Remarks on Some Basic Issues in Quantum Mechanics*

Berthold-Georg Englert

Max-Planck-Institut für Quantenoptik , Hans-Kopfermann-Straße 1 , D-85478 Garching, Atominstitut der Österreichischen Universitäten , Stadionallee 2 , A-1020 Wien, Austria, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA

Z. Naturforsch . 54 a, 11-32 (1999); received October 26, 1998

Two-way interferometers with which-way detectors are not only of importance in physical research, they are also a useful teaching device. A number of basic issues can be illustrated and discussed, even at the level of undergraduate teaching. Among these issues are: the physical meaning of a state vector; entangled systems ; Einstein-Podolsky-Rosen correlations ; statistical operators and the as-if realities associated with them; quantum erasure ; Schrödinger's cat ; and, finally, wave-particle duality .

Contents 1 . Introduction

1. Introduction

1.1. Motivation , objective, and outline

1.2. A confession

2. Two-way Interferometers

3. Which-way Detection , EPR Correlations

4. Examples of Interferometers with Which-way Marking

5. Entanglement and Correlations

6. Mixtures , Blends , and As-if-realities

6.1. Blends correspond to as-if-realities

6.2. All blends are equal

7. Quantum Erasure

8. PostFestum

9. What Does a Quantum Eraser Erase ?

9.1. More realistic which-way markers

9.2. The as-if-reality of quantum erasure

9.3. Schrödinger's cat

9.3.1. Interferences between live and dead cats? No!

9.3.2. Selfadjoint operators and physical observables

9.3.3. Additional remarks 10. Wave-particle Duality

10.1. Distinguishability of the ways

10.2. A digression: Asymmetric interferometers

10.3. An inequality

* Dedicated to Professor Georg Süßmann on occasion of his 70th birthday .

Reprint requests to Dr. B.-G. Englert , Garching ; Fax: +49 89 329 05 200, E-mail: bge@mpq.mpg.de.

/. /. Motivation, objective, and outline

More than once, I have come across statements such as "The formalism of quantum mechanics is well established, and there are a number of good textbooks that treat the subject satisfactorily . But the physical interpretation of the formalism is still under debate, and there are a number of questions that still lack a satisfactory answer ." I disagree for two reasons.

First, I cannot imagine teaching the formalism independently of the interpretation. Before one can write the Schrödinger equation on the blackboard , one must have already explained the significance of the wave function that is differentiated1 . Otherwise the equation is meaningless and has no right to appear in a physics lecture.

Second, all the allegedly unanswerable questions show up in discussions of measurements on quantum objects, and I am convinced that there are no questions in this context that cannot be dealt with in an introductory course on quantum mechanics. It is the objective of these notes to supply a concise exposition that is based on which-way detection in simple two-way interferometers .

After a brief repetition of the basics of two-way interferometers in Sect . 2, we turn to which-way detection in Section 3. There we meet Einstein-Podolsky-Rosen correlations and recall, in passing, what is

'Of course, the interpretation of the wave function, say, may be subject to refinements when our understanding evolves , but there is an agreed-upon interpretation at each stage of the development.

0932-0784 / 99 / 0100-0011 $ 06.00 Verlag der Zeitschrift für Naturforschung, Tübingen www.znaturforsch.com

12

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

meant by phrases like: "The system is in a certain state" . In Sect . 4, three experimental set-ups are sketched that illustrate different methods of incorporating devices for which-way detection into standard interferometers . After dealing with the notion of entanglement in Sect . 5, we are ready for a discussion of the physical meaning of the statistical operator (Section 6) . The ground is then prepared for a short lesson about quantum erasure in Section 7. Some objections are shown to be invalid, and that brings up the subject of state reduction (Section 8) . When offering additional remarks on quantum erasure , in Sect . 9, we run into Schrodinger's cat . Finally, Sect . 10 deals with wave-particle duality in a quantitative manner .

The material is well suited for an undergraduate course because the mathematical aspects are rather elementary. Some basic knowledge about Hubert space vectors , selfadjoint and unitary operators , and the like suffices for Sections 2 - 9. In Sect . 10, some properties of the trace-class norm are made use of; although this might not be standard course material, it can be supplied easily.

The presentation is entirely in the language of quantum kinematics , no dynamical equations are employed. The students must only know that the transition from "before" to "after" is effected by a unitary transformation . Detailed temporal evolution plays no role, but the causal order in which things happen is relevant, of course.

This is not to say that time dependences can always be ignored in discussions of quantum measurements. All measurements take time (and they also happen inside a certain spatial region ). Some phenomena , such as the quantum Zeno effect , can only be understood by paying careful attention to the evolution. This subject matter is, however, beyond the scope of these notes, and I refer the reader to Schenzle's instructive article [1].

Occasionally, I remark on things that will strike the experienced reader as rather elementary. These remarks are meant for those undergraduates who study the material without a teacher's guidance.

1.2. A confession

In the opening paragraphs I have already offered a personal opinion. Perhaps I should confess more thoroughly where I stand.

Quantum mechanics, as I understand it, is solidly founded on experimental findings2 and the theoretical conclusions drawn by Planck , Einstein , Bohr , Heisenberg , Pauli , Schrödinger , and Dirac , to name the main contributors. One can learn the subject from the classic textbooks by Dirac [3], Böhm [4], and Gottfried [5]; von Neumann's book [6] puts particular emphasis on mathematical aspects ; a modern text that is much to my liking is the one by Ballentine [7]3. In addition, Schrodinger's seminal essay of 1935 [8] best remembered for a marginal issue , the cat example is recommended reading, and so are Süßmann's [9] and van Kampen's [10] remarks on quantum measurements.

Quantum mechanics works (and so does its rel-ativistic extension , renormalized quantum field theory ). During the seven decades since its conception , we have not become aware of a single observational fact in disagreement with quantum mechanical predictions . Although this large body of evidence lends strong support to the judgment that quantum mechanics provides for a consistent picture of the physical world, the logical possibility of a future failure is not excluded, of course. As soon as such a failure will have occurred, we'll be living through exciting times and something profoundly new will be learned. For the time being, however, there is no need for modifications of quantum mechanics.

Some are bound to disagree with the last sentence because they feel uneasy with the fundamentally probabilistic world view of quantum mechanics, and an intrinsically deterministic universe is philosophically more appealing to them. The pseudo-classical mechanics invented by Böhm [11,12] is the prime example. By construction, Bohmian mechanics agrees with quantum mechanics as far as its experimentally testable aspects are concerned, and has additional elements (namely hypothetical particle trajectories) that are of no consequence . In addition, these trajectories possess extremely implausible properties [13,14] that invalidate the realistic interpretation intended by Böhm .

2Attempts to infer quantum mechanics from some mathematical statements are misguided, in my opinion, irrespective of how convincing the arguments may appear at first glance. A recent example is Fröhner's undertaking [2] who believes, so it seems, that all of quantum mechanics follows from the Riesz-Fejer theorem .

'There are, of course, other books worthy of recommendation. Nothing is implied by not mentioning them.

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

13

Quantum mechanics deals with the behavior of atomic systems and, in particular, with measurements on them. For the concept of a measurement to be meaningful, it is necessary as Heisenberg and Bohr have emphasized that the physical world can be divided into the atomic system under study, whose quantum properties are important, and the rest, whose quantum properties are irrelevant. The measuring apparatus (a photographic plate , say, or a Geiger counter ) is part of the "rest." Of course, this does not deny that the apparatus exhibits quantum features itself the contrary is true: the photographic process and the mechanism of the Geiger counter rely on quantum processes but only that these aspects are presently irrelevant.

Consider, for example, a magnetic silver atom that passes through the inhomogeneous magnetic field of a Stern-Gerlach apparatus and then hits a glass plate; photographic development eventually reveals whether the atom was deflected up or down. Here we are studying the quantum properties of atomic magnetism . Do the quantum mechanical details of the chemistry of the photographic process matter ? No, they don't. Our conclusions concerning the magnetic atoms are not altered if we use another detection process instead of the glass plate.

We can, therefore, ignore the technical details of the detection and simply speak of a detector for the atoms . In a next step we recognize that the detector actually determines the position of the atom . So we learn that the crucial element of the apparatus is the inhomogeneous magnetic field because it encodes the information about the magnetic moment in the center-of-mass motion. This leads us to regarding one center-of-mass coordinate as the basic physical quantity carrying the measurement result. That coordinate is then the quantum marker4 in this example. The glass plate et cetera amplifies the datum recorded by the marker, and as a result of the amplification we can register the outcome and tell our colleagues what has been found.

One could insist that the measurement is completed only after the amplification has turned the information stored in the center-of-mass state (the marker state) into a macroscopically recognizable signal . But it is

4 What I call the marker here is alternatively termed meter or even detector in the literature. Since macroscopic devices are frequently associated both with meters (such as thermometers ) and with detectors (such as Geiger counters ). I suggest to use the less provocative word "marker" instead.

clear that the essential step is the one in which the information is encoded in the center-of-mass motion.

The amplification is part of the final measurement of the marker state, the so-called "reading of the marker." Prior to the amplification we could, in this example, manipulate the center-of-mass motion with the intention to read the marker in a different way. This freedom is particularly important in Sects . 7 and 9.

The possibility of dividing the physical world into the quantum system of interest and the (quasi-)-classical rest is a fundamental empirical fact . Without this division, one could not speak sensibly about measurements on quantum systems, because there wouldn't be any classical systems for reference. The division is not only possible, it is also necessary5.

Whereas it is a healthy attitude of working physicists to simply accept this division as an empirical fact , one can, and should, ask

Can quantum mechanics explain why the quantum/classical division is possible?

For reasons such as the ones discussed in Sect . 9.3,1 am confident that the answer will be "yes" eventually. But, admittedly, the case is not closed as yet. There are others who have seemingly convinced themselves that quantum mechanics does not rise to this challenge. The late Bell was arguably the most outspoken advocate of this point of view. He put "measurement" on his list of forbidden words [15, 16] in fact, he declared it "the very worst word on this list" and claimed that quantum mechanics "carries within itself the seed of its own destruction" [17]. Myself, I find it impossible to agree with Bell and recognize my own convictions in the replies by van Kampen [ 18], by the late Peierls [19], and by Gottfried [20].

The interference of alternatives is characteristic of quantum systems; classical alternatives do not interfere6. It should be clear that there is a border regime where quantum interferences are faint, and therefore a sharp boundary cannot exist. Nevertheless, one is almost always sure whether certain degrees of freedom are on the quantum side or on the classical side. And in the case of doubt, one plays it

5As a consequence , I have no use for a "wave function of the universe." But that's a side issue.

6In anticipation of the discussion in Sect . 9.3.2 I note that, therefore, there are no physical observables associated with selfad-joint operators that are sensitive to interferences between classical alternatives .

14

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

safe and enlarges the quantum system one "shifts the Heisenberg cut" as the phrase goes.

Quantum mechanics has been accused of being incomplete or inexact or simply ill-defined because there is no formal procedure that would draw a unique dividing line between the quantum system and the classical rest . These charges are unfounded. It is the nature of the Heisenberg cut that it can be shifted to some extent, but not arbitrarily. There are always certain degrees of freedom that are undoubtedly on one side of the cut or on the other. One can reasonably, and in a generally agreed-upon way, speak of day and night without ever assigning to dawn a uniquely defined instant in time7.

An affirmative answer to (1) needs a mechanism that explains why alternatives have different interference properties on the opposite sides of the quantum/classical border . In one approach, exemplified by the Ghirardi-Rimini-Weber scheme [21], the dynamical equations of quantum mechanics are modified; in view of their ad-hoc nature , however, such schemes are hardly convincing. I have much more sympathy for the other, conservative, approach that does not give up lightly what has been hard won; it takes quantum mechanics at its face value and searches for the said mechanism by investigating the quantum properties of systems with very many degrees of freedom. The decoherence process studied by Zurek [22] and others is very likely an essential ingredient.

An important aspect of the question (1) concerns the occurrence of factual events , that is: local interactions that have surely happened. Rain is falling independently of any manipulation by an experimenter . The formation of each individual rain drop depends on quantum processes at its initial stage; once it is formed, a drop is a classical object and undoubtedly in existence. More elementary than this formation process are scattering events in which a few particles participate only. In a recent proposal by Haag [23], the emphasis is shifted from measurements and their results to events as (one of) the fundamental concept(s) . This intriguing program has not been worked out as yet to the extent necessary for a final judgment; I do think, however, that it opens a new front at which (1) can be attacked.

2. Two-way Interferometers

In a double-slit interferometer the pattern on the screen originates in the state vector

|V> = ^(|5,) + |S2 , (2)

where \S\), \Si) symbolize the amplitudes of slit 1 and slit 2, respectively. These amplitudes are orthonormal,

(Sj\Sk)=6jk, (3)

and thus \tp) is properly normalized. The corresponding statistical operator (vulgo the density matrix)

$) = WM = ^(^>(i|+^>(2|)

slit 1 slit 2

+U\Sl)(S2\ + \S2)(Sl\) (4)

l v v '

cross terms

is half the sum of the single-slit contributions plus cross terms, which give rise to the double-slit interference pattern .

It is clear that this mathematical structure is common to all (symmetric) two-way interferometers , such as double-slit interferometers (for light , electrons , neutrons , or atoms ), or Mach-Zehnder interferometers , both of the optical and of the neutron kind , or biprism interferometers , with light or electrons , or Stern-Gerlach interferometers for magnetic atoms , or Ramsey-Borde interferometers for two-level atoms , or photon-pair interferometers , et cetera8. We shall continue to speak of slit 1, slit 2, the screen, ..., but the reader should keep in mind that the double-slit interferometer is just a stand-in for all two-way interferometers . Further, rather than speaking of particles or waves we use the noun quanton (suggested by Bunge as reported by Levy-Leblond [25]) as a generic term for the interfering quantum object (photon , electron , neutron , atom,...). The subscript Q in (4) anticipated this terminology .

In the spirit of this implicit generality we shall not pay attention to the subtleties of the pattern formation on the screen (such as the keeping apart of the single-slit diffraction pattern from the double-slit interference pattern that we are interested in, or the necessity

8By now there is an extensive literature on interferometers . The review articles in reference [24] report the state of the art.

I owe this telling analogy to Briegel .

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

15

to integrate over the various arrival times ). Instead we exhibit the interference pattern in the probability p((/>) of finding the superposition

\S(4>)) = ={\S]) + \S2)e"f), (5)

where 4> is the interferometric phase difference9 . In the state characterized by (2) or (4), this probability is

/ >) = \(s(m)\2 = M\s{4>)){S{4>)\p%))

= l-(l+cos4>). (6)

Its

, and since the extreme values of pi0)() are

(0) _ , (0) _ 0 (7)

/'max ' ' rnun u' v' J

the visibility V(0) of these fringes ,

(0) _ (0) 1)(0) _ ^max /"min _ i ,o\

Kmax T Pmin

is as large as it can possibly be.

Perhaps unnecessarily I note that the piece of information "the quanton has a 50:50 chance of passing through slit 1 or slit 2" is not sufficient to determine p{Q of (4). All statistical operators of the form

pZ = ^(|5i)(S,| + |S2)(S2|)

+^(|Si)e'(S2| +152)(5,|) (9)

are consistent with this information, not only the e = 1 version of (4). The positivity of Pq\ requires |e| < 1 but no further restrictions apply to the complex number e. Knowledge of the degree of coherence between the two quantum alternatives "through slit 1" and "through slit 2" as well as of their phase relation is needed to fix the value of e.

3. Which-way Detection , EPR Correlations

Now let us consider the more complicated situation in which the interferometer is supplemented

9The value of 4> is determined, for example, by the site where the quanton hits the screen of a double-slit interferometer , or by the difference in the optical path lengths of a Mach-Zehnder interferometer ; see Section 4.

by a device for which-way (WW) detection . A few quantum degrees of freedom of this device constitute the WW marker10. Eventually an amplification will enable the experimenter to read the marker, thereby extracting the WW information stored in the marker state.

The quanton interacts with the marker. Ideally, two orthonormal states \M\ ) and |M2) of the marker get then correlated with the amplitudes \S\ ) and |S2) of the two slits, so that the combined system of interfering quanton and WW marker has the state vector

|*) = -^(|Af,,S,) + |M2,52)) (10)

and a corresponding statistical operator "

P = |#)(#| (ID

= ]-{\MuSl){Ml,S1\ + \M2,S2)(M2,S2\)

+i(|M1,S1)(M2,S2| + |M2,52)(M,,S1|).

Again we recognize contributions that refer to one way only or to both ways.

The purpose of the WW marker should be obvious. If we find the marker in state |Mi) then the quanton is surely in state \S\ ), and likewise for \M2) and \S2). In other words:

Knowing that the WW marker is in state

|M|) or IM2) is tantamount to knowing that the (12)

quanton is in state |5|) or \Si), respectively.

Therefore, it is now possible to manipulate the quanton for example, by making a measurement on it without losing the information whether the quanton is in state |5i ) or |S2), because this WW information is stored in the WW marker .

Correlations of the Einstein-Podolsky-Rosen (EPR) type [26] are exploited here. Their reality can be demonstrated experimentally, possibly in the manner Alice and Bob do it in Section 8. But one must not be led astray by extreme operationalism : The EPR correlations exist irrespective of their observation by a human observer , they are a physical property of

10More than "a few" lead to the situation discussed in Section 9.3. 1' The letter P is the upper case of p, not of p.

16

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

the combined quanton-marker system in the state |#) of(10)12.

The phrase "the quanton is in state |5i )" is shorthand for "the statistical properties of (future) measurements on the quanton are correctly predicted by the statistical operator pq = \S\) (S\ |". This mini-malistic interpretation of state vectors and statistical operators is quite sufficient. One might, of course, try to go beyond this minimalism and give additional on-tological meaning to the state vector, perhaps responding to a philosophical impetus . In doing so, one should however remember van Kampen's caveat : Whoever endows |Si) (or any other state vector |...) ) with more meaning than is needed for computing observable phenomena is responsible for the consequences (Theorem IV in [10]). Similarly, the phrase "the WW marker is found in \M\)" abbreviates a statement such as "the result of a measurement of the observable M = \M\){M\\ - \M2)(M2\ is the eigenvalue

M' = r13.

Having the WW information safely stored we can now think of looking for the interference pattern . Inasmuch as knowing the way gives evidence of the quanton's particle aspects , whereas the observation of interference fringes manifests the quanton's wave aspects , we expect that complementarity prevents us from getting both. Indeed, the quanton's statistical operator that is needed in the trace of (6), obtained by tracing P over the marker's degree(s) of freedom,

PQ = trto{P} = ^(|S,)(S,| + |52)(A2l). (13)

no longer contains the cross terms of (4); now the pattern

is just the fringeless background of the single-slit contributions .

The fringes disappear whether we read the WW marker to learn about the way or not. The transition from p(Q of (4) to pq of (13) (termed the " collapse of the wave function" in some jargon ) is not caused by the acquisition of human knowledge; surely, "it has nothing to do with the mind" ( Wheeler as quoted by Haag [32]).

4. Examples of Interferometers with Which-way Marking

A first example for an interferometer with a which-way detection device , the thought experiment of [33], is sketched in Figure 1. The two slits of a Young interferometer are illuminated by atomic de Broglie waves . Prior to reaching the slits , the atoms pass through resonators ; the de Broglie waves are precollimated (by some equipment not shown in the figure) so that the partial amplitudes fit through the entrance and exit holes of the resonators . The resonators are initially

15,)

|52>

p(Q{\S(4>))(S(]-

(14)

l2Mermin has recently adv ertised his " Ithaca interpretation " [27] whose central theme is the assertion that correlations (of the EPR type ) are real and that they are the only real thing. I have much sympathy for this point of view.

Measurement" means a traditional iwi Neumann measurement , where the possible outcomes are the eigenvalues of the observable in question. Aharonov and Vaidman . in collaboration with Albert and Anandan , have recently introduced the intriguing and useful concepts of weak measurement |28 , 29] and protective measurement [30. 31], where other properties of the measured observable are more relevant. At their final stage, however, a von Neumann measurement determines the position, say. of a pointer variable . I am grateful for the enjoyable and instructive discussions with Aharonov on these matters.

resonators slit plate

screen

Fig. 1. Atomic de Broglie waves , indicated by symbolic wave trains , are collimated so that they pass through resonators before reaching the slits of a Young's double-slit interferometer . At most one atom at a time is in the apparatus . The atom emits a photon into the resonator it traverses. The atom is the quanton ; the relevant photonic degrees of freedom constitute the marker. The wave trains are shown at three different instants : before entering the resonators ; after emerging from them, but before reaching the slits ; after having been diffracted at the slits . The state vector \$ ) of (10) refers to the latter instant.

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

17

Mi, Si)

\M2 h

E^HMO

\SW)

SEPP

|S(* + *))

Fig. 3. A source of entangled photon pairs SEPP emits polarization-entangled EPR pairs . One photon , the quanton , propagates to the right and enters a Mach-Zehnder interferometer through a polarizing beam splitter PBS . Vertical polarization (v) is reflected, horizontal polarization (h) is transmitted. The latter is changed to vertical also by a HWP . The second photon of the pair, whose polarization is the marker, propagates upwards and is detected behind another PBS .

The polarization of the photon is used for WW marking. For this purpose, all entering photons are polarized horizontally, say. A half-wave plate (HWP) in the \S\) arm of the interferometer turns the polarization from horizontal to vertical. Here, the photon's polarization degree-of-freedom constitutes the marker in accordance with

Fig. 2. A polarized photon enters a Mach-Zehnder interferometer through a half-transparent mirror HTM . The transmitted amplitude passes through a half-wave plate HWP which changes the polarization to an orthogonal one. The quanton is made up by the orbital degrees of freedom of the photon , the polarization is the marker. A phase shifter PS and a second HTM combine the quanton states so that the superpositions \S(4>)) and \S(4> + n)) of (5) emerge at the two output ports.

empty, and matters are arranged such that each atom emits with certainty a photon into a privileged mode of the resonator it traverses.

This is then the situation: The atom is the quanton , and its center-of-mass degrees offreedom are the ones relevant for the interferometer . The marker is made up by the photonic degrees of freedom of the selected resonator modes . The quanton states \S\ ) and \Si) are the diffracted de Broglie waves , as indicated in Fig. 1, and

|Mi) = "one photon in the 1st resonator,

none in the 2nd", \Mi) = "one photon in the 2nd resonator, (15)

none in the lsl",

specifies the significance of the marker states. The value of the interferometric phase

of (5) is determined by the site at which the atom hits the screen.

In the second example of Fig. 2 we have a Mach-Zehnder interferometer for light, operated in a one-photon-at-a-time fashion . The entering photon encounters a half-transparent mirror (HTM) ; the two resulting parts of the photon's orbital amplitude are the quanton states \S\ ) and |S2). A phase shifter (PS) and a second HTM make the superposition \S( leave at the symmetric output port and the orthogonal one, that is: \S( + n)), at the asymmetric one.

|Mi) = " vertically polarized" , | A/2) = " horizontally polarized" .

(16)

It is, of course, necessary to detect the photons at the output together with their polarization .

A Mach-Zehnder interferometer is also employed in the third example (Fig . 3), which is a variant of problem 9-6 in Ballentine's textbook [7]. Here we begin with a source of entangled photon pairs (SEPP)14 . It emits EPR pairs of photons going to the right or going up, so that one photon is polarized horizontally and the other one vertically . The emitted pairs are in a state such as

V2

(|h_,vt) + |v_,hr ,

(17)

where, for instance, h_ means " horizontally polarized " photons goes to the right.

l4Such sources are actually available, see [34]; the recent tele-portation experiments [35, 36] made use of SEPPs .

18

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

Of each pair, the photon that goes to the right is the quanton . It enters a Mach-Zehnder interferometer with a polarizing beam splitter (PBS) at the entrance, which reflects vertically polarized photons and transmits horizontally polarized ones . The transmitted amplitude passes through a HWP , so that the quanton photon is surely polarized vertically when it reaches the PS and HTM that probe for |5(0)> and \S(4> + tt)).

The polarization of the up-going photon is used for WW marking. The states \M\ ) and |M2) stand for vertical and horizontal polarization , respectively, and can be probed by ordinary photodetection behind another PBS .

The three examples exhibit three different methods for WW marking. In Fig. 1 another physical system (the resonators with their photon modes ) is used. By contrast, in Fig. 2 the same physical object the photon serves both as quanton (orbital motion) and as marker (polarization) . In these two examples, the WW marking is done during the passage of the quanton through the apparatus , and one could operate the interferometer without WW marking. Not so in the third example, where the quanton and the marker are created jointly (by the SEPP ) and are in the entangled quanton-and-marker state to begin with. As a consequence , the way through the interferometer can be known even before the quanton reaches the entry port. In other words, the way is predictable in the set-up of Fig. 3, and therefore this set-up exemplifies the situation considered by Jaeger , Shimony , and Vaidman in [37].

The thought experiment of Fig. 1 will most likely never be realized, but there are other, more realistic, schemes in which a privileged photon mode of a resonator is used for WW marking. In particular, Ramsey interferometers for atoms in Rydberg states are well suited for this purpose. One can either use a resonant interaction , as in the proposal of [38], or a dispersive interaction , as in the experiment reported in [39]. The systematic loss of the fringe visibility has been observed in this experiment, but WW information has not been extracted as yet.

Kwiat and Schwindt of Los Alamos National Laboratory have recently built the interferometer of Fig. 2 [40]. They have not only succeeded in demonstrating the systematic dependence of the fringe visibility on the parameters, but have also obtained WW information from the final polarization state (in the quantitative manner discussed in Section 10). This two-fold challenge has also been met by Dürr , Nonn ,

and Rempe of the University of Konstanz who have built an atom interferometer that uses hyperfine sub-levels for the WW marking [41, 42].

Finally, concerning Fig. 3, this should be a relatively easy experiment if a SEPP is at hand. An actual realization could be very rewarding, in particular because the feature of "late choice" (see Sect . 7) can be incorporated rather simply.

5. Entanglement and Correlations

As long as there is no further interaction between the quanton and the WW marker , the statistical operator pq of (13) obtains independently of the history of the WW marker after the quanton-marker interaction . The statistical properties of all measurements that could be performed on the quanton are correctly predicted by this pq . Accordingly, this statistical operator characterizes the state of the quanton after its interaction with the WW marker just like Pq* of (4) did before the interaction.

Likewise we have a statistical operator for the WW marker ,

PtA = trQ{P} = |(|Af,)(Af,| + |M2)(M2|) , (18)

which characterizes the marker state after the interaction. The (direct) product of pm and pq ,

PmPq = ^(|M1,S,)(M1,51|+|M2,52){M2,S2| (19)

+ |A/,,S2)(A/1,52| + |M2,S|)(M2,S,|).

differs from P of (12). This is no surprise, of course, because the quanton and the marker are entangled (German: verschränkt or verheddert , as coined by Schrödinger [8]) if P of (12) applies, but they would not be entangled if the product pm pq represented their statistical properties. As a consequence of this entanglement , joint probabilities of measurements on both the quanton and the marker are not simply equal to the product of the individual probabilities . For instance, the probability for finding the quanton in state \S\ ) is 50%, and this is also the probability for finding the WW marker in state |M2); but the joint probability for finding the quanton in state \S\ ) and also the marker in state |M2) is 0% not 25% as would be the case if (19) were the statistical operator. This exemplifies the general situation:

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

19

Entanglement between different degrees of freedom results in correlations between the results (20) of measurements on the subsystems .

And vice versa: if measurement results are correlated, then the subsystems must be entangled15.

We note that the entanglement of two subsystems can be of a purely classical nature. For the quanton-and-marker system this would be the case, for example, if its statistical operator were given by

]-{\MuSl)(Ml,Sl\ + \M2,S2)(M2,S2\), (21)

where the cross terms of (12) are missing. The corresponding situation would be this one: Either the state \M\,S\) is realized or the state |Af2,52) (with equal probability) , but we don't know which one. In other words: (21) represents a classical mixture of disentangled states . Inasmuch as statement (12) is as true for (21) as it is for (12), the statistical operator (21) surely describes an entangled quanton-and-marker system . But, owing to the absence of the cross terms in (21), the correlations are here not of the quantum-mechanical EPR type .

6. Mixtures , Blends , and As-if-realities

6.1. Blends correspond to as-if-realities

If the interferometer is operated such that only one way is realized with certainty , then one has pq = |5i)(5i| or pq = |52)(52|, of course. The statistical operator pq of (13) is a 50:50 blend of these cases. Does this mean that the quanton is either in state \S\ ) or in state \S2) but we simply don't know in which one? This interpretation suggests itself and does not lead to inconsistencies . One must, however, be aware that it represents only an as-if-reality. The statistical predictions resulting from pq of (13) are such that it appears as if the quanton were in | Si) or | S2). Equally well one can regard this pq as a 50:50 blend of another pair of orthonormal states , such as

|S,) = |Si)cos0 + |S2)eivsin0,

|S2) = |S2)cosi? -|S,)e-i'sini? (22)

15 There may also be correlations of a different kind for measurements on a single degree of freedom in the sense that the expectation values of two observables A and B and of their symmetrized product AB + BA are such that AB + BAA) does not equal twice the product of (A) and (B). Such correlations are ubiquitous, but they are clearly of quite a different nature than the EPR ones .

because

PQ = ^(|51)(S1| + |S2)(S2|) (23)

holds not only for the -ß = 0 case of (13) but for all values of the parameters fl and quanton were either in state |Si) or in IS2), but we don't know in which one.

In (13) and (23) we have a simple example for the general observation that a statistical operator can be mingled from (projectors to) pure states in a plethora of different choices, with a corresponding abundance of equally good as-if-realities . Following Süßmann [9] we say that there are many blends (German : Gemenge ), exemplified by the right-hand sides of (13) and (23), that make up one and the same mixture (German : Gemisch ), exemplified by the identical left-hand sides .

In general terms, any decomposition of a statistical operator into a convex sum of projectors (the statistical operators of pure states),

P = ^2 wkPk (24a)

fc

with

p2k=Pk, tr{pk} = 1, wk > 0, ^wk = 1, (24b)

fc

identifies one of the blends that compose the given mixture p . Each blend is associated with an interpretation in terms of an as-if-reality: "The system is in one of the states pk, with statistical weights wk , but we don't know in which one." which yields a consistent picture of the phenomena nothing more, and nothing less.

It is not necessary that the pk's are mutually orthogonal in (24). If they happen to be, then we are dealing with (one of) the spectral decomposition(s) of the mixture p. Such blends are mathematically particular, but not physically.

Further we note that the sums in (24) could be integrals. For example,

r11 f2nd&

Pq=/ dtfsin(2tf)/ -^|S,)(Si| (25) Jo Jo 2lT

blends pq from all possible projectors |Si)(Si| with statistical weights that are uniform in cos(2i? ) and

20

B.-G. Engiert Remarks on Some Basic Issues in Quantum Mechanics

6.2. All blends are equal

Repeated measurements on quantons prepared in the same mixture cannot, by any means, distinguish one blend, or one as-if-reality, from the other. Whereas it is a matter of personal taste or preconceptions whether one regards this impossibility as an indication that quantum mechanics does not provide for a complete description, it is certainly not a matter of opinion that this impossibility is fundamental16. For, if one could tell one blend from the other (hypothetical nonlinear additions to the Schrödinger equation would enable one to achieve this) then one could send signals at arbitrary speed [43] and get in violent conflict with Einsteinian causality .

The marker-quanton entanglement in (10) and (12) and, in particular, the resulting equivalence stated in (12) invite the surmise that the d = 0 blend (13) is, in some sense, more physical than the ti 4 0 ones of (23), irrespective of what is said in the preceding paragraph . After all, couldn't it be that, paraphrasing Orwell , all as-if-realities are equal but some are more equal than others? No, certainly not, because we can cast the state vector (10) into the form

\*) = ^={\Ml,Sl) + \M2,S2)) (26)

where

\MX) = |M,)cos^ + |M2)e-^sin^, \M2) = |A/2)costf-|Mi)e^sin!? (27)

are the superpositions of the marker states that correspond to (22). Accordingly, the correlations between \M\) and \S\) as well as \M2) and \S2) are equally strong for all values of $ and EPR correlations :

Irrespective of the chosen setting of the parameters d and WW marker is in state \M\ ) or \Mi) is tantamount to knowing that the quanton is in state \S\ ) or \Si), respectively.

As a consequence , in repeated experiments that are all accounted for by the statistical operator of (12), the experimenter can choose, at his discretion, how

16Ballentine discusses , in Sect . 9-4 of his textbook [7], an instructive example of two equivalent as-if-realities and reports a historical incident that illustrates well the confusion that may arise when the as-if character is not appreciated.

he wants to analyze the data. Each quanton can be identified as the member of either the subensemble specified by | Si) or the one specified by |52), whereby the values of -9 and

quanton to the next at least in principle, if not in practice. This sorting of quantons into subensembles is done with the aid of a measurement of the marker state that is: a certain "reading of the marker" a measurement that discriminates between \M\) and \M2), and does not involve an observation of the quanton itself.

Let us take a look at the interference pattern from the point of view offered by the as-if-reality that goes with the blend of (23). To avoid ambiguities we restrict -d and

17. We have

p() = 5|2 + ^\(S(4>)\S2)\2 = \ (29) with

|(S(2 = ^ [l+sin(2tf)cos(

and

|(S(0)|S2)|2 = i[l - sin(2tf)cos(0 - v? )] . (30b)

Thus it is as if p(4>) = \ obtains because there are "really" two patterns, each with a fringe visibility of V = sin(2tf) and the fringes shifted by

fringeless sum of (29), the crests of (30a) meet the troughs of (30b) and vice versa.

7. Quantum Erasure

We are now prepared for discussing the kind of sorting that goes by the name of quantum erasure . Consider the following scenario . A quanton is sent through the interferometer equipped with the WW detection device , so that (12) gives a correct account of the statistical predictions of the experiment. The interferometer phase

is set to a chosen value, and the observable

O(0 = |S(0))(S(0)| (31)

l7This d range is only half of the range of integration in (25) because the replacements i? n/2 - x).

would just exchange |S|)(5,| and |S2){52|.

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

21

is measured. In accordance with (14) the measurement results 0 and 1 are equally probable. Another measurement determines whether the WW marker is in state \M\ ) or in |M2). We adopt this color coding: Finding \M\ ) identifies the quanton as a member of the blue subensemble , and the quanton belongs to the yellow subensemble if |M2) is found. We have d = 0 here, and (30 a and b) tell us that in both subensem-bles the eigenvalues of 0(0) occur with the same frequency. In other words: The blue quantons exhibit no interference fringes , nor do the yellow ones.

Of course, this is not surprising, but expected because the blue quantons are those which are known to have taken the way through slit 1 and the yellow ones the way through slit 2. We are just repeating what is stated in (12).

Alternatively, the experimenter can decide to make another measurement on the marker, one that distinguishes |Mi ) from |M2) of (27) for a setting of d and distinction of

|M+) = -^(|M,) + |M2)) (32a)

and

|M_) = -J=(|M2)-|Mi)). (32b)

In the set-up of Fig. 3, for example, such a |M+)/|M_) distinction is done by turning the polarization-sensitive beam splitter in the marker part by 45 .

We extend the color coding: Finding the marker state \M+ ) puts the quanton into the red subensemble , and finding |M_) puts it into the green one. According to (30a), the red quantons exhibit the original interference pattern of (6) with perfect fringe visibility , and the green quantons yield the antifringes .

Again, this shouldn't come as a surprise because we are simply facing a special case of (28), namely this one: Each red quanton is known to be in the state

|S+) = -J=(|S,) + |S2 (33a)

and each green one is in

|5_> = -^(|52)-|S, . (33b)

For the red quantons and the green ones it is unknown through which slit they passed in fact not just

unknown, but rather unknowable . As a consequence , the alternatives interfere.

As soon as the |M+)/|M_)-distinguishing measurement has been made, it is no longer possible to make the |Mi)/|M2) distinction, because these two measurements are incompatible. There is hardly a point in looking for \M\) or |M2) when it is already known that the marker is in state |M+ ), say, since we learn nothing about the state of the marker prior to the |M+)/|M_)-distinguishing measurement . In a manner of speaking, the |M+)/|M_)-distinguishing measurement has erased whatever WW information was there before. This circumstance gave the name quantum erasure (QE) to the red/green sorting [33,44,45]. We shall therefore speak of the QE measurement when the |M+)/|M_) distinction is made, and of the WW measurement in case of the |Mi)/|M2) distinction.

Inasmuch as WW information is particlelike and interference fringes are evidence for wavelike properties , one could also say that the WW sorting processes the data such that a particle experiment is effectively carried out, whereas the QE sorting amounts to doing a wave experiment. The freedom of deciding at a very late stage, possibly after the performance of the inter-ferometric 0(0) measurement , whether the quanton in question will be part of the particle experiment or of the wave experiment, demonstrates that QE offers a particular realization of experiments with delayed choice, which were first discussed by von Weizsäcker in 1941 [46] and became popular from 1978 on as a result of Wheeler's efforts [47].

In the set-up of Fig. 3, for example, the marker photon , which propagates upwards from the SEPP , can be sent on a detour through a long fiber. The experimenter can then first check at which output port the quanton photon emerges from the Mach-Zehnder interferometer , and then choose one of the sortings . This late-choice aspect of QE is especially intriguing.

For each quanton traversing the interferometer two measurements are made, the measurement of 0(0) on the quanton and the WW or the QE measurement on the WW marker . Does the joint probability of getting 0(0) = 1 and ending up in the red ensemble, say, depend on the order in which the two measurements are performed? Surely it doesn't because the two measurements refer to different degrees of freedom; the corresponding operators commute . Therefore, the experimenter can first measure 0(0) and then decide whether the quanton in question should

22

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

get a red/green label (QE sorting) or a blue/yellow label (WW sorting) . This freedom enables the experimenter to sort the data in fancy manners if he likes to do so. For instance, quantons that hit one half of the screen of a two-slit interferometer could be labeled by blue and yellow, and those hitting the other half by red and green. In view of (28), it is clear that much more complicated sorting schemes are conceivable.

8. PostFestum

No one doubts that these joint probabilities are independent of the order in which the 0(0) measurement and the sorting measurement are done for each individual quanton . Nevertheless, objections have been raised against the statement that the temporal order is truly without significance, and thus against the freedom of a late choice between the WW and the QE sorting . In particular, it has been argued both in private and in public; recently and eloquently by Mohrhoff [48] that the reading of the WW marker must be done before the 0() measurement is performed. Or, put differently, one has to sort the quanton into one of the color-coded subensembles first and then determine its contribution to the interference pattern . When 0() is measured first, so the critics assert , nothing can be learned from a subsequent, postfestum reading of the WW marker .

I disagree [49] because this critique is at odds with the objective nature of the EPR correlations (28) that link quanton states to corresponding marker states .

Consider the situation in which 0(0) of (31) has been measured and the eigenvalue 1 found. This characterizes a subensemble , for which the statistical operator of the marker is

with

|M(0)) = -^=(|M1) + |M2)e-^). (35)

The conditional probabilities for the outcomes of the WW measurement are then

red: |(M+|M(0))|2 = (1 + cos0), green : |(M_|M(0))|2 = (1 -cos0). (37)

So, this subensemble contains equal numbers of blue and yellow quantons but the relative frequency of red and green ones will be biased if cos 0^0.

Once more, there is no surprise. As soon as the eigenvalue 1 of 0(0) has been found, it is known that the quanton is in state |S(0)) which, according to (28), is tantamount to knowing that the marker is in state |M(0)), as is confirmed by (34). And the probabilities that the quanton is blue, yellow, red, or green are the ones stated above.

The said critics do not dispute these facts, but they give them a twist to arrive at their central argument, which is essentially as follows. Having found the eigenvalue 1 of 0(0), the statistical operator of the joint quanton-marker system is no longer given by P of (12) but rather by the one of the subensemble characterized by the measurement result, that is

P(*> = |M(0),5(0))(M(0),S(0)|

(38)

As indicated, this factors into a marker part with pm of (34) and a quanton part with pq equal to 0(0) of (31), and therefore there is no entanglement between the marker and the quanton in P('*). In other words: The equivalence stated in (28) is not true for P**', only for P. And so, the critics conclude, after the 0(0) measurement has been performed, measurements on the marker will no longer tell us anything about the state of the quanton .

Superficially it appears that a valid objection has been raised. Actually, however, the critics are missing the point.

The purpose of the WW marker is to store information about the state of the quanton . Manipulations of, such as measurements on, the quanton do not affect what is stored in the WW marker . Perhaps the following scene from the lab illustrates the issue. Student Alice has the job of making measurements on the quanton , student Bob reads the WW marker . Two experimental situations, El and E2, are of interest:

El: Bob finds the marker in state \M\ ) and tells Alice about it. Then Alice makes a |5i)/|S2)-distin-guishing measurement on the quanton as a test of the

blue: |(yellow :

2' M2\M())\2 = \,

(36)

and those of the QE measurement are

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

23

prediction that the quanton is in state \S\ ). Indeed, this is what she will always find under the circumstances stated, and so Alice confirms what is said in (28).

E2: Bob finds the marker in state \M\ ) and tells Alice about it. But before the message arrives, Alice measures 0(4>). Upon Bob's request to confirm that the quanton is in state \ i ), she replies: Sorry, too late. In many repetitions of these circumstances, Alice will confirm that the results of the 0(0) measurement are statistically consistent with Bob's findings about the marker state.

E2 has occurred and the supervisor turns to Alice with the question: "Unfortunately Bob's message has arrived after your 0(0) measurement. If you had made the \S\)/\ 2) distinction, as in El, instead of measuring 0(0), what would you have found? " Undoubtedly she answers: "I would have found \S\), of course! " because this is the logical implication of the empirical experience gained in El. Naturally, the supervisor is pleased.

The day after, Alice falls ill and Chuck takes her place. At the end of the day the supervisor comes to the lab and asks Chuck the very same question to which Alice had responded so pleasingly . Chuck, however, gives a different answer; he says: " Textbooks on quantum mechanics warn against making statements about the hypothetical outcome of measurements that haven't been performed and can no longer been made. Therefore, I'd say there is no sensible answer to your question." The supervisor isn't happy at all with this reply, because Chuck ignores the lesson of El.

Very often, statements about the outcomes of measurements that could have been performed but have not been made actually, do not make much sense very often, but not always. Alice is on safe ground here because of her El experience . Her answer is not imprudent, but demonstrates a thorough understanding of the characteristics of the experiment.

Does Alice's answer "I would have found \S\), of course! " depend on the time when Bob makes his measurement? No, it doesn't because the correlations of (28) are reciprocal, so that the roles of Alice and Bob can be interchanged in El. According to the post-festum critics , however, Alice's answer should depend on the timing. When her measurement is first, she should give Chuck's answer . The critics' fallacy is thus brought to light: Their reasoning is at variance with the empirical reality of the EPR correlations stated in (28) and confirmed by El. Case closed.

The critics are led astray by regarding the state reduction that turns P of (12) into P^1 of (38) as a physical process, not as the mental process it is. This point of view necessarily requires that the original state vector | P ) of (10) as well as the reduced one, l^1*') of (38), are regarded as real physical objects, rather than as the book-keeping devices that they are. We recall the recommended minimalistic interpretation : The state vector | P ) serves the sole purpose of summarizing concisely our knowledge about the entangled quanton-and-marker system ; in conjunction with the known dynamics, it enables us to make correct predictions about the statistical properties of future measurements. Whoever endows | #) with more meaning than that...

The notion of state reduction that just came up is nothing mysterious. It is dictated by the rules of correct book-keeping . We begin with a statistical operator P that refers to a certain ensemble (here : of quantons entangled with WW markers ) and summarizes what we know about it. Then a measurement result (here : "0(0) equals 1") is used to identify a subensemble . Probabilistic predictions concerning this subensemble cannot be based on the original P, but must rely on a suitably refined Psub (here : P^') that accounts for the defining properties of the subensemble . In other words: PSUb yields the correct conditional probabilities , conditioned on the said measurement result . The transition P > Psub is the corresponding state reduction . Clearly, it is not a physical process, but a mental one that simply reflects the change in our knowledge about the system. In addition, state reduction is not a specialty of quantum mechanics; it is a technical device of all statistical theories: "when I toss a coin the 50-50 probability distribution changes abruptly if I look at the outcome" van Kampen n [18]).

9. What Does a Quantum Eraser Erase ?

9.1. More realistic which-way markers

The analysis of Sects . 2 and 7 owes its simplicity to the idealization that the pure states |Afi ) and \Mi) suffice for an appropriate description of the WW marker . We lift this restriction now and suppose more generally that the WW marker is initially prepared in a state characterized by the statistical operator p^ . The quanton is initially in the state \ip ) of (2), so that Pq' of (4) is the initial statistical operator , for which we now write more compactly

24

p = Ua* a+ aa1+<, + **)

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics ,im The significance of the complex number

where

C = lrM{pM}=lrM{UlpUl}

(48)

2)(Sl

^ = |S,)(52|

(40)

is revealed by a glance at the resulting interference pattern ,

p() = tvQ{0(4>)pQ} =l-[\+ Re(e-J*C)] .(49) Inasmuch as

are obvious analogs of spin-flip operators .

The net effect of the interaction that creates the entanglement between marker and quanton is accounted for by two unitary operators , U\ and U2, one for each way, that is:

"max "min

} = ^(1 |C|) (50)

.(')

Pff

Wi^' for way 1, Ul$U2 = $ for way 2.

(41)

are the extreme values of p(), the fringe visibility V equals the modulus of C,

Inasmuch as erV and cjcj^ are the projectors that select the ways, the unitary operator that transforms the joint marker-quanton system is

V = \C\

(51)

and the argument of C determines the location of the crests and troughs .

A well functioning WW detection device is such that />M' and pM' can be kept apart. (The relevant numerical measure is introduced in Sect . 10; here we'll get around without technicalities of this kind.) It is clear that both the initial marker state p^ and the unitary operators U\ , U2 must be chosen judiciously to achieve well distinguishable final marker states .

A particularly unfortunate choice is exempli fied by

P^ = \{\M,){M,\ + \M2){M2\) (52a)

in conjunction with

Ux = I,

U2 = |M1)(M2| + |M2)(M,| + ---, (52b)

where the ellipsis indicates those irrelevant pieces of U2 that act on marker states orthogonal to both \M\) and \M2). Here one gets

(I) (2) (0) ,c~ ,

Pm =Pm =Pm (52c)

so that no WW information is available, and

Pm = \{\M2){M{\ + |M2)(M,|) , (52d)

implying

C = 0,V = 0, (52e)

= a*aUl +aa*U2,

(42)

U

M&Q

and the entangled state

p = [/t JO),

r

(43)

jm&qPm Pq ^m&q

results from the marker-quanton interaction . Its explicit form is

P = ^(v^Pm+^Pm+vPm + ^pIa) (44)

slit 1

slit 2

where the single-slit contributions involve the marker states pM' and p$ corresponding to them, and the cross terms contain

,5m = UlpU

(45)

and its adjoint . Of course, the statistical operator of (12) is a particular realization of (44); for example, the choice p$ = |Mi)(.1/i| in conjunction with U\ = 1 and U2 such that (A/, \U2 = (M2\ turns (44) into (12). Partial traces produce the statistical operators for the WW marker ,

PM=trQ{P} = l(^ + ^), and for the quanton

(46)

PQ = trM{P} = ^(t<7 + (T(Tt+Ca+CVt).(47) is bad news too: No fringes .

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

25

9.2. The as-if-reality of quantum erasure

This simple example illustrates graphically that the lack of WW information does not ensure good fringe visibility , nor does a fringeless pattern indicate that WW information has become available. The implication works only in the opposite directions:

Full fringe visibility precludes any WW information, and the acquisition of complete WW knowledge (53) enforces the disappearance of interference fringes .

Intermediate situations are the subject of Section 10.

In Sect . 7 we mentioned that the data sorting called quantum erasure got its name because in the course of performing QE the WW information is lost it is " erased" . Now we are facing a new situation in which the interference fringes are gone, but no WW information has been gained to compensate for the loss. Is a data sorting of the QE type still possible although there is no WW information that could be erased?

Yes, QE is still possible. We justify this affirmative answer in the general context of (44). To perform QE one makes a measurement on the marker that distinguishes the states \av) whose defining property is

t/,|a) = U2\av) eia" . (54)

These |a) 's are the eigenstates of the unitary operator Ulu{ and the phase factors exp(ia) are the respective eigenvalues . When the marker is found in the state |a), the corresponding subensemble of quantons is characterized by

p> = I(ffV + a(Tt + eia a + e~iQV) . (55)

Since this is essentially pq of (47) with C -+ exp(ia), the interference pattern of this subensemble is [cf. (49)]

P("\) = ^[l+cos(0-a)]. (56)

So, when the quantons are sorted according to the result of the |a)-distinguishing measurement , then each subensemble shows an interference pattern with unit fringe visibility . Therefore, the said measurement is a QE measurement . This analysis does not make use of any special properties of U\ and U2, and makes no reference at all to p^\ Consequently, it is indeed true

that QE can be performed even in the unfortunate case specified by (52).

We have thus seen quite explicitly that QE is possible even if there is no WW information to be erased. If it's not the WW information, then what does a quantum eraser erase?

The answer is based on the observation that there is an as-if-reality to pq of (47) that goes with the Pq''s of (55):

PQ = X>('%). (57)

V

where the weights wM are, of course, just the probabilities for finding the marker in the respective \av) states ,

w{u) = (au\PM\au). (58)

The identities

wM=e-ia (av\pM\av)=ei0"'{av\pl\au) (59)

are the essential ingredients in showing that the right-hand sides of (47) and (57) are the same. Equations (56) and (57) tell us that there is (at least) one as-if-reality to pq in which each alternative exhibits an interference pattern with unit fringe visibility .

The question, What does a quantum eraser erase? , asked in the title of this section, is therefore answered as follows:

A quantum eraser removes the cover that hides

the as-if-reality of alternatives with maximal (60)

fringe visibility .

If there is WW information stored in the marker beforehand, then it is erased when QE is performed. In general, however, the availability of WW information is not a precondition for QE18.

9.3. Schrodinger's cat

9.3.1. Interferences between live and deadcats ? No!

The affirmative "Yes, QE is still possible." is reassuring, but it is not a claim of practical feasibility . On the contrary, the requirement of distinguishing

''Therefore, one could lament that quantum erasure is a misnomer . But, who knows a more fitting term?

26

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

the eigenstates \av ) of U\UX from each other can and will be prohibitively difficult under the typical circumstances of an experiment. In particular, if the marker is itself a macroscopic piece of the WW detection device so that p^ as well as U\ and /2 make reference to very many degrees of freedom, QE is simply impossible, and the interference fringes cannot be retrieved.

Such is the situation in Schrodinger's (in)famous cat example [8], where a radioactive atom is the quan-ton (|Si) : excited atom , |S2): ground-state atom) that gets entangled with a Geiger counter , a hammer, some poisonous gas , ..., and finally the cat. The marker consists of all this equipment plus those parts of the environment with which the Geiger counter , ..., the cat are interacting during the period of interest19. It is clear that any attempt to do QE on a macroscopic marker like this one, with the aim of finding interferences between \S\) and |S2) is bound to fail, simply because U\ and ? 72 cannot be known with the necessary precision, if for no other reason.

Before turning to Schrodinger's cat problem , let us briefly mention a technical point. In general, the unitary evolution that entangles the atom (= quanton) with the marker (= cat plus ...) is not of the simple form assumed in Section 9.1. Rather, we begin with pq' = ffV = |5i)(5i| (atom excited) and an initial marker state p^ that represents our very limited knowledge about the Geiger counter ,..., the cat plus the environment. Then

UM&Q = ai(7Vl+aW2+(TVi + a^V4 (61)

is acting where V\,..., V4 affect only the marker variables . This results in a final P of the form (44) with

Pm V/z^V, live catt),

Pm viPmV2 (dead cat), (62)

,5m oc VjVmV , ("live&dead" cat).

Now, whereas (7m&q is unitary, the operators V\ ,..., V4 need not be unitary themselves and, as a rule, they will not be. Then it is possible, and indeed plausible , that the cross term pm vanishes20, although both pM* and pM' are nonzero, and QE cannot be done

l9That includes the quantized radiation field with its infinity of photon degrees of freedom.

20In the jargon of reference [22] one could say: The entangled system suffers from decoherence .

in the first place. An elementary example for this situation (with no relevance for Schrodinger's cat ! ) is provided by a marker that has only two possible states \M\ ) and IM2); then

Vi = V4 = |M2)(M,|, V2 = V3 = |M,)(M2| (63)

are such that Um&q is unitary, and Pm = 0 obtains for Pm of (52) while pM> and pM> project to |M,) and IA/2). respectively.

In a popular jargon any entangled state of a quantum degree of freedom and a macroscopic marker is called a Schrödinger cat21. In this sense (44) represents a cat if the experimenter can distinguish between pM* and pM* without further ado . It thus seems that " Schrödinger cat " is just another word for an entangled system . And so one should think that everything worth saying has been said about his cat, in the more than sixty years since Schrödinger published his Generalbeichte (general confession) . Nevertheless, there is continuing interest in the subject and the question

Why do we never see interferences between ( .

the dead and the live cat?

is still being asked, which repeats (1), in essence.

Let us answer this question at the example of the Schrödinger cat (44) . A look at pm of (46), which is simply half the sum of pM' (live cat) and pM* (dead cat), justifies this immediate reply:

Because there are no interference terms in if^u\

the final state of the cat.

It is true that there are such terms in P of (44), viz . the contributions involving pm (provided that pm does not vanish to begin with). But if pM' and pM }are macroscopically different , then the visibility V is surely zero, and the statistical operator pq of (47) contains no trace of these interference terms either.

In other words: Neither a measurement on the marker (the cat) alone, nor a measurement on the quanton alone is sensitive to the presence of the interference terms in P. One would have to measure a joint observable (which one? ), or better: form subensem-bles of cats according to the outcomes of suitable measurements on the quanton . Consider, for example,

-'Sometimes the term is also applied to superpositions of macroscopically different states of a single quantum degree of freedom; see, for example, reference [39]. This usage is misleading and should be discouraged, the more so because such superpositions are common in standard interferometric devices .

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

27

the quanton observable Q = a + cr* with eigenvalues Q' = 1. This yields subensembles characterized by

pff^/C + ^i^ + PM)- (65)

[Since p^' and p^' are assumed to be macroscopically different , p"m must be traceless; see Section 10. Thus, p^' is properly normalized to unit trace.] Cross terms are present here, and so one could expect that each subensemble would exhibit interferences between pjj,' and pM between the live state and the dead state of each cat plus ..., so to say.

Now, just like the cross terms in pq of (47) are noticeable only if an appropriate observable is measured, such as 0(0) of (31), the demonstration of these life/death interferences requires a corresponding marker observable . Naming this observable is easy if the marker is as simple as in Sects . 2-8, but utterly impossible for a macroscopic device that deserves to be called a Schrodinger cat . Phrased in words that summarize to some extent the findings of Süßmann [9] and Peres [50]: The cross terms (pM + pM) in (65) are ineffective, they are of no phenomeno-logical consequences ; the phenomenology associated with p^' is indiscernible from the one that goes with Pm = j(pm(+) + Pm(-)) of (46).

9.3.2. Selfadjoint operators and physical observables

"Naming this observable " is not only a matter of identifying a selfadjoint operator that is sensitive to the cross terms, which task is not so difficult, but rather a matter of finding an observable , that is: a physical quantity that can be measured. Whereas we take for granted that there is a corresponding selfadjoint operator to each observable22 (at least if we pay the price of some idealizations ), there is no reason why we should have an observable to each selfadjoint operator .

Here I am disagreeing with Dirac because I think that he is asking for too much in his well-known

"This excuses sloppy formulations such as "the quanton observable Q " in the stead of something more precise such as "the selfadjoint operator Q that corresponds to the x component of the quanton's spin vector " and the like.

statement (page 37 in [3], wording adapted to the present conventions23 ):

"The question now presents itself Can every selfadjoint operator be measured? The answer theoretically is yes. In practice it may be awkward, or perhaps even beyond the ingenuity of the experimenter , to devise an apparatus which could measure some particular selfadjoint operator , but the theory always allows one to imagine that the measurement can be made24."

The logical development of quantum mechanics does not need the axiom of a one-to-one correspondence between observables and selfadjoint operators . And, isn't it much more plausible , in view of the very few fundamental interactions in physics, that only a small subset of all thinkable selfadjoint operators correspond to physical observables ?

A much referred-to statement to the same extent can be found in von Neumann's book (page 167 of the German edition of [6], page 313 in the English translation):

" Den physikalischen Größen eines quantenmechanischen Systems sind , wie wir wissen , die hypermaximalen Hermiteschen Operatoren eindeutig zugeordnet [...] , und es ist zweckmäßig anzunehmen , daß diese Zuordnung eine ein-eindeutige ist d. h. daß wirklich jeder hypermaximale Hermitesche Operator einer physikalischen Größe entspricht . (In III.3. machten wir hiervon gelegentlich auch Gebrauch)25 ."

23Dirac's terminology is different from the modern one; in particular, his 'real dynamical variable' is today's 'Hermitean operator' and his 'observables' are today's 'selfadjoint operators.'

24This succinct quote is taken from the 4th Ed. (1958) of Dirac's seminal textbook ; the corresponding section of the 3rd Ed. (1947) is worded identically . The 2nd Ed. (1935) contains statements that amount to the same (see pages 28-30, 37 and 38). but there is nothing analogous in the 1st Ed. (1930).

It is also remarkable that Dirac knew about the difference between Hermitean and selfadjoint operators and that he appreciated its physical significance at the time when he completed the 2nd edition (November 1934), but not when he wrote the 1st edition. Presumably, this is evidence for lessons learned from von Neumann's book of 1932 [6].

A different attitude can also be encountered. When the mathematical physicist Friedrichs visited Heisenberg in the early 1930s and told him (proudly. I imagine) that the mathematicians had made an important contribution to the development of quantum mechanics by clarifying the said difference, Heisenberg responded with the question: "Is there one? " (I owe this charming anecdote to Haag. )

25In Beyer's translation : "There corresponds to each physical quantity of a quantum mechanical system, a unique hypermaximal Hermitian operator , as we know [...], and it is convenient to assume that this correspondence is one-to-one that is, that actually

28

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

The remark in parentheses sounds as if the one-to-one correspondence which, as I said above, is not needed as a building block of quantum mechanics were used by von Neumann in an important argument. A look at his section III.3, however, reveals that this is not the case. What he has actually made use of in this section is a much weaker property , namely that if X and Y are the selfadjoint operators of two physical quantities that can be measured simultaneously, then there are also observables that correspond to the linear combinations xX + j/Y with numerical coefficients x and y. Indeed, if you can simultaneously measure X and Y then you have already measured all such linear combinations. Surely, this very special case of simultaneously observable physical quantities cannot be regarded as evidence in support of the general claim of a one-to-one correspondence.

Harkening back to what is said about the Heisenberg cut in the Introduction, we note a similar (and related26) situation here. No formal criterion is at hand that would enable us to judge whether any given selfadjoint operator corresponds to a physical observable27 . And for reasons analogous to the ones that deny a rigorously definable location for the Heisenberg cut, such a criterion cannot exist.

9.3.3. Additional remarks

In addition to this notorious problem of identifying an appropriate physical quantity of the marker to be measured, in correlation with the outcome of the Q measurement on the quanton that identifies the subensembles of (65), there is the challenge to reproduce the initial conditions implicit in p{$ with the precision that is necessary to avoid a complete washing-out of the interference pattern looked for. In conclusion, one must agree with Schrödinger's judgment that the notion of a superposition state of a live and a dead cat is burlesk (ludicrous).

A remark on the notion of " superposition of two statistical operators" , such as p$ and p^, is in order. Superpositions of two (normalized) state vectors ,

each hypermaximal operator corresponds to a physical quantity. (We also made occasional use of this in III.3.)" A " hypermaximal Hermitian " operator is a selfadjoint one.

26Recall footnote 6.

27Of course, a physical observable must not be in conflict with the conservation of electric charge in other words, it must be gauge invariant but this condition, plus a couple of similarly elementary ones, is not enough.

|^i) and |^>2). are familiar textbook matter : Linear combinations \ip ) = \ipi)a\ + |^2)<*2. vvith complex coefficients , are also acceptable state vectors ; the requirement

|a,|2 + |Q2|2 + 2Re(a;a2(Vi|^2>) = 1 (66a)

ensures proper normalization . Superpositions of two statistical operators, p\ and pi, are constructed analogously . First one finds (Hilbert-Schmidt) operators A\ and Aj such that p\ = A\Ax and pi = a\A2, then A = a\A\ + aiAi yields the superposition p = A1 A. Here the normalization is enforced by the analogous restriction

|a1|2 + |a2|2 + 2Re(a;Q2tr{A)A2}) = l(66b)

on a 1 and 02. For example, the statistical operators Pu of (46) and p^] of (65) are superpositions of p^

and p$. If pi = \i>i)(ip\\ and p2 = |^2)(02| represent pure states, the state-vector superpositions yield p = \ip){ip\ which are particular statistical-operator superpositions; the latter kind is more general, however. Of course, there is no guarantee that arbitrary superpositions p of two physical states p\ and pi are also physical, as is demonstrated by Schrödinger's cat example .

10. Wave-particle Duality

10.1. Distinguishability of the ways

For 0 < \C\ < 1, we have fringes of reduced visibility in (49). Is there also a limited amount of WW information available ? Inasmuch as an interference pattern is a manifestation of the wave aspects of the quanton , whereas WW knowledge documents its particle aspects , we are heading for a quantitative statement about wave-particle duality . The principle of complementarity28 implies that wave and particle aspects are mutually exclusive, in the sense of (53), but it says nothing quantitative about the possible compromises .

We must read the WW marker to extract WW information, that is to say: we must measure a marker observable W [with (nondegenerate) eigenvalues W and eigenstates \W)] and see what we can infer from the measurement result.

28 A technical definition is given in [33].

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

29

Suppose that the eigenvalue W is found. In view of (46), this happens with the probability

{W\PM\W) = \(W\$\W) + \{W\$\W). (67)

v----------v----------' >--------------------'

slit 1 slit 2

Unless the contribution of one of the slits vanishes, we cannot be certain about the way. But we know which way to bet on, namely on the one that contributes most to the probability (W|pm|W)29. After many repetitions , the betting odds are given by the " likelihood yv f r guessing the way right,"

W = ^Maxj^W'l^lW), \(W\$\W)}. (68)

W *------------v------------' > y------------'

slitl slit2

The value of vv depends, of course, on the observable W that we choose to measure.

Since we guess right for 50% of the quantons when betting at random, w > \ must hold. And if we are lucky and know the way for each quanton with certainty , then w = 1. It is therefore natural to quantify the acquired WW knowledge by the number

/Cw = 2w - 1, 0 < K\n < 1 , (69)

so that

/Cw = 0 : no WW knowledge,

(70) K\N = 1 : full WW knowledge.

As a consequence of (68), yv 's given by30

^ = ^\(wM-p^)\W)\. (71)

w

The largest possible value of /Cyv is the distinguishability V of the ways,

V = Maxw/Cw (72)

Inasmuch as the distinguishability V represents Nature's information about the ways whereas the knowledge IC\j\i is what Man can learn from measuring the observable W , the inequality

29This betting strategy was introduced by Wootters and Zurek [51].

30The identity Max{x, y] = A(x + y) + A i - y\ is used in the transition from (68) to (71).

W < V (73)

states an obvious hierarchy : Man cannot be smarter than Nature. In passing we note that the numbers w> w . and T> quantify information or knowledge without invoking an entropic concept of some kind.

The equal sign holds in (73) when the eigenstates | W) of W are also eigenstates of p^ - Pm'; then the moduli of the eigenvalues of p$ - p are summed in (71), so that the distinguishability is explicitly given by

D^MlPÜ'-Päl}- C74 )

Thus, mathematically speaking, V is the distance between jp$ and jPm in the trace-class norm .

10.2. A digression : Asymmetric interferometers

Which-way information of a different kind is available in asymmetric interferometers where the a priori probabilities of the alternatives "through slit 1" and "through slit 2" are different. Then the way is predictable to some extent, so that we have some WW knowledge even without any WW marking.

In case of such an asymmetry , the initial statistical operator of the quanton is of the general form

PQ.asym = W\(Ti a + W2(Tat + y/wi W2 (fff + fV ) ,(75)

where wi and W2 are the respective probabilities for the two ways (w\ + w2 = 1, of course). The parameter e plays the same role here as in equation (9), which is the w\ = W2 = | version of (75). When betting on the more probable way , we get an a priori likelihood of

a phon =Max{w,,w2} = r(l +V) , (76)

which identifies the predictability V of the ways,31

V = \w\ W2I (77)

With a WW detection device in place, corresponding expressions for yv and V are found. They are

31 If something is known about the process of formation of the mixture pj ? ' m. that is: if a particular blend can be physically distinguished, then the a priori likelihood and thus the predictability can be larger; see reference [37] for a discussion of such situations. Equations (78) and (80) are equally valid under these circumstances.

30

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

obtained from (71) and (74), respectively by the replacements p^} - vnp and \p -+ w2p^. The inequality (73) is then supplemented by

J0),

The equal sign holds in (81) if pM represents a pure state, because then we have32

pft' = |M(0 )

V w,

(Oh

(78) pW = |M(1,)(M(1,| with |MiX)) = U\\M'

which sets an obvious lower bound on the knowledge

For the statistical operator (75) the fringes have an a priori visibility Vb that is given by

p = |M(2))(M(2)| with |M(2,) = t/2t|M<0)), Pm = |M,2))(M(I)|, (83a)

and the non-zero eigenvalues of pM' - pM' are given by (l -|(M(1)|M(2,)|2)1/2,sothat

Vo = 2^/w\W2 k

(79)

(0)

V = \J\ - |(M<'>|M<2>)|2 (83b)

As a consequence of the positivity of pQ asym which requires |e| < 1the predictability V and the a priori visibility Vq must obey the inequality

and

V = |trM{pM}| = |(M(1)|M(2,)|. (83c)

V2 + V02 < 1 .

(80)

Indeed, the upper limit of (81) is reached.

For a proof of the duality relation (81) we follow the strategy of [59]; situations that are more general than the ones considered here in particular, the extension to asymmetric interferometers and the complications arising from quanton-marker couplings of the form (61) are dealt with in [60, 61]. First, we

This observation has been made implicitly or explicitly by a number of authors in various physical contexts in which alternatives can become predictable to some extent. I am aware of [37, 51 - 58]; the measurements on neutrons and photons reported in [52, 53] and [56], respectively, are consistent with (80).

10.3. An inequality

The fringe visibility V of (51) and the distinguisha-bility V of (74) quantify the wave aspects of the quan-ton and its particle aspects , respectively, and so the stage is set for the quantitative statement about wave-particle duality . It reads [59]

(0)

employ the spectral decomposition of pM , C = Y,mkWk){Mk\

k

with

(84a)

mk > 0, Y, mk = 1 - {M3\Mk) = 6jk , (84b)

A,-

in pM' and pM' to arrive at

Pm' - Pg' = "U-(K1,> ,| - K>> >|)

A:

(85) where

V1 + V2 < 1 . Clearly, the extreme cases of (53), viz

V = 1 implies V = 0,

V = 1 implies V = 0,

(81)

'f>

(0h

(86)

r<2>\ - Jvt

(82)

M[-)=Ul\MD

Then we make use of the triangle inequality tr{|pa-A>|} a|} + tr{|pb|},

(87)

are an immediate consequence of (81), and there is room for the example of equations (52) in which V = 0 and V = 0.

32Note that |M(") and |M12') need not be orthogonal to each other. An analogous remark applies to |M") and |m2') in (86).

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

31

valid for any two trace-class operators pa and pb, to establish

V < ^m,trM{||M<1))(Mi1)| - |M<2>>(M<2)||} .

(88)

The lesson learned at the example of equations (83) enables us to evaluate these simpler traces , and we find

Vk^-\(Ml \M? )\2. (89)

k

It is convenient to express the amplitudes (Mkl)\Mk2)) in terms of two angle variables dk and k,

(M^M'2^ sin tWfc (90)

with 0 < flk < 7r/2 and 0 < y>/t < 2tt, so that

V <^2mkcostik. (91)

k

Likewise, the visibility is given by

V = | ^m^M^lM*2')! = | mt sin^e^|.

k k

(92) In conjunction with (91) this yields

P2+V2<^mJ^mt[--], (93)

3 k

where [ ] = cos #_, cos dk + sin " dj sin dk cos(yp ., -ipk) is the scalar product of two unit vectors in spherical coordinates, so that [ ] < 1, and we get

V2 + V2< (>fc)2 = (MpIS'})2 = 1 - (94)

fc

which is (81), indeed.

We emphasize that the proof of the duality relation (81) does not invoke an uncertainty relation of the Heisenberg-Robertson kind [62, 63], that is:

6X6Y>l-\(i[X,Y])\ (95)

for the spreads of two observables A' and Y and the expectation value of their commutator , and the same remark applies to the more general treatments in

[60,61]. Indeed, the mathematics used in demonstrating (95) is quite different and more elementary: One notes that the expectation value (A* A) is nonnegative and exploits this fact for A = 6Y(X - (X)) i6X (Y (Y)). The conclusion

The duality relation (81) and the uncertainty relation (95) are logically independent statements. Both ._ ,.

are consequences of the rules of the game we call quantum mechanics, but one does not imply the other.

is justified by this observation.

One should appreciate that (81) leaves a lot of room for simultaneous manifestations of wave and particle properties between the two extreme cases of (53) or (82). For example, we could have /Cyv = V = 80% so that we can guess the way right for (1 + V)/2 = 90% of the quantons while building up an interference pattern with a fringe visibility of as much as V = \/l - V2 = 60%. And even if we know the way with a confidence of (1 + T>)/2 = 99%, we may still have well visible fringes with V = 20%. In the related, yet somewhat different, context of (80) Greenberger and Yasin [57] remark that this " amazing result testifies to the power of the superposition principle " a good line to end on.

Ackno w led gerne n ts

My own understanding of the matters discussed in these notes has developed under the shaping influence of many colleagues. I am particularly grateful for the insights gained in discussions with J. A. Ber-gou , H.-J. Briegel , R. Glauber , R. Haag , W. E. Lamb Jr ., H. Rauch , A. Schenzle , A. Schlüter , M. O. Scully , G. Süßmann , H. Walther , C. F. von Weizsäcker , and last, not least the late J. Schwinger .

Part of this work was done at the Atominstitut in Vienna. I would like to thank Helmut Rauch and his collaborators for the hospitable environment they have provided, and the Technical University of Vienna for financial support.

Dedication

It is a great pleasure indeed to dedicate these notes to Professor Georg Süßmann on the occasion of his 70th birthday. His powerful performance in discussions, always having an enlightening example or analogy at his disposal, can shake anybody's self-confidence but then one is amply rewarded by emerging with convictions that are battle tested.

32

B.-G. Englert Remarks on Some Basic Issues in Quantum Mechanics

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Technical Terms

TTcount
quanton57
Phys23
WW marker23
B.-G. Englert Remarks19
QE12
Lett11
Sect11
interference pattern10
WW10
subensemble10
interferometer10
B.-G. Englert10
photon9
pq9
experimenter9
Heisenberg8
consequence8
quantum erasure8
polarization7
fringe visibility7
Mach-Zehnder interferometer7
quantons6
observable6
Inasmuch6
resonators5
slits5
interference fringes5
fringes5
entanglement5
Amer5
correlations5
amplification5
atom5
Geiger counter5
photons5
apparatus4
notion4
Dirac4
atoms4
probability4
statistical operator pq4
wave-particle duality4
SEPP4
physical observables4
selfadjoint operator4
supervisor4
QE measurement4
eigenvalue 13
blends3
subsystems3
outcomes3
detector3
superpositions3
critics3
Bhm3
quantum eraser3
WW detection device3
Schrdinger3
propagates3
particle aspects3
single-slit contributions3
eigenvalues3
selfadjoint operators3
G. Smann3
PQ3
double-slit interferometer3
resonator3
Sects3
duality relation (81)3
inequality3
exemplified3
distinguishability3
certainty3
remark3
formalism3
as-if-realities3
which-way detection3
Correlations3
eds3
yields2
EPR correlations2
quanton photon2
van Kampen2
asymmetric interferometers2
unit fringe visibility2
affirmative answer2
interferences2
state |5i2
classical alternatives2
astray2
priori likelihood2
impossibility2
crests2
subensembles2
troughs2
Alice's answer2
Quantum Eraser Erase2
sorting2
inhomogeneous magnetic field2
correspond2
N. G. van Kampen2
two-way interferometers2
magnetic atoms2
plausible2
predictability2
Schrdinger equation2
data sorting2
initial marker state p^2
neutrons2
electrons2
entangled quanton-and-marker system2
quantitative statement2
joint probability2
superposition2
entangled state2
Entanglement2
characterizes2
Superpositions2
joint probabilities2
double-slit interference pattern2
Professor Georg Smann2
observable W2
Asymmetric interferometers2
Wave-particle Duality2
Zurek2
Einstein-Podolsky-Rosen correlations2
initial statistical operator2
particles2
Haag2
amplitudes2
positivity2
statistical predictions2
eigenstates2
observables2
Smann2
macroscopic marker2
Gottfried2
WW measurement2
HWP2
C. F. von Weizscker2
quanton observable Q2
D. Bhm2
Vaidman2
Sei2
dynamical equations2
von Neumann's book2
objections2
Which-way Marking2
grateful2
transmitted amplitude2
probabilities2
Aharonov2
particular realization2
PBS2
phenomena2
orthogonal2
interference terms2
trace-class norm2
particle experiment2
repetitions2
EPR Correlations2
Interferometers2
mathematical aspects2
Bohr2
quantitative manner2
Blends2
selfadjoint2
entangled system2
Mathematical Foundations2
atom emits2
S. Drr2
As-if-realities2
unitary operators2
jargon2
Schrodinger's cat2
respective eigenvalues1
phase factors exp(ia)1
unitary operator Ulu_{{}1
Grangier1
single quantum degree1
good fringe visibility1
fringeless pattern1
implication works1
graphically1
Partial traces1
eigenvalues Q'1
adjoint1
complete WW knowledge (53)1
disappearance1
macroscopically different states1
Orsay1
distinguishes1
standard interferometric devices1
erased"1
QE type1
defining property1
doctoral thesis (Universite de Paris-Sud1
Schrodinger's cat problem1
unitary evolution1
Schrdinger cat1
entangles1
Generalbeichte (general confession)1
marker states p_{M}'1
radioactive atom1
quan-ton (|Si)1
gets entangled1
poisonous gas1
atom (= quanton)1
marker variables1
infinity1
photon degrees1
relevance1
decoherence1
quantized radiation field1
popular jargon1
,5m oc VjVmV1
ado1
quantum degree1
Schrodinger's (in)famous cat example1
macroscopic piece1
weights w^{M}1
identities1
alternative exhibits1
hides1
unfortunate case1
marker (the cat)1
form subensem-bles1
joint observable (which1
|a)-distinguishing measurement1
p_{M}^{} }are macroscopically different1
precondition1
interferences between _{(}1
practical feasibility1
distinguishing1
misnomer1
affirmative1
Schrdinger cat (44)1
9.3. Schrodinger's cat1
deadcats1
viz1
corresponding subensemble1
erV1
Kluwer1
prediction1
Bob's findings1
Appraisal1
logical implication1
|5i)/|S2)-distin-guishing measurement1
Manipulations1
equivalence stated1
eds.. Themes1
Superficially1
valid objection1
empirical experience1
pleasingly1
reasoning1
Bohmian Mechanics1
variance1
empirical reality1
EPR correlations stated1
critics' fallacy1
Chuck's answer1
Textbooks1
hypothetical outcome1
El experience1
post-festum critics1
pq equal1
quanton part1
eloquently1
QE sorting1
Mohrhoff1
color-coded subensembles first1
Uncertainty Relations (Balatonfred 1997)1
temporal order1
individual quanton1
complicated sorting schemes1
PostFestum1
one doubts1
sorting measurement1
critics assert1
postfestum reading1
eds.. Preludes1
conditional probabilities1
yellow quantons1
tantamount1
joint quanton-marker system1
theoretical physics1
corresponding marker states1
critique1
EPR correlations (28)1
Squeezed States1
link quanton states1
original state vector |P1
book-keeping devices1
yields subensembles1
modulus1
cjcj^1
projectors1
unitary operator1
Pff1
spin-flip operators1
statistical operator p^1
compactly1
resulting interference pattern1
obvious analogs1
joint marker-quanton system1
functioning WW detection device1
ellipsis1
unfortunate choice1
irrelevant pieces1
marker states orthogonal1
=Pm =Pm (52c)1
distinguishable final marker states1
Acta Phys1
(The relevant numerical measure1
technicalities1
judiciously1
Slov1
restriction1
pure states |Afi1
certain ensemble (here1
statistical operator P1
quantons entangled1
WW markers1
summarizes1
correct book-keeping1
correct predictions1
J. T Cushing1
recommended minimalistic interpretation1
state vector |P1
summarizing concisely1
measurement result (here1
Probabilistic predictions1
50-50 probability distribution changes1
corresponding state reduction1
realistic which-way markers1
simplicity1
idealization1
Wave-particle duality1
said measurement result1
suitably refined P_{sub} (here1
defining properties1
correct conditional probabilities1
conditioned1
marker-quanton interaction1
statement (page 371
pft'1
replacements p^^{}} - vnp1
pW1
statistical operator (75)1
priori visibility Vb1
B.-G. Englert Remarks1
mixture pj1
parameter e1
respective probabilities1
probable way1
Wiss1
yv1
Akad1
non-zero eigenvalues1
Soc1
various physical contexts1
predictable1
distinguisha-bility1
quan-ton1
implicitly1
quanton-marker couplings1
priori visibility Vq1
complications1
Bayer1
Abh1
PQ.asym1
asymmetry1
vv1
slitl slit21
acquired WW knowledge1
/Cyv1
Nature's information1
Physics World 3(8)1
W *------------v------------' > y------------'1
mechanics1
eigenvalue W1
slit 1 slit 21
contributes1
likelihood yv f^{r}1
Exact Quantum Mechanics1
Wootters1
digression1
jPm1
Which-way information1
priori probabilities1
WW knowledge1
D^MlP'-Pl}- C741
moduli1
obvious hierarchy1
numbers w> w1
invoking1
entropic concept1
Philos1
spectral decomposition1
H.-J. Briegel1
Englewood Cliffs 19901
W. E. Lamb Jr1
A. Schlter1
Atominstitut1
shaping influence1
2^{nd} ed.. World Scientific1
Ackno w1
superposition principle1
Naturwissenschaften 231
gerne1
E. Schrdinger1
collaborators1
hospitable environment1
J. von Neumann1
Fundamentals1
Mathematische Grundlagen der Quantenmechanik1
Springer1
R. T Beyer1
F. H. Frhner1
Contemp1
enlightening example1
analogy1
anybody's self-confidence1
convictions1
amazing result testifies1
Greenberger1
dj sin d_{k} cos(yp1
3 k1
Proc1
scalar product1
unit vectors1
k k1
angle variables d_{k}1
analogous remark1
immediate consequence1
two trace-class operators p_{a}1
equations (83) enables1
simpler traces1
spherical1
uncertainty relation1
uncertainty relation (95)1
exploits1
Both ._1
simultaneous manifestations1
particle properties1
expectation value (A* A)1
commutator1
Heisenberg-Robertson kind1
spreads1
observables A'1
expectation value1
marker observable W [with (nondegenerate) eigenvalues W1
possible compromises1
wie wir wissen1
Den physikalischen Gren eines quantenmechanischen Systems sind1
hypermaximalen Hermiteschen Operatoren eindeutig zugeordnet [...]1
und es ist zweckmig anzunehmen1
da diese Zuordnung eine ein-eindeutige ist d. h. da wirklich jeder hypermaximale Hermitesche Operator einer physikalischen Gre entspricht1
von Neumann's book (page 1671
P. D. D. Schwindt1
Nature London (submitted)1
axiom1
small subset1
thinkable selfadjoint operators correspond1
referred-to statement1
(In III.3. machten wir hiervon gelegentlich auch Gebrauch)^{25}1
Appl1
mathematicians1
mathematical physicist Friedrichs1
charming anecdote1
Haag.1
^{25}In Beyer's translation1
Hermitean1
identically1
'real dynamical variable'1
'observables'1
^{24}This succinct quote1
Dirac's seminal textbook1
particular selfadjoint operator1
ingenuity1
Peres1
Phrased1
phenomeno-logical consequences1
phenomenology1
9.3.2. Selfadjoint operators1
Schrodinger cat1
macroscopic device1
K. Drhl1
unit trace.] Cross terms1
appropriate observable1
life/death interferences1
corresponding marker observable1
corresponding selfadjoint operator1
idealizations1
wording1
vector1
present conventions^{23}1
T Nonn1
answer theoretically1
quanton's1
selfadjoint operator Q1
disagreeing1
Nature London 3951
excuses sloppy formulations1
stead1
F. De Martini1
unique hypermaximal Hermitian operator1
ensures proper normalization1
acceptable state vectors1
analogously1
Aj1
superposition p1
vvith complex coefficients1
^{ar}e familiar textbook matter1
(normalized) state vectors1
statistical operators"1
hypermaximal operator1
hypermaximal Hermitian1
physical observable1
normalization1
analogous restriction1
WW knowledge documents1
manifestation1
complementarity^{28}1
quantitative1
Physics World 4(1)1
WW information available1
10.1. Distinguishability1
statistical operators Pu1
state-vector superpositions yield p1
arbitrary superpositions p1
Schrdinger's cat example1
superposition state1
Comm1
Nature London 3901
observable physical quantities1
Harkening1
similar (and related^{26}) situation1
formal criterion1
numerical coefficients x1
linear combinations xX + j/Y1
two-slit interferometer1
parentheses1
von Neumann1
weaker property1
physical observable^{27}1
reasons analogous1
initial conditions implicit1
J.-M. Levy-Leblond1
complete washing-out1
Atom Interferometry1
Schrdinger's judgment1
reproduce1
correlation1
rigorously definable location1
criterion1
notorious problem1
appropriate physical quantity1
macroscopically different1
paraphrasing Orwell1
question (1) concerns1
decoherence process1
occurrence1
factual events1
manipulation1
ad-hoc nature1
Ghirardi-Rimini-Weber scheme1
classical rest1
inexact1
assigning1
uniquely1
quantum/classical border1
individual rain drop1
fundamental concept(s)1
double-slit interferometers (for light1
(symmetric) two-way interferometers1
Mach-Zehnder interferometers1
neutron kind1
biprism interferometers1
mathematical structure1
l v _{v} 1
intriguing program1
Two-way Interferometers1
corresponding statistical operator (vulgo1
slit 1 slit 21
enlarges1
selfad-joint operators1
Geiger counters1
detectors (such1
provocative word1
intention1
(quasi-)-classical rest1
thermometers1
macroscopic devices1
cetera amplifies1
quantum marker^{4}1
datum1
center-of-mass state (the marker state)1
macroscopically recognizable signal1
fundamental empirical fact1
sensibly1
quantum interferences1
interference1
sharp boundary1
quantum side1
^{6}In anticipation1
Peierls1
own convictions1
physicists1
empirical fact1
quantum/classical division1
outspoken advocate1
Stern-Gerlach interferometers1
Ramsey-Borde interferometers1
manipulate1
corresponding statistical operator1
Einstein-Podolsky-Rosen (EPR) type1
manner Alice1
extreme operationalism1
correlated1
quanton interacts1
Which-way Detection1
two quantum alternatives1
quanton hits1
optical path lengths1
which-way (WW) detection1
human observer1
combined quanton-marker system1
other state vector |...)1
van Kampen's caveat1
observable phenomena1
consequences (Theorem IV1
observable M1
philosophical impetus1
additional on-tological meaning1
shorthand1
mini-malistic interpretation1
state vectors1
minimalism1
coherence1
complex number e. Knowledge1
neutron1
electron1
subscript _{Q}1
terminology1
implicit generality1
(photon1
interfering quantum1
photon-pair interferometers1
two-level atoms1
noun quanton (suggested1
Bunge1
Levy-Leblond1
subtleties1
single-slit diffraction pattern1
<p dependence exhibits interference fringes1
interferometric phase difference^{9}1
' rnun ^{u}' v' J1
Kmax ^{T} Pmin1
pZ1
various arrival times1
Briegel1
necessity1
interferometers1
reference [24] report1
telling analogy1
basic physical quantity1
photographic process matter1
equation1
is differentiated^{1}1
unanswerable questions1
concise exposition1
simple two-way interferometers1
blackboard1
satisfactory answer1
Reprint requests1
70th birthday1
Garching1
good textbooks1
subject satisfactorily1
brief repetition1
refinements1
statistical operator (Section 6)1
standard interferometers1
state reduction (Section 8)1
offering additional remarks1
Hubert space vectors1
incorporating devices1
illustrate different methods1
understanding evolves1
agreed-upon interpretation1
Tbingen www.znaturforsch.com1
certain state"1
10.1. Distinguishability1
9.3.2. Selfadjoint operators1
Schrdinger's cat1
as-if realities1
Contents 11
1. Introduction1
1.1. Motivation1
entangled systems1
which-way detectors1
Hans-Kopfermann-Strae 11
Max-Planck-Institut fr Quantenoptik1
Atominstitut der sterreichischen Universitten1
Stadionallee 21
Two-way interferometers1
1.2. A confession1
2. Two-way Interferometers1
9. What1
8. PostFestum1
9.1. More realistic which-way markers1
9.2. The as-if-reality1
9.3. Schrdinger's cat1
7. Quantum Erasure1
6.1. Blends correspond1
3. Which-way Detection1
4. Examples1
5. Entanglement1
6. Mixtures1
suffices1
quantum kinematics1
experimentally testable aspects1
Bohmian mechanics1
additional elements (namely hypothetical particle trajectories)1
trajectories1
implausible properties1
pseudo-classical mechanics1
appealing1
quantum mechanical predictions1
disagreement1
uneasy1
fundamentally probabilistic world view1
intrinsically deterministic universe1
invalidate1
infer quantum mechanics1
magnetic silver atom1
apparatus exhibits quantum features1
Stern-Gerlach apparatus1
atomic magnetism1
quantum mechanical details1
measuring apparatus (a photographic plate1
Quantum mechanics deals1
mathematical statements1
convincing1
Riesz-Fejer theorem1
other books worthy1
single observational fact1
conception1
Schenzle's instructive article1
quantum Zeno effect1
undergraduates1
confession1
opening paragraphs1
certain spatial region1
time dependences1
effected1
unitary transformation1
Detailed temporal evolution1
causal order1
experimental findings^{2}1
theoretical conclusions1
van Kampen's [10] remarks1
marginal issue1
Quantum mechanics works (and1
rel-ativistic extension1
renormalized quantum field theory1
Schrodinger's seminal essay1
liking1
Planck1
Einstein1
Pauli1
classic textbooks1
eigenvalue1
quanton's particle aspects1
convex sum1
decomposition1
projectors (the statistical operators1
given mixture p1
statistical weights w_{k}1
identical left-hand sides1
Gemisch1
good as-if-realities1
corresponding abundance1
blends (German1
Gemenge1
mixture (German1
spectral decomposition(s)1
sums1
marker-quanton entanglement1
Einsteinian causality1
resulting equivalence stated1
surmise1
ti 4 0 ones1
other (hypothetical nonlinear additions1
indication1
blends pq1
possible projectors |Si)(Si|1
B.-G. Engiert Remarks1
preconceptions1
plethora1
(projectors to) pure states1
verschrnkt1
PmPq1
verheddert1
coined1
individual probabilities1
PtA1
statistical operator characterizes1
uses hyperfine sub-levels1
atom interferometer1
actual realization1
(see Sect1
quanton-marker interaction1
quanton-and-marker system1
state |Af2,52) (with equal probability)1
orthonormal states1
inconsistencies1
expectation values1
symmetrized product AB + BA1
EPR ones1
6.1. Blends correspond1
Mixtures1
classical mixture1
disentangled states1
statistical operator (21)1
quantum-mechanical EPR type1
preceding paragraph1
Berthold-Georg Englert1
circumstance1
|M+)/|M_)-distinguishing measurement1
name quantum erasure (QE)1
red/green sorting1
wavelike properties1
state |M+1
unknowable1
green quantons yield1
perfect fringe visibility1
antifringes1
red quanton1
red quantons1
WW sorting processes1
QE sorting amounts1
quanton traversing1
late-choice aspect1
corresponding operators commute1
red/green label (QE sorting)1
blue/yellow label (WW sorting)1
sortings1
detour1
inter-ferometric 0(0) measurement1
von Weizscker1
Wheeler's efforts1
marker photon1
original interference pattern1
red quantons exhibit1
ambiguities1
discriminates1
fringeless sum1
Quantum Erasure1
following scenario1
as-if character1
equivalent as-if-realities1
Irrespective1
^{16}Ballentine discusses1
textbook1
instructive example1
interferometer phase <p1
replacements1
distinguishes |Mi1
blue quantons1
distinction1
polarization-sensitive beam splitter1
red subensemble1
blue quantons exhibit1
subensem-bles1
<p ip 7r1
measurement results 01
blue subensemble1
yellow subensemble1
Konstanz1
Rempe1
one atom1
Young's double-slit interferometer1
relevant photonic degrees1
different instants1
E^HMO1
collimated1
symbolic wave trains1
pointer variable1
von Neumann measurement1
instructive discussions1
resonators slit plate1
Atomic de Broglie waves1
entangled photon pairs SEPP emits polarization-entangled EPR pairs1
polarizing beam splitter PBS1
half-wave plate HWP1
half-transparent mirror HTM1
orbital degrees1
phase shifter PS1
quanton states1
polarized photon1
photon's polarization degree-of-freedom1
Vertical polarization (v)1
horizontal polarization (h)1
second photon1
half-wave plate (HWP)1
measured observable1
protective measurement1
collapse1
(termed1
Wheeler1
which-way detection device1
sketched1
fringeless background1
marker's degree(s)1
quanton's wave aspects1
interference fringes manifests1
complementarity1
quanton's statistical operator1
tracing P1
Young interferometer1
atomic de Broglie waves1
possible outcomes1
traditional iwi Neumann measurement1
Anandan1
intriguing1
weak measurement |281
EPR type1
correlations (of1
de Broglie waves1
partial amplitudes fit1
Ithaca interpretation1
assertion1
center-of-mass degrees offreedom1
photonic degrees1
marker (polarization)1
quanton (orbital motion)1
entangled quanton-and-marker state1
Jaeger1
Shimony1
photon modes1
physical system (the resonators1
up-going photon1
transmits horizontally polarized ones1
horizontal polarization1
ordinary photodetection1
three examples exhibit1
privileged photon mode1
Ramsey interferometers1
systematic dependence1
Los Alamos National Laboratory1
final polarization state (in1
Drr1
Nonn1
Schwindt1
Kwiat1
Rydberg states1
resonant interaction1
dispersive interaction1
systematic loss1
reflects vertically polarized photons1
polarizing beam splitter (PBS)1
photon's orbital amplitude1
half-transparent mirror (HTM)1
phase shifter (PS)1
symmetric output port1
asymmetric1
entering photon encounters1
one-photon-at-a-time fashion1
selected resonator modes1
diffracted de Broglie waves1
interferometric phase <p1
atom hits1
vertically polarized"1
horizontally polarized"1
horizontally polarized1
emitted pairs1
^{l4}Such sources1
recent tele-portation experiments1
SEPPs1
vertically1
one photon1
third example (Fig1
variant1
Ballentine's textbook1
entangled photon pairs (SEPP)^{14}1
fancy manners1

Person names

NameCount
M. O. Scully7
H. Walther6
Y. Aharonov5
L. Vaidman4
H. Rauch3
Z. Naturforsch3
J. Phys3
R. Haag3
J. Bell3
W. H. Zurek3
A. Zeilinger2
L. Vaidmann2
P. G. Kwiat2
K. Mattle2
H. Weinfurter2
J. Summhammer2
K. Gottfried2
R. Glauber2
G. Rempe2
Z. Phys2
A. Schenzle2
J. A. Wheeler2
Prentice Hall2
S. Haroche1
D. Tuppinger1
N. Y. Acad1
J. M. Raimond1
D. D. Schwindt1
A. Peres1
N. Gisin1
W. K. Wootters1
U. Mohrhoff1
T. Nonn1
L. Mandel1
North Holland1
L. Van-Hove1
H. Feshbach1
A. De-Shalit1
A. R. Marlow1
D. Han1
I. Man'ko1
S. Kim1
J. Janszky1
R. J. Finkelstein1
S. Deser1
A. Yasin1
D. M. Greenberger1
R. Schieder1
A. Prieur1
C. Wunderlich1
W. Heisenberg1
S. Goldstein1
A. Fine1
H. P. Robertson1
P. Mittelstaedt1
S. Branca1
M. Nauenberg1
P. R. Holland1
J. D. Trimmer1
R. Peierls1
G. C. Ghirardi1
P. Berman1
T. Weber Phys1
A. Rimini1
L. E. Ballentine1
A. M. Dirac1
A. Here1
P. One1
P. And1
J. A. Ber-gou1
W. E. Lamb1
Helmut Rauch1
J. Schwinger1
A. Einstein1
B. Podolsky1
A. Shimony1
G. Jaeger1
S. Popescu1
M. Brune1
E. Hagley1
X. Maitre1
J. Dreyer1
L. Hardy1
F. De1
D. Z. Albert1
N. D. Mermin1
N. Rosen1
J. Anandan1
D. Bouwmeester1
D. Boschi1
M. Eibl1
A. Maali1

Locations

Locationcount
Kampen4
London4
New York3
Singapore2
Berlin2
Amsterdam1
Konstanz1
Austria1
USA1
Wien1

Personname im Literaturverzeichnis

M. O. Scully6
Y. Aharonov5
H. Walther5
L. Vaidman4
W. H. Zurek3
J. Phys3
J. Bell3
L. Vaidmann2
H. Weinfurter2
Z. Naturforsch2
A. Zeilinger2
R. Haag2
J. Summhammer2
H. Rauch2
K. Mattle2
K. Gottfried2
Prentice Hall2
Z. Phys2
P. G. Kwiat2
J. A. Wheeler2
G. Rempe2
R. Schieder1
D. D. Schwindt1
A. Prieur1
W. K. Wootters1
U. Mohrhoff1
N. Gisin1
A. Peres1
T. Nonn1
N. Y. Acad1
D. Tuppinger1
P. Mittelstaedt1
W. Heisenberg1
North Holland1
L. Van-Hove1
H. Feshbach1
A. R. Marlow1
D. Han1
I. Man'ko1
S. Kim1
J. Janszky1
A. De-Shalit1
R. J. Finkelstein1
S. Haroche1
L. Mandel1
A. Yasin1
H. P. Robertson1
A. Fine1
S. Deser1
S. Goldstein1
D. M. Greenberger1
G. Jaeger1
T. Weber Phys1
A. Rimini1
G. C. Ghirardi1
P. Berman1
A. Einstein1
N. Rosen1
B. Podolsky1
R. Peierls1
M. Nauenberg1
A. Schenzle1
R. Glauber1
A. M. Dirac1
L. E. Ballentine1
P. R. Holland1
J. D. Trimmer1
N. D. Mermin1
D. Z. Albert1
E. Hagley1
M. Brune1
A. Shimony1
J. Dreyer1
X. Maitre1
C. Wunderlich1
A. Maali1
S. Popescu1
L. Hardy1
D. Bouwmeester1
J. Anandan1
M. Eibl1
D. Boschi1
F. De1
S. Branca1
J. M. Raimond1

Personname im Text

Person im TextHäufigkeit
A. Schenzle1
H. Rauch1
M. O. Scully1
H. Walther1
Helmut Rauch1
J. Schwinger1
W. E. Lamb1
R. Haag1
P. One1
P. And1
A. Here1
J. A. Ber-gou1
R. Glauber1
Z. Naturforsch1

Technische Terme im Literaturverzeichnis mit Häufigkeit

Technischer TermHäufigkeit
Phys23
Lett11
B.-G. Englert9
Amer5
eds3
Sei2
Mathematical Foundations2
N. G. van Kampen2
S. Drr2
G. Smann2
D. Bhm2
K. Drhl1
F. De Martini1
T Nonn1
Nature London 3951
Appl1
P. D. D. Schwindt1
Grangier1
Nature London (submitted)1
Acta Phys1
eds.. Themes1
Kluwer1
eds.. Preludes1
theoretical physics1
Uncertainty Relations (Balatonfred 1997)1
Squeezed States1
Appraisal1
Bohmian Mechanics1
Nature London 3901
Orsay1
Slov1
Wave-particle duality1
J. T Cushing1
doctoral thesis (Universite de Paris-Sud1
Physics World 4(1)1
R. T Beyer1
Springer1
Englewood Cliffs 19901
2^{nd} ed.. World Scientific1
E. Schrdinger1
Mathematische Grundlagen der Quantenmechanik1
J. von Neumann1
C. F. von Weizscker1
Contemp1
F. H. Frhner1
Fundamentals1
Naturwissenschaften 231
Proc1
Physics World 3(8)1
Exact Quantum Mechanics1
mechanics1
Comm1
Atom Interferometry1
Wiss1
Akad1
Philos1
Soc1
Abh1
Bayer1
J.-M. Levy-Leblond1

Technische Terme im Text

Technischer TermHäufigkeit
quanton57
WW marker23
B.-G. Englert Remarks19
QE12
Sect11
subensemble10
interference pattern10
interferometer10
WW10
pq9
experimenter9
photon9
consequence8
Heisenberg8
quantum erasure8
fringe visibility7
polarization7
Mach-Zehnder interferometer7
Inasmuch6
quantons6
observable6
slits5
interference fringes5
resonators5
fringes5
atom5
photons5
entanglement5
amplification5
correlations5
Geiger counter5
wave-particle duality4
physical observables4
QE measurement4
statistical operator pq4
apparatus4
notion4
Dirac4
atoms4
supervisor4
probability4
SEPP4
selfadjoint operator4
superpositions3
subsystems3
certainty3
Schrdinger3
which-way detection3
blends3
Bhm3
single-slit contributions3
remark3
selfadjoint operators3
Sects3
quantum eraser3
particle aspects3
detector3
duality relation (81)3
eigenvalues3
inequality3
distinguishability3
PQ3
Correlations3
outcomes3
critics3
eigenvalue 13
propagates3
WW detection device3
double-slit interferometer3
resonator3
formalism3
as-if-realities3
exemplified3
EPR correlations2
jargon2
neutrons2
interferences2
affirmative answer2
Zurek2
classical alternatives2
van Kampen2
inhomogeneous magnetic field2
magnetic atoms2
particles2
Haag2
positivity2
state |5i2
superposition2
electrons2
amplitudes2
double-slit interference pattern2
astray2
Entanglement2
entangled state2
data sorting2
eigenstates2
unit fringe visibility2
initial marker state p^2
initial statistical operator2
particular realization2
probabilities2
repetitions2
Alice's answer2
macroscopic marker2
plausible2
observable W2
asymmetric interferometers2
priori likelihood2
predictability2
quantitative statement2
Superpositions2
entangled system2
interference terms2
quanton observable Q2
particle experiment2
WW measurement2
orthogonal2
quanton photon2
characterizes2
joint probabilities2
transmitted amplitude2
PBS2
Vaidman2
grateful2
atom emits2
HWP2
joint probability2
entangled quanton-and-marker system2
sorting2
subensembles2
crests2
troughs2
correspond2
impossibility2
statistical predictions2
observables2
yields2
Aharonov2
Smann2
two-way interferometers2
objections2
As-if-realities2
Schrodinger's cat2
mathematical aspects2
quantitative manner2
Interferometers2
Bohr2
Quantum Eraser Erase2
phenomena2
EPR Correlations2
Blends2
Schrdinger equation2
dynamical equations2
Gottfried2
Which-way Marking2
Professor Georg Smann2
selfadjoint2
unitary operators2
Einstein-Podolsky-Rosen correlations2
Asymmetric interferometers2
von Neumann's book2
Wave-particle Duality2
trace-class norm2
6. Mixtures1
implication works1
complete WW knowledge (53)1
disappearance1
unfortunate case1
QE type1
unitary operator Ulu_{{}1
fringeless pattern1
phase factors exp(ia)1
respective eigenvalues1
corresponding subensemble1
5. Entanglement1
distinguishes1
defining property1
|a)-distinguishing measurement1
erased"1
marker states orthogonal1
(The relevant numerical measure1
technicalities1
judiciously1
functioning WW detection device1
joint marker-quanton system1
cjcj^1
projectors1
unitary operator1
distinguishable final marker states1
unfortunate choice1
adjoint1
Partial traces1
graphically1
marker states p_{M}'1
marker-quanton interaction1
ellipsis1
irrelevant pieces1
=Pm =Pm (52c)1
good fringe visibility1
practical feasibility1
popular jargon1
quantum degree1
ado1
Schrdinger cat1
relevance1
decoherence1
quantized radiation field1
infinity1
photon degrees1
3. Which-way Detection1
Generalbeichte (general confession)1
interferences between _{(}1
joint observable (which1
form subensem-bles1
macroscopically different states1
single quantum degree1
marker (the cat)1
2. Two-way Interferometers1
Schrdinger cat (44)1
viz1
p_{M}^{} }are macroscopically different1
,5m oc VjVmV1
marker variables1
affirmative1
erV1
distinguishing1
misnomer1
deadcats1
9.3. Schrodinger's cat1
identities1
alternative exhibits1
hides1
precondition1
macroscopic piece1
Schrodinger's (in)famous cat example1
Schrodinger's cat problem1
unitary evolution1
entangles1
atom (= quanton)1
4. Examples1
poisonous gas1
radioactive atom1
quan-ton (|Si)1
gets entangled1
weights w^{M}1
obvious analogs1
9.3. Schrdinger's cat1
corresponding marker states1
conditional probabilities1
9.2. The as-if-reality1
yellow quantons1
link quanton states1
EPR correlations (28)1
color-coded subensembles first1
critics assert1
postfestum reading1
critique1
tantamount1
9.1. More realistic which-way markers1
Manipulations1
|5i)/|S2)-distin-guishing measurement1
prediction1
9. What1
valid objection1
Superficially1
joint quanton-marker system1
quanton part1
pq equal1
equivalence stated1
Mohrhoff1
eloquently1
detour1
marker photon1
sortings1
late-choice aspect1
quanton traversing1
Wheeler's efforts1
von Weizscker1
10.1. Distinguishability1
QE sorting amounts1
inter-ferometric 0(0) measurement1
9.3.2. Selfadjoint operators1
corresponding operators commute1
red/green label (QE sorting)1
sorting measurement1
individual quanton1
temporal order1
QE sorting1
one doubts1
PostFestum1
blue/yellow label (WW sorting)1
fancy manners1
two-slit interferometer1
complicated sorting schemes1
Bob's findings1
8. PostFestum1
conditioned1
correct conditional probabilities1
said measurement result1
corresponding state reduction1
50-50 probability distribution changes1
defining properties1
suitably refined P_{sub} (here1
WW markers1
summarizes1
measurement result (here1
Probabilistic predictions1
realistic which-way markers1
simplicity1
resulting interference pattern1
standard interferometric devices1
spin-flip operators1
Pff1
compactly1
6.1. Blends correspond1
idealization1
pure states |Afi1
restriction1
statistical operator p^1
quantons entangled1
certain ensemble (here1
7. Quantum Erasure1
post-festum critics1
Chuck's answer1
critics' fallacy1
El experience1
hypothetical outcome1
logical implication1
empirical experience1
pleasingly1
Textbooks1
reasoning1
variance1
summarizing concisely1
correct predictions1
correct book-keeping1
statistical operator P1
state vector |P1
recommended minimalistic interpretation1
empirical reality1
EPR correlations stated1
original state vector |P1
book-keeping devices1
modulus1
macroscopically different1
respective probabilities1
PQ.asym1
parameter e1
probable way1
Atominstitut der sterreichischen Universitten1
asymmetry1
WW knowledge1
digression1
Which-way information1
Stadionallee 21
priori probabilities1
Hans-Kopfermann-Strae 11
yv1
priori visibility Vb1
non-zero eigenvalues1
priori visibility Vq1
Max-Planck-Institut fr Quantenoptik1
statistical operator (75)1
pW1
mixture pj1
B.-G. Englert Remarks1
replacements p^^{}} - vnp1
pft'1
jPm1
D^MlP'-Pl}- C741
likelihood yv f^{r}1
contributes1
W *------------v------------' > y------------'1
slitl slit21
vv1
slit 1 slit 21
eigenvalue W1
complementarity^{28}1
quantitative1
possible compromises1
marker observable W [with (nondegenerate) eigenvalues W1
entangled systems1
acquired WW knowledge1
numbers w> w1
invoking1
entropic concept1
moduli1
obvious hierarchy1
Wootters1
/Cyv1
which-way detectors1
Nature's information1
Two-way interferometers1
complications1
quanton-marker couplings1
superposition principle1
amazing result testifies1
Ackno w1
gerne1
shaping influence1
Greenberger1
particle properties1
exploits1
uncertainty relation (95)1
Both ._1
simultaneous manifestations1
H.-J. Briegel1
W. E. Lamb Jr1
enlightening example1
analogy1
anybody's self-confidence1
convictions1
hospitable environment1
collaborators1
A. Schlter1
G. Smann1
C. F. von Weizscker1
Atominstitut1
expectation value (A* A)1
commutator1
analogous remark1
immediate consequence1
two trace-class operators p_{a}1
equations (83) enables1
simpler traces1
spectral decomposition1
quan-ton1
implicitly1
various physical contexts1
predictable1
distinguisha-bility1
angle variables d_{k}1
k k1
Heisenberg-Robertson kind1
spreads1
observables A'1
expectation value1
uncertainty relation1
spherical1
3 k1
dj sin d_{k} cos(yp1
scalar product1
unit vectors1
as-if realities1
WW knowledge documents1
thinkable selfadjoint operators correspond1
small subset1
referred-to statement1
von Neumann's book (page 1671
Den physikalischen Gren eines quantenmechanischen Systems sind1
1. Introduction1
axiom1
present conventions^{23}1
answer theoretically1
ingenuity1
particular selfadjoint operator1
wie wir wissen1
hypermaximalen Hermiteschen Operatoren eindeutig zugeordnet [...]1
Dirac's seminal textbook1
identically1
Hermitean1
mathematical physicist Friedrichs1
^{24}This succinct quote1
'observables'1
und es ist zweckmig anzunehmen1
da diese Zuordnung eine ein-eindeutige ist d. h. da wirklich jeder hypermaximale Hermitesche Operator einer physikalischen Gre entspricht1
(In III.3. machten wir hiervon gelegentlich auch Gebrauch)^{25}1
'real dynamical variable'1
wording1
statement (page 371
macroscopic device1
corresponding marker observable1
Schrodinger cat1
Phrased1
Peres1
life/death interferences1
appropriate observable1
eigenvalues Q'1
yields subensembles1
WW sorting processes1
unit trace.] Cross terms1
phenomeno-logical consequences1
phenomenology1
stead1
selfadjoint operator Q1
quanton's1
vector1
excuses sloppy formulations1
disagreeing1
9.3.2. Selfadjoint operators1
1.1. Motivation1
corresponding selfadjoint operator1
idealizations1
mathematicians1
charming anecdote1
^{ar}e familiar textbook matter1
physical observable1
vvith complex coefficients1
acceptable state vectors1
ensures proper normalization1
hypermaximal Hermitian1
hypermaximal operator1
superposition state1
statistical operators"1
Schrdinger's cat1
(normalized) state vectors1
analogously1
Aj1
Schrdinger's cat example1
10.1. Distinguishability1
WW information available1
manifestation1
arbitrary superpositions p1
state-vector superpositions yield p1
superposition p1
normalization1
analogous restriction1
statistical operators Pu1
Schrdinger's judgment1
complete washing-out1
weaker property1
linear combinations xX + j/Y1
numerical coefficients x1
observable physical quantities1
von Neumann1
parentheses1
Haag.1
^{25}In Beyer's translation1
unique hypermaximal Hermitian operator1
Contents 11
Harkening1
similar (and related^{26}) situation1
appropriate physical quantity1
correlation1
reproduce1
initial conditions implicit1
notorious problem1
criterion1
formal criterion1
physical observable^{27}1
reasons analogous1
rigorously definable location1
1.2. A confession1
unknowable1
manipulate1
corresponding statistical operator1
theoretical conclusions1
Einstein-Podolsky-Rosen (EPR) type1
manner Alice1
Planck1
correlated1
optical path lengths1
Einstein1
which-way (WW) detection1
quanton interacts1
experimental findings^{2}1
extreme operationalism1
state vectors1
minimalism1
additional on-tological meaning1
philosophical impetus1
mini-malistic interpretation1
confession1
opening paragraphs1
human observer1
combined quanton-marker system1
shorthand1
quanton hits1
Which-way Detection1
reference [24] report1
telling analogy1
Briegel1
various arrival times1
interferometers1
necessity1
terminology1
implicit generality1
subtleties1
single-slit diffraction pattern1
Schrodinger's seminal essay1
liking1
Pauli1
complex number e. Knowledge1
coherence1
two quantum alternatives1
pZ1
classic textbooks1
interferometric phase difference^{9}1
<p dependence exhibits interference fringes1
' rnun ^{u}' v' J1
Kmax ^{T} Pmin1
van Kampen's caveat1
other state vector |...)1
possible outcomes1
traditional iwi Neumann measurement1
causal order1
Detailed temporal evolution1
unitary transformation1
EPR type1
correlations (of1
de Broglie waves1
partial amplitudes fit1
Ithaca interpretation1
assertion1
effected1
Anandan1
instructive discussions1
resonators slit plate1
Atomic de Broglie waves1
symbolic wave trains1
pointer variable1
von Neumann measurement1
intriguing1
weak measurement |281
protective measurement1
measured observable1
time dependences1
certain spatial region1
interference fringes manifests1
quanton's wave aspects1
complementarity1
quanton's statistical operator1
quanton's particle aspects1
undergraduates1
observable phenomena1
consequences (Theorem IV1
observable M1
eigenvalue1
tracing P1
marker's degree(s)1
which-way detection device1
sketched1
Young interferometer1
atomic de Broglie waves1
Wheeler1
collapse1
Schenzle's instructive article1
fringeless background1
quantum Zeno effect1
(termed1
subscript _{Q}1
neutron1
sensibly1
fundamental empirical fact1
physicists1
empirical fact1
quantum/classical division1
(quasi-)-classical rest1
experimentally testable aspects1
detectors (such1
Geiger counters1
provocative word1
intention1
outspoken advocate1
own convictions1
quantum side1
^{6}In anticipation1
selfad-joint operators1
appealing1
sharp boundary1
quantum interferences1
Bohmian mechanics1
Peierls1
interference1
pseudo-classical mechanics1
thermometers1
macroscopic devices1
Stern-Gerlach apparatus1
mathematical statements1
atomic magnetism1
quantum mechanical details1
convincing1
magnetic silver atom1
Quantum mechanics deals1
measuring apparatus (a photographic plate1
Riesz-Fejer theorem1
apparatus exhibits quantum features1
photographic process matter1
infer quantum mechanics1
datum1
additional elements (namely hypothetical particle trajectories)1
center-of-mass state (the marker state)1
macroscopically recognizable signal1
cetera amplifies1
quantum marker^{4}1
invalidate1
implausible properties1
trajectories1
basic physical quantity1
enlarges1
inexact1
double-slit interferometers (for light1
van Kampen's [10] remarks1
marginal issue1
Mach-Zehnder interferometers1
(symmetric) two-way interferometers1
mathematical structure1
slit 1 slit 21
l v _{v} 1
rel-ativistic extension1
Quantum mechanics works (and1
neutron kind1
biprism interferometers1
Levy-Leblond1
interfering quantum1
(photon1
electron1
Bunge1
noun quanton (suggested1
Stern-Gerlach interferometers1
Ramsey-Borde interferometers1
two-level atoms1
photon-pair interferometers1
corresponding statistical operator (vulgo1
renormalized quantum field theory1
Ghirardi-Rimini-Weber scheme1
ad-hoc nature1
decoherence process1
uneasy1
fundamentally probabilistic world view1
quantum/classical border1
classical rest1
assigning1
uniquely1
intrinsically deterministic universe1
question (1) concerns1
occurrence1
fundamental concept(s)1
intriguing program1
Two-way Interferometers1
conception1
single observational fact1
disagreement1
factual events1
manipulation1
quantum mechanical predictions1
individual rain drop1
collimated1
Young's double-slit interferometer1
statistical weights w_{k}1
given mixture p1
unanswerable questions1
spectral decomposition(s)1
sums1
concise exposition1
projectors (the statistical operators1
Gemisch1
identical left-hand sides1
decomposition1
convex sum1
blends pq1
possible projectors |Si)(Si|1
Einsteinian causality1
marker-quanton entanglement1
resulting equivalence stated1
surmise1
other (hypothetical nonlinear additions1
indication1
B.-G. Engiert Remarks1
equation1
preconceptions1
is differentiated^{1}1
mixture (German1
Gemenge1
quantum-mechanical EPR type1
Mixtures1
6.1. Blends correspond1
inconsistencies1
statistical operator (21)1
disentangled states1
refinements1
quanton-and-marker system1
state |Af2,52) (with equal probability)1
classical mixture1
brief repetition1
orthonormal states1
corresponding abundance1
good as-if-realities1
Berthold-Georg Englert1
blends (German1
plethora1
(projectors to) pure states1
expectation values1
simple two-way interferometers1
symmetrized product AB + BA1
EPR ones1
ti 4 0 ones1
preceding paragraph1
distinction1
polarization-sensitive beam splitter1
red subensemble1
red quantons exhibit1
distinguishes |Mi1
blue quantons1
yellow subensemble1
subensem-bles1
blue quantons exhibit1
70th birthday1
original interference pattern1
perfect fringe visibility1
|M+)/|M_)-distinguishing measurement1
circumstance1
name quantum erasure (QE)1
red/green sorting1
state |M+1
other books worthy1
green quantons yield1
antifringes1
red quanton1
red quantons1
blue subensemble1
measurement results 01
equivalent as-if-realities1
as-if character1
satisfactory answer1
subject satisfactorily1
instructive example1
textbook1
paraphrasing Orwell1
blackboard1
Irrespective1
^{16}Ballentine discusses1
good textbooks1
discriminates1
Reprint requests1
interferometer phase <p1
replacements1
<p ip 7r1
following scenario1
Quantum Erasure1
ambiguities1
fringeless sum1
Garching1
B.-G. Englert1
understanding evolves1
agreed-upon interpretation1
half-transparent mirror (HTM)1
entering photon encounters1
photon's orbital amplitude1
phase shifter (PS)1
symmetric output port1
one-photon-at-a-time fashion1
atom hits1
photonic degrees1
selected resonator modes1
diffracted de Broglie waves1
interferometric phase <p1
asymmetric1
vertically polarized"1
vertically1
emitted pairs1
horizontally polarized1
^{l4}Such sources1
one photon1
entangled photon pairs (SEPP)^{14}1
horizontally polarized"1
third example (Fig1
variant1
Ballentine's textbook1
center-of-mass degrees offreedom1
statistical operator (Section 6)1
entangled photon pairs SEPP emits polarization-entangled EPR pairs1
polarizing beam splitter PBS1
Vertical polarization (v)1
horizontal polarization (h)1
E^HMO1
different instants1
one atom1
quantum kinematics1
suffices1
relevant photonic degrees1
Hubert space vectors1
second photon1
half-wave plate HWP1
orbital degrees1
phase shifter PS1
quanton states1
state reduction (Section 8)1
half-transparent mirror HTM1
offering additional remarks1
half-wave plate (HWP)1
photon's polarization degree-of-freedom1
polarized photon1
recent tele-portation experiments1
SEPPs1
Konstanz1
atom interferometer1
uses hyperfine sub-levels1
actual realization1
Rempe1
Nonn1
Los Alamos National Laboratory1
systematic dependence1
final polarization state (in1
Drr1
(see Sect1
illustrate different methods1
verheddert1
coined1
Tbingen www.znaturforsch.com1
individual probabilities1
verschrnkt1
PmPq1
quanton-marker interaction1
statistical operator characterizes1
PtA1
certain state"1
Schwindt1
Kwiat1
ordinary photodetection1
three examples exhibit1
physical system (the resonators1
photon modes1
horizontal polarization1
up-going photon1
polarizing beam splitter (PBS)1
reflects vertically polarized photons1
transmits horizontally polarized ones1
standard interferometers1
quanton (orbital motion)1
marker (polarization)1
resonant interaction1
dispersive interaction1
systematic loss1
incorporating devices1
Rydberg states1
Ramsey interferometers1
entangled quanton-and-marker state1
Jaeger1
Shimony1
privileged photon mode1
wavelike properties1
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