classical terms; Planck's constant does not occur in this
description.



Together, the two postulates induce the following reasoning. In every
phenomenon the interaction between the object and the apparatus
comprises at least one quantum. But the description of the phenomenon
must use classical notions in which the quantum of action does not
occur. Hence, the interaction cannot be analysed in this description.
On the other hand, the classical character of the description allows
us to speak in terms of the object itself. Instead of saying:
"the interaction between a particle and a photographic plate has
resulted in a black spot in a certain place on the plate", we
are allowed to forgo mentioning the apparatus and say: "the
particle has been found in this place". The experimental
context, rather than changing or disturbing pre-existing properties of
the object, defines what can meaningfully be said about the
object.



Because the interaction between object and apparatus is left out in
our description of the phenomenon, we do not get the whole picture.
Yet, any attempt to extend our description by performing the
measurement of a different observable quantity of the object, or
indeed, on the measurement apparatus, produces a new phenomenon and we
are again confronted with the same situation. Because of the
unanalyzable interaction in both measurements, the two descriptions
cannot, generally, be united into a single picture. They are what Bohr
calls complementary descriptions:



> 
> 
> [the quantum of action]...forces us to adopt a new mode of
> description designated as complementary in the sense that any given
> application of classical concepts precludes the simultaneous use of
> other classical concepts which in a different connection are equally
> necessary for the elucidation of the phenomena. (Bohr 1929: 10)
> 
> 
> 



The most important example of complementary descriptions is provided
by the measurements of the position and momentum of an object. If one
wants to measure the position of the object relative to a given
spatial frame of reference, the measuring instrument must be rigidly
fixed to the bodies which define the frame of reference. But this
implies the impossibility of investigating the exchange of momentum
between the object and the instrument and we are cut off from
obtaining any information about the momentum of the object. If, on the
other hand, one wants to measure the momentum of an object the
measuring instrument must be able to move relative to the spatial
reference frame. Bohr here assumes that a momentum measurement
involves the registration of the recoil of some movable part of the
instrument and the use of the law of momentum conservation. The
looseness of the part of the instrument with which the object
interacts entails that the instrument cannot serve to accurately
determine the position of the object. Since a measuring instrument
cannot be rigidly fixed to the spatial reference frame and, at the
same time, be movable relative to it, the experiments which serve to
precisely determine the position and the momentum of an object are
mutually exclusive. Of course, in itself, this is not at all typical
for quantum mechanics. But, because the interaction between object and
instrument during the measurement can neither be neglected nor
determined the two measurements cannot be combined. This means that in
the description of the object one must choose between the assignment
of a precise position or of a precise momentum.



Similar considerations hold with respect to the measurement of time
and energy. Just as the spatial coordinate system must be fixed by
means of solid bodies so must the time coordinate be fixed by means of
unperturbed, synchronised clocks. But it is precisely this requirement
which prevents one from taking into account of the exchange of energy
with the instrument if this is to serve its purpose. Conversely, any
conclusion about the object based on the conservation of energy
prevents following its development in time.



The conclusion is that in quantum mechanics we are confronted with a
complementarity between two descriptions which are united in the
classical mode of description: the space-time description (or
coordination) of a process and the description based on the
applicability of the dynamical conservation laws. The quantum forces
us to give up the classical mode of description (also called the
"causal" mode of description by
 Bohr[4]:
 it is impossible to form a classical picture of what is going on when
radiation interacts with matter as, e.g., in the Compton effect.



> 
> 
> Any arrangement suited to study the exchange of energy and momentum
> between the electron and the photon must involve a latitude in the
> space-time description sufficient for the definition of wave-number
> and frequency which enter in the relation [\(E = h\nu\) and \(p =
> h\sigma\)]. Conversely, any attempt of locating the collision between
> the photon and the electron more accurately would, on account of the
> unavoidable interaction with the fixed scales and clocks defining the
> space-time reference frame, exclude all closer account as regards the
> balance of momentum and energy. (Bohr 1949: 210)
> 
> 
> 



A causal description of the process cannot be attained; we have to
content ourselves with complementary descriptions. "The
viewpoint of complementarity may be regarded", according to
Bohr, "as a rational generalization of the very ideal of
causality".



In addition to complementary descriptions Bohr also talks about
complementary phenomena and complementary quantities. Position and
momentum, as well as time and energy, are complementary
 quantities.[5]




We have seen that Bohr's approach to quantum theory puts heavy
emphasis on the language used to communicate experimental
observations, which, in his opinion, must always remain classical. By
comparison, he seemed to put little value on arguments starting from
the mathematical formalism of quantum theory. This informal approach
is typical of all of Bohr's discussions on the meaning of
quantum mechanics. One might say that for Bohr the conceptual
clarification of the situation has primary importance while the
formalism is only a symbolic representation of this situation.



This is remarkable since, finally, it is the formalism which needs to
be interpreted. This neglect of the formalism is one of the reasons
why it is so difficult to get a clear understanding of Bohr's
interpretation of quantum mechanics and why it has aroused so much
controversy. We close this section by citing from an article of 1948
to show how Bohr conceived the role of the formalism of quantum
mechanics:



> 
> 
> The entire formalism is to be considered as a tool for deriving
> predictions, of definite or statistical character, as regards
> information obtainable under experimental conditions described in
> classical terms and specified by means of parameters entering into the
> algebraic or differential equations of which the matrices or the
> wave-functions, respectively, are solutions. These symbols themselves,
> as is indicated already by the use of imaginary numbers, are not
> susceptible to pictorial interpretation; and even derived real
> functions like densities and currents are only to be regarded as
> expressing the probabilities for the occurrence of individual events
> observable under well-defined experimental conditions. (Bohr 1948:
> 314)
> 
> 
> 


### 3.2 Bohr's view on the uncertainty relations



In his Como lecture, published in 1928, Bohr gave his own version of a
derivation of the uncertainty relations between position and momentum
and between time and energy. He started from the relations



\[\tag{13} E = h\nu \text{ and } p = h/\lambda\]


 which connect the notions of energy \(E\) and momentum
\(p\) from the particle picture with those of frequency \(\nu\) and
wavelength \(\lambda\) from the wave picture. He noticed that a wave
packet of limited extension in space and time can only be built up by
the superposition of a number of elementary waves with a large range
of wave numbers and frequencies. Denoting the spatial and temporal
extensions of the wave packet by \(\Delta x\) and \(\Delta t\), and
the extensions in the wave number \(\sigma := 1/\lambda\) and
frequency by \(\Delta \sigma\) and \(\Delta \nu\), it follows from
Fourier analysis that in the most favorable case \(\Delta x \Delta
\sigma \approx \Delta t \Delta \nu \approx 1\), and, using (13), one
obtains the relations 



\[\tag{14} \Delta t \Delta E \approx \Delta x \Delta p \approx h\]


Note that \(\Delta x, \Delta \sigma\), etc., are not standard
deviations but unspecified measures of the size of a wave packet. (The
original text has equality signs instead of approximate equality
signs, but, since Bohr does not define the spreads exactly the use of
approximate equality signs seems more in line with his intentions.
Moreover, Bohr himself used approximate equality signs in later
presentations.) These equations determine, according to Bohr: 



> 
> 
> the highest possible accuracy in the definition of the energy and
> momentum of the individuals associated with the wave field. (Bohr
> 1928: 571).
> 



He noted, 



> 
> 
> This circumstance may be regarded as a simple symbolic expression of
> the complementary nature of the space-time description and the claims
> of causality.
>  (*ibid*).[6]
> 
> 
> 
> 



We note a few points about Bohr's view on the uncertainty
relations. First of all, Bohr does not refer to *discontinuous
changes* in the relevant quantities during the measurement
process. Rather, he emphasizes the possibility of *defining*
these quantities. This view is markedly different from
Heisenberg's view. A draft version of the Como lecture is even
more explicit on the difference between Bohr and Heisenberg:



> 
> 
> These reciprocal uncertainty relations were given in a recent paper of
> Heisenberg as the expression of the statistical element which, due to
> the feature of discontinuity implied in the quantum postulate,
> characterizes any interpretation of observations by means of classical
> concepts. It must be remembered, however, that the uncertainty in
> question is not simply a consequence of a discontinuous change of
> energy and momentum say during an interaction between radiation and
> material particles employed in measuring the space-time coordinates of
> the individuals. According to the above considerations the question is
> rather that of the impossibility of defining rigorously such a change
> when the space-time coordination of the individuals is also
> considered. (Bohr 1985: 93)
> 
> 
> 



Indeed, Bohr not only rejected Heisenberg's argument that these
relations are due to discontinuous disturbances implied by the act of
measuring, but also his view that the measurement process
*creates* a definite result:



> 
> 
> The unaccustomed features of the situation with which we are
> confronted in quantum theory necessitate the greatest caution as
> regard all questions of terminology. Speaking, as it is often done of
> disturbing a phenomenon by observation, or even of creating physical
> attributes to objects by measuring processes is liable to be
> confusing, since all such sentences imply a departure from conventions
> of basic language which even though it can be practical for the sake
> of brevity, can never be unambiguous. (Bohr 1939: 24)
> 
> 
> 



Nor did he approve of an epistemological formulation or one in terms
of experimental inaccuracies:



> 
> 
> [...] a sentence like "we cannot know both the momentum and
> the position of an atomic object" raises at once questions as to
> the physical reality of two such attributes of the object, which can
> be answered only by referring to the mutual exclusive conditions for
> an unambiguous use of space-time concepts, on the one hand, and
> dynamical conservation laws on the other hand. (Bohr 1948: 315; also
> Bohr 1949: 211)
> 
> 
> 
> It would in particular not be out of place in this connection to warn
> against a misunderstanding likely to arise when one tries to express
> the content of Heisenberg's well-known indeterminacy relation by
> such a statement as "the position and momentum of a particle
> cannot simultaneously be measured with arbitrary accuracy".
> According to such a formulation it would appear as though we had to do
> with some arbitrary renunciation of the measurement of either the one
> or the other of two well-defined attributes of the object, which would
> not preclude the possibility of a future theory taking both attributes
> into account on the lines of the classical physics. (Bohr 1937:
> 292)
> 
> 
> 



Instead, Bohr always stressed that the uncertainty relations are first
and foremost an expression of complementarity. This may seem odd since
complementarity is a dichotomic relation between two types of
description whereas the uncertainty relations allow for intermediate
situations between two extremes. They "express" the
dichotomy in the sense that if we take the energy and momentum to be
perfectly well-defined, symbolically \(\Delta E = \Delta p\) = 0, the
position and time variables are completely undefined, \(\Delta x =
\Delta t = \infty\), and vice versa. But they also allow intermediate
situations in which the mentioned uncertainties are all non-zero and
finite. This more positive aspect of the uncertainty relation is
mentioned in the Como lecture:



> 
> 
> At the same time, however, the general character of this relation
> makes it possible to a certain extent to reconcile the conservation
> laws with the space-time coordination of observations, the idea of a
> coincidence of well-defined events in space-time points being replaced
> by that of unsharply defined individuals within space-time regions.
> (Bohr 1928: 571)
> 
> 
> 



However, Bohr never followed up on this suggestion that we might be
able to strike a compromise between the two mutually exclusive modes
of description in terms of unsharply defined quantities. Indeed, an
attempt to do so, would take the formalism of quantum theory more
seriously than the concepts of classical language, and this step Bohr
refused to take. Instead, in his later writings he would be content
with stating that the uncertainty relations simply defy an unambiguous
interpretation in classical terms:



> 
> 
> These so-called indeterminacy relations explicitly bear out the
> limitation of causal analysis, but it is important to recognize that
> no unambiguous interpretation of such a relation can be given in words
> suited to describe a situation in which physical attributes are
> objectified in a classical way. (Bohr 1948: 315)
> 
> 
> 



Finally, on a more formal level, we note that Bohr's derivation
does not rely on the commutation relations
 (1)
 and
 (5),
 but on Fourier analysis. These two approaches are equivalent as far
as the relationship between position and momentum is concerned, but
this is not so for time and energy since most physical systems do not
have a time operator. Indeed, in his discussion with Einstein (Bohr
1949), Bohr considered time as a simple classical variable. This even
holds for his famous discussion of the "clock-in-the-box"
thought-experiment where the time, as defined by the clock in the box,
is treated from the point of view of classical general relativity.
Thus, in an approach based on commutation relations, the
position-momentum and time-energy uncertainty relations are not on
equal footing, which is contrary to Bohr's approach in terms of
Fourier analysis. For more details see (Hilgevoord 1996 and 1998).


## 4. The Minimal Interpretation



In the previous two sections we have seen how both Heisenberg and Bohr
attributed a far-reaching status to the uncertainty relations. They
both argued that these relations place fundamental limits on the
applicability of the usual classical concepts. Moreover, they both
believed that these limitations were inevitable and forced upon us.
However, we have also seen that they reached such conclusions by
starting from radical and controversial assumptions. This entails, of
course, that their radical conclusions remain unconvincing for those
who reject these assumptions. Indeed, the operationalist-positivist
viewpoint adopted by these authors has long since lost its appeal
among philosophers of physics.



So the question may be asked what alternative views of the uncertainty
relations are still viable. Of course, this problem is intimately
connected with that of the interpretation of the wave function, and
hence of quantum mechanics as a whole. Since there is no consensus
about the latter, one cannot expect consensus about the interpretation
of the uncertainty relations either. Here we only describe a point of
view, which we call the "minimal interpretation", that
seems to be shared by both the adherents of the Copenhagen
interpretation and of other views.



In quantum mechanics a system is supposed to be described by its wave
function, also called its quantum state or state vector. Given the
state vector \(\ket{\psi}\), one can derive probability
distributions for all the physical quantities pertaining to the
system, usually called its observables, such as its position,
momentum, angular momentum, energy, etc. The operational meaning of
these probability distributions is that they correspond to the
distribution of the values obtained for these quantities in a long
series of repetitions of the measurement. More precisely, one imagines
a great number of copies of the system under consideration, all
prepared in the same way. On each copy the momentum, say, is measured.
Generally, the outcomes of these measurements differ and a
distribution of outcomes is obtained. The theoretical momentum
distribution derived from the quantum state is supposed to coincide
with the hypothetical distribution of outcomes obtained in an infinite
series of repetitions of the momentum measurement. The same holds,
*mutatis mutandis*, for all the other physical quantities
pertaining to the system. Note that no simultaneous measurements of
two or more quantities are required in defining the operational
meaning of the probability distributions.



The uncertainty relations discussed above can be considered as
statements about the spreads of the probability distributions of the
several physical quantities arising from the same state. For example,
the uncertainty relation between the position and momentum of a system
may be understood as the statement that the position and momentum
distributions cannot both be arbitrarily narrow--in some sense of
the word "narrow"--in any quantum state. Inequality
 (9)
 is an example of such a relation in which the standard deviation is
employed as a measure of spread. From this characterization of
uncertainty relations follows that a more detailed interpretation of
the quantum state than the one given in the previous paragraph is not
required to study uncertainty relations as such. In particular, a
further ontological or linguistic interpretation of the notion of
uncertainty, as limits on the applicability of our concepts given by
Heisenberg or Bohr, need not be supposed.



Of course, this minimal interpretation leaves the question open
whether it makes sense to attribute precise values of position and
momentum to an individual system. Some interpretations of quantum
mechanics, e.g., those of Heisenberg and Bohr, deny this; while
others, e.g., the interpretation of de Broglie and Bohm insist that
each individual system has a definite position and momentum (see the
entry on
 Bohmian mechanics).
 The only requirement is that, as an empirical fact, it is not
possible to prepare pure ensembles in which all systems have the same
values for these quantities, or ensembles in which the spreads are
smaller than allowed by quantum theory. Although interpretations of
quantum mechanics, in which each system has a definite value for its
position and momentum are still viable, this is not to say that they
are without strange features of their own; they do not imply a return
to classical physics.



We end with a few remarks on this minimal interpretation. First, it
may be noted that the minimal interpretation of the uncertainty
relations is little more than filling in the empirical meaning of
inequality
 (9).
 As such, this view shares many of the limitations we have noted above
about this inequality. Indeed, it is not straightforward to relate the
spread in a statistical distribution of measurement results with the
*inaccuracy* of this measurement, such as, e.g., the resolving
power of a microscope, or of a *disturbance* of the system by
the measurement. Moreover, the minimal interpretation does not address
the question whether one can make *simultaneous* accurate
measurements of position and momentum.



As a matter of fact, one can show that the standard formalism of
quantum mechanics does not allow such simultaneous measurements. But
this is not a consequence of relation
 (9).
 Rather, it follows from the fact that this formalism simply does not
contain any observable that would accomplish such a task. The
extension of this formalism that allows observables to be represented
by positive-operator-valued measures or POVM's, does allow the
formal introduction of observables describing joint measurements (see
also
 section 6.1).
 But even here, for the case of position and momentum, one finds that
such measurements have to be "unsharp", which entails that
they cannot be regarded as simultaneous accurate measurements.



If one feels that statements about inaccuracy of measurement, or the
possibility of simultaneous measurements, belong to any satisfactory
formulation of the uncertainty principle, one will need to look for
other formulations of the uncertainty principle. Some candidates for
such formulations will be discussed in
 Section 6.
 First, however, we will look at formulations of the uncertainty
principle that stay firmly within the minimal interpretation, and
differ from
 (9)
 only by using measures of uncertainty other than the standard
deviation.


## 5. Alternative measures of uncertainty



While the standard deviation is the most well-known quantitative
measure for uncertainty or the spread in the probability distribution,
it is not the only one, and indeed it has distinctive drawbacks that
other such measures may lack. For example, in the definition of the
standard deviations
 (11)
 one can see that that the probability density function 
\(\abs{\psi(q)}^2\) is weighed by a quadratic factor \(q^2\) that
puts increasing emphasis on its tails. Therefore, the value of
\(\Delta\_\psi \bQ\) will depend predominantly at how this
density behaves at the tails: if these falls off very quickly, e.g.,
like a Gaussian, it will be small, but if the tails drop off only
slowly the standard deviation may be very large, even when most
probability is concentrated in a small interval.



The upshot of this objection is that having a lower bound on the
product of the standard deviations of position and momentum, as the
Heisenberg-Kennard uncertainty relation
 (9)
 gives, does not by itself rule out a state where *both* the
probability densities for position and momentum are extremely
concentrated, in the sense of having more than \((1- \epsilon)\) of
their probability concentrated in a a region of size smaller than
\(\delta\), for any choice of \(\epsilon, \delta >0\). This means,
in our view, that relation
 (9)
 actually fails to express what most physicists would take to be the
very core idea of the uncertainty principle.



One way to deal with this objection is to consider alternative
measures to quantify the spread or uncertainty associated with a
probability density. Here we discuss two such proposals.


### 5.1 Landau-Pollak uncertainty relations



The most straightforward alternative is to pick some value \(\alpha\)
close to one, say \(\alpha = 0.9\), and ask for the width of the
smallest interval that supports the fraction \(\alpha\) of the total
probability distribution in position and similarly for momentum:



\[\begin{align\*} 
\tag{15} W\_{\alpha}(\bQ, \psi) &:=
 \inf\_{\abs{I}} \left\{ I: \int\_I {\abs{\psi(q)}}^2 dq \geq \alpha \right\} \\ 
\notag W\_{\beta}(\bP,\psi) &:= \inf\_I \left\{\int\_I \abs{\tilde\psi(p)}^2 dp \geq \beta \right\} 
\end{align\*}\]


In a previous work (Uffink and Hilgevoord 1985) we called such
measures *bulk widths*, because they indicate how concentrated
the "bulk" (i.e., fraction \(\alpha\) or \(\beta\)) of the
probability distribution is. Landau and Pollak (1961) obtained an
uncertainty relation in terms of these bulk widths. 



\[\begin{align\*}
\tag{16} W\_\alpha (\bQ, \psi) W\_\beta (\bP, \psi) &\geq 
 2\pi \hbar \left( \alpha \beta - \sqrt{(1-\alpha)(1-\beta)} \right)^2 \\
\notag &\mbox{if } \alpha+ \beta \geq 1/2 
\end{align\*}\label{LP}\]


This Landau-Pollak inequality shows that if the choices of \(\alpha,
\beta\) are not too low, there is a state-independent lower bound on
the product of the bulk widths of the position and momentum
distribution for any quantum state.



Note that bulk widths are not so sensitive to the behavior of the
tails of the distributions and, therefore, the Landau-Pollak
inequality is immune to the objection above.Thus, this inequality
expresses constraints on quantum mechanical states not contained in
relation
 (9).
 Further, by the well-known Bienayme-Chebyshev inequality, one
has 



\[\begin{align\*} 
\tag{17} W\_\alpha (\bQ,\psi) &\leq \frac{2}{\sqrt {1- \alpha}} \Delta\_\psi \bQ \\ 
\notag W\_\beta (\bP, \psi) &\leq \frac{2}{\sqrt {1- \beta}} \Delta\_\psi \bP 
\end{align\*}\]


 so that inequality
 (16)
 implies (by choosing \(\alpha,\beta\) optimal) that \( \Delta\_\psi
\bQ \Delta\_\psi \bP \geq 0.12 \hbar \). This,
obviously, is not the best lower bound for the product of standard
deviations, but the important point is here that the Landau-Pollak
inequality
 (16)
 in terms of bulk widths *implies* the existence of a lower
bound on the product of standard deviations, while conversely, the
Heisenberg-Kennard equality
 (9)
 does *not imply* any bound on the product of bulk widths. A
generalization of this approach to non-commuting observables in a
finite-dimensional Hilbert space is discussed in Uffink 1990.


### 5.2 Entropic uncertainty relations



Another approach to express the uncertainty principle is to use
entropic measures of uncertainty. The foremost example of these is the
*Shannon entropy*, which for the position and momentum
distribution of a given state vector \(\ket{\psi}\) may be
defined as: 



\[\begin{align\*} 
\tag{18} H(\bQ, \psi) &:= -\int \abs{\psi(q)}^2 \ln \abs{\psi(q)}^2 dq \\ 
\notag H(\bP, \psi) &:= -\int \abs{\tilde{\psi}(p)}^2 \ln \abs{\tilde{\psi}(p)}^2 dp 
\end{align\*}\]


 One can then show (see Beckner 1975;
Bialinicki-Birula and Micielski 1975) that 



\[\tag{19} H(\bQ, \psi) + H(\bP,\psi) \geq \ln (e \pi \hbar) \]


A nice feature of this entropic uncertainty relation is that it
provides a strict improvement of the Heisenberg-Kennard relation. That
is to say, one can show (independently of quantum theory) that for any
probability density function \(p(x)\) 



\[\tag{20} -\int\! p(x) \ln p(x) dx \leq \ln (\sqrt{2 \pi e} \Delta x )\]



Applying this to the inequality
 (19)
 we get: 



\[\tag{21}
\frac{\hslash}{2} \leq(2\pi e)^{-1} \exp (H(\bQ, \psi) + H(\bP,\psi)) \leq 
 \Delta\_\psi \bQ\Delta\_\psi \bP
\]


 showing that the entropic uncertainty relation
implies the Heisenberg-Kennard uncertainty relation. A drawback of
this relation is that it does not completely evade the objection
mentioned above, (i.e., these entropic measures of uncertainty can
become as large as one pleases while \(1-\epsilon\) of the probability
in the distribution is concentrated on a very small interval), but the
examples needed to show this are admittedly more far-fetched.



For non-commuting observables in a \(n\)-dimensional Hilbert space,
one can similarly define an entropic uncertainty in the probability
distribution \(\abs{\braket{a\_i}{\psi}}^2\) for a
given state \(\ket{\psi}\) and a complete set of
eigenstates \(\ket{a\_i}\), \( (i= 1, \ldots n)\), of the
observable \(\bA\): 



\[\tag{22} 
H(\bA ,\psi) := -\sum\_{i=1}^n \abs{\braket{a\_i}{\psi}}^2 \ln \abs{\braket{a\_i}{\psi}}^2
\]


 and \(H(\bB,\psi)\) similarly in terms of the probability distribution 
\(\abs{\braket{b\_j}{\psi}}^2\) for a complete set
of eigenstates \(\ket{b\_j}\), (\(j =1, \ldots, n\)) of
observable \(\bB\). Then we obtain the uncertainty relation
(Maassen and Uffink 1988):



\[\tag{23} 
H( bA, \psi) + H(\bB, \psi) \geq 2 \ln \max\_{i,j} \abs{\braket{a\_i}{b\_j}},
\]


 which was further generalized and improved
by (Frank and Lieb 2012). The most important advantage of these
relations is that, in contrast to Robertson's inequality
 (12),
 the lower bound is a positive constant, independent of the state.


## 6. Uncertainty relations for inaccuracy and disturbance



Both the standard deviation and the alternative measures of
uncertainty considered in the previous subsection (and many more that
we have not mentioned!) are designed to indicate the width or spread
of a single given probability distribution. Applied to quantum
mechanics, where the probability distributions for position and
momentum are obtained from a given quantum state vector, one can use
them to formulate uncertainty relations that characterize the spread
in those distribution for any given state. The resulting inequalities
then express limitations on what state-preparations quantum mechanics
allows. They are thus expressions of what may be called a
*preparation uncertainty principle*:



> 
> 
> In quantum mechanics, it is impossible to prepare any system in a
> state \(\ket{\psi}\) such that its position and momentum
> are both precisely predictable, in the sense of having both the
> expected spread in a measurement of position and the expected spread
> in a momentum measurement arbitrarily small.
> 
> 
> 



The relations
 (9,
 16,
 19)
 all belong to this category; the mere difference being that they
employ different measures of spread: viz. the standard deviation, the
bulk width or the Shannon entropy.



Note that in this formulation, there is no reference to simultaneous
or joint measurements, nor to any notion of accuracy like the
resolving power of the measurement instrument, nor to the issue of how
much the system in the state that is being measured is
*disturbed* by this measurement. This section is devoted to
attempts that go beyond the mold of this preparation uncertain
principle.


### 6.1 The recent debate on error-disturbance relations



We have seen that in 1927 Heisenberg argued that the measurement of
(say) position must necessarily disturb the conjugate variable (i.e.,
momentum) by an amount that is inversely proportional to the
inaccuracy of measurement of the former. We have also seen that this
idea was not maintained in the Kennard's uncertainty relation
 (9),
 a relation that was embraced by Heisenberg (1930) and most textbooks.



A rather natural question thus arises whether there are further
inequalities in quantum mechanics that would address
Heisenberg's original thinking more directly, i.e., that do deal
with how much one variable is disturbed by the accurate measurement of
another. That is, we will look at attempts that would establish a
claim which may be called a *measurement uncertainty
principle*.



> 
> 
> In quantum mechanics, there is no measurement procedure by which one
> can accurately measure the position of a system without disturbing it
> momentum, in the sense that some measure of inaccuracy in position and
> some measure of the disturbance of momentum of the system by the
> measurement cannot both be arbitrarily small.
> 
> 
> 



This formulation of the uncertainty principle has always remained
controversial. Uncertainty relations that would express this alleged
principle are often called "error-disturbance" relations
or "noise-disturbance" relations We will look at two
recent proposals to search for such relations: Ozawa (2003) and Busch,
Lahti, and Werner (2013).



In Ozawa's approach, we assume that a system \(\cal S\) of
interest in state \(\ket{\psi}\) is coupled to a
measurement device \(\cal M\) in state \(\ket{\chi}\), and
their interaction is governed by a unitary operator \(U\). On the
Hilbert space of the joint system the observable \(\bQ\) of
the system \(\cal S\) we are interested in is represented by



\[\tag{24} \bQ\_{\rm in} = \bQ \otimes \mathbb{1}\]


 The measurement interaction will allow us to perform an
(inaccurate) measurement of this quantity by reading off a pointer
observable \(\boldsymbol{Q'}\) of the measurement device after the
interaction. Hence this inaccurate observable may be represented as



\[\tag{25}
\bQ'\_{\rm out} = U^\dagger( \mathbb{1} \otimes \bQ') U\]


The measure of noise in the measurement of \(\bQ\) is then
chosen as: 



\[\tag{26} 
\epsilon\_\psi(\bQ) := \expval{(\bQ'\_{\rm out} - \bQ\_{\rm in})^2}\_{\psi \otimes \chi}^{1/2}\]


A comparison of the initial momentum 
 \(\bP\_{\rm in} = \bP \otimes \mathbb{1}\) 
and the final momentum after the measurement 
 \(\bP\_{\rm out} = U^\dagger (\bP \otimes \mathbb{1})U\) is made by choosing 
a measure of the disturbance of \(\bP\) by the measurement procedure: 



\[\tag{27} 
\eta\_\psi(\bP):= \expval{(\bP\_{\rm in} - \bP\_{\rm out})^2}\_{\psi\otimes\chi}^{1/2}
\]


 Ozawa
obtained an inequality involving those two measures, which, however,
is more involved than previous uncertainty relations. For our
purposes, however, the important point is that Ozawa showed that the
product \(\epsilon\_\psi (\bQ) \eta\_\psi (\bP)\)
has no positive lower bound. His conclusion from this was that
Heisenberg's noise-disturbance relation is violated.



Yet, whether Ozawa's result indeed succeeds in formulating
Heisenberg's qualitative discussion of disturbance and accuracy
in the microscope example has come under dispute. See Busch, Lahti and
Werner (2013, and 2014 (Other Internet Resources)), and Ozawa (2013,
Other Internet Resources).



An objection raised in this dispute is that a quantity like
\(\expval{(\bQ'\_{\rm out} - \bQ\_{\rm in})^2}^{1/2}\) tells us very
little about how good the observable \({\bQ'}\_{\rm out}\) can stand in
as an inaccurate measurement of \(\bQ\_{\rm in}\). The main point to
observe here is that these operators generally do not commute, and
that measurements of \(\bQ'\_{\rm out}\), of \(\bQ\_{\rm in}\) and of
their difference will require altogether three different measurement
contexts. To require that \(\epsilon\_\psi(\bQ)\) vanishes, for
example, means only that the state prepared belongs to the linear
subspace corresponding to the zero eigenvalue of the operator 
\(\bQ'\_{\rm out} - {\bQ}\_{\rm in}\), and therefore that 
\(\expval{\bQ'\_{\rm out}}\_\psi = \expval{\bQ\_{\rm in}}\_\psi\), 
but this does not preclude that the probability
distribution of \(\bQ\_{\rm out}\) in state \(\psi\) might be wildly
different from that of \(\bQ\_{\rm in}\). But then no one would think
of \(\bQ\_{\rm out}\) as an accurate measurement of \(\bQ\_{\rm in}\)
so that the definition of \(\epsilon\_\psi(\bQ)\) does not express what
it is supposed to express. A similar objection can also be raised
against \(\eta\_\psi (\bP)\).



Another observation is that Ozawa's conclusion that there is no
lower bound for his error-disturbance product for is not at all
surprising. That is, even without probing the system by a measurement
apparatus, one can show that such a lower bound does not exist. If the
initial state of a system is prepared at time \(t=0\) as a Gaussian
quasi-monochromatic wave packet with \(\expval{\bQ\_0}\_\psi =0\) and 
evolves freely, we can use a time-of-flight
measurement to learn about its later position. Ehrenfest's
theorem tells us: \(\expval{\bQ\_t}\_\psi = \frac{t}{m} \expval{\bP}\_\psi\).



Hence, as an approximative measurement of the position
\(\bQ\_t\), one could propose the observable
\(\bQ'\_t = \frac{t}{m}\bP\). It is known that
under the stated conditions (and with \(m\) and \(t\) large) this
approximation holds very well, i.e., we do not only have 
\(\expval{\bQ'\_t -\bQ\_t}\_\psi =0\), but also 
\(\expval{(\bQ'\_t -\bQ\_t)^2} \approx 0\),
as nearly as we please. But since \(\bQ'\_t\) is just the
momentum multiplied by a constant, its measurement will obviously not
disturb the momentum of the system. In other words, for this example,
one has \(\epsilon\_\psi (\bQ)\) as small as we please with
zero disturbance of the momentum. Therefore, any hopes that there
could be a positive lower bound for the product \(\epsilon\_\psi
(\bQ) \eta\_\psi (\bP)\) seem to be dashed, even
with the simplest of measurement schemes, i.e. a free evolution.



Ozawa's results do not show that Heisenberg's analysis of
the microscope argument was wrong. Rather, they throw doubt on the
appropriateness of the definitions he used to formalize
Heisenberg's informal argument.



An entirely different analysis of the problem of substantiating a
measurement uncertainty relation was offered by Busch, Lahti, and
Werner (2013). These authors consider a measurement device \(\cal M\)
that makes a joint unsharp measurement of both position and momentum.
To describe such joint unsharp measurements, they employ the extended
modern formalism that characterizes obervables not by self-adjoint
operators but by positive-operator-valued measures (POVM's). In
the present case, this means that the measurement procedure is
characterized by a collection of positive operators, \(M(p,q)\), where
the pair \(p,q\) represent the outcome variables of the measurement,
with 



\[\tag{28}
M(p,q) \geq 0, \iint \! dp dq \, M(p,q) =\mathbb{1} .\]


The two marginals of this POVM, 



\[\tag{29}
\begin{align\*}
M\_1(q) &= \int\! dp M(p,q)\\
M\_2(p) &= \int\! dq M(p,q)
\end{align\*}
\]


 are also POVM's in
their own right and represent the unsharp position \(Q'\) and unsharp
momentum \(P'\) observables respectively. (Note that these do
*not* refer to a self-adjoint operator!)



For a system prepared in a state \(\ket{\psi}\), the joint
probability density of obtaining outcomes \((p,q)\) in the joint
unsharp measurement
 (28)
 is then 



\[\tag{30}
\rho(p,q) := \expvalexp{M(p,q)}{\psi},\]

 while the marginals of this joint probability
distribution give the distributions for \(Q'\) and \(P'\).



\[\begin{align\*} 
\tag{31} \mu'(q) &: = \int \! dp \, \rho(p,q) = \expvalexp{M\_1(q)}{\psi} \\
\notag \nu'(p) &:= \int \! dp \, \rho(p,q) = \expvalexp{M\_2(q)}{\psi} 
\end{align\*}\]


 Since a joint sharp measurement of position and momentum
is impossible in quantum mechanics, these marginal distributions
 (31)
 obtained from \(M\) will differ from that of ideal measurements of
\(\bQ\) and of \(\bP\) on the system of interest
in state \(\ket{\psi}\). However, one can indicate how
much these marginals deviate from separate exact position and momentum
measurements on the state \(\ket{\psi}\) by a pairwise
comparison of
 (31)
 to the exact distributions 



\[\begin{align\*}
\tag{32} \mu(q) &:= \abs{\braket{q}{\psi}}^2 \\
\notag \nu(p) &:= \abs{\braket{p}{\psi}}^2
\end{align\*}\]


In order to do so, BLW propose a distance function \(D\) between
probability distributions, such that \(D(\mu, \mu')\) tells us how
close the marginal position distribution \(\mu'(q)\) for the unsharp
position \(Q'\) is to the exact distribution \(\mu(q)\) in a sharp
position measurement, and likewise, \(D(\nu ,\nu')\) tells us how
close the marginal momentum distribution \(\nu'(p)\) for \(P'\) is to
the exact momentum distribution \(\nu(p)\).



The distance they chose is the Wasserstein-2 distance, a.k.a. (a
variation on) the earth-movers distance.



*Definition* (Wasserstein-2 distance)
  

Let \(\mu(x)\) and \(\mu'(y)\) be any two probability distributions on
the real line, and \(\gamma(x,y)\) any joint probability distribution
that has \(\mu'\) and \(\mu\) as its marginals. Then: 



\[\tag{33} 
D(\mu, \mu') := \inf\_\gamma \left(\iint (x-y)^2 \gamma (x,y) dx dy \right)^{1/2}\]


Applying this definition to the case at hand, i.e.  pairwise to the
quantum mechanical distributions \(\mu'(q)\) and \(\mu(q)\) and to
\(\nu'(p)\) and \(\nu(p)\) in
 (31)
 and
 (32),
 BLW's final step is to take a supremum over all possible input
states \(\ket{\psi}\) to obtain 



\[\tag{34}
\begin{align\*}
\Delta(Q, Q') & = \sup\_{\ket{\psi}} D(\mu, \mu') \\
\Delta(P, P') & = \sup\_{\ket{\psi}} D(\nu, \nu')
\end{align\*}
\]

 From these
definitions, they obtain 



\[\tag{35}\Delta(Q, Q') \Delta (P,P') \geq \frac{\hbar}{2}\]


Arguing that \(\Delta(Q, Q')\) provides a sensible measure for the
inaccuracy or noise about position, and \(\Delta(P, P')\) for the
disturbance of momentum by any such joint unsharp measurement, the
authors conclude, in contrast to Ozawa's analysis, that an
error-disturbance uncertainty relation does hold, which they take as
"a remarkable vindication of Heisenberg's
intuitions" in the microscope thought experiment.



In comparison of the two, there are a few positive remarks to make
about the Busch-Lahti-Werner (BLW) approach. First of all, by focusing
on the distance
 (33)
 this approach is comparing entire *probability distributions*
rather than just the expectations of operator differences. When this
distance is very small, one is justified to conclude that the
distribution has changed very little under the measurement procedure.
This brings us closer to the conclusion that the error or disturbance
introduced is small. Secondly, by introducing a supremum over all
states to obtain \(\Delta( Q, Q')\), it follows that when this latter
expression is small, the measured distribution \(\mu'\) differs only
little from the exact distribution \(\mu\) *whatever the state of
the system* is. As the authors argue, this means that
\(\Delta(Q,Q')\) can be seen as a figure-of-merit of the measurement
device alone, and in this sense analogous to the resolving power of a
microscope.



But we also think there is an undesirable feature of the BLW approach.
This is due to the supremum over states appearing *twice*, both
in \(\Delta(Q,Q')\) and in \(\Delta(P,P')\). This feature, we argue,
deprives their result from practical applicability.



To elucidate: In concrete applications, one would prepare a system in
some state (not exactly known) and perform a given joint measurement
\(M\) of \(Q'\) and \(P'\). If it is given that, say, \(\Delta(Q,Q')\)
is very small, one can safely infer that \(Q\) has been measured with
small inaccuracy, since this guarantees that the measured position
distribution differs very little from what an exact position
measurement would give, regardless of the state of the system. Now,
one would like to be able to infer that in this case the disturbance
of the momentum \(P\) from \(P'\) must be considerable *for the
state prepared*. But the BLW only gives us: 



\[
\Delta(P, P') = \sup\_{\ket{\psi}} D(\nu, \nu') \geq 
 \frac{\hbar}{2 \Delta(Q, Q')}
\]


 and this
does not imply anything for the state in question! Thus, the BLW
uncertainty relation does not rule out that for some states it might
be possible to perform a joint measurement in which both \(D(\mu,
\mu')\) and \(D(\nu, \nu')\) are very small, and in this sense have
negligibe error and disturbance. It seems premature to say that this
vindicates Heisenberg's intuitions.



Summing up, we emphasize that there is no contradiction between the
BLW analysis and the Ozawa analysis: where Ozawa claims that the
product of two quantities might for some states be less than the usual
limit, BLW show that product of different quantities will satisfy this
limit. The dispute is not about mathematically validity, but about how
reasonable these quantities are to capture Heisenberg's
qualitative considerations. The present authors feel that, in this
dispute, Ozawa's analysis fail to be convincing. On the other
hand, we also think that the BLW uncertainty relation is not
satisfactory. Also, we would like to remark that both protagonists
employ measures that are akin to standard deviations in being very
sensitive to the tail behavior of probability distributions, and thus
face a similar objection as raised in
 section 5.
 The final word in this dispute on whether a measurement uncertainty
principle holds has not been reached, in our view.