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Can wave–particle duality be based on the uncertainty relation?
Stephan Du¨rr and Gerhard Rempea)Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany ͑Received 28 July 1999; accepted 27 January 2000͒ Wave and particle properties of a quantum object cannot be observed simultaneously. In particular,the fringe visibility in an interferometer is limited by the amount of which-way information whichcan be obtained. This limit is set by the recently discovered duality relation. So far, all derivationsof the duality relation are independent of Heisenberg’s uncertainty relation. Here we demonstratethat it is alternatively possible to derive the duality relation in the form of an uncertainty relation forsome suitably chosen observables. 2000 American Association of Physics Teachers. I. INTRODUCTION
This result was confirmed experimentally in Refs. 21 and 22.
None of the derivations of Eqs. ͑1͒ and ͑2͒ cited above Wave–particle duality refers to the fact that a quantum involves any form of the uncertainty relation. It therefore object can exhibit either wave or particle properties, depend- seems that ‘‘the duality relation is logically independent of ing on the experimental situation. In a double-slit experi- the uncertainty relation.’’ 13 In this article, we will show, ment, for example, the object must pass through both slits however, that for arbitrary which-way schemes, Eqs. ͑1͒ and simultaneously in order to create an interference pattern.
͑2͒ can always be derived in the form of a Heisenberg– This testifies to the object’s wave nature. On the other hand, Robertson uncertainty relation for some suitably chosen ob- performing a which-way experiment reveals which of the servables ͑which will turn out to be different from position slits each object passes through, manifesting its particle na- ture. However, performing a which-way experiment un-avoidably destroys the interference pattern.
This was illustrated in various gedanken experiments, II. PREDICTABILITY
such as Einstein’s recoiling slit1 or Feynman’s light In this section, we consider a two-beam interferometer microscope.2 In order to explain the loss of interference in without a which-way marker, as shown in Fig. 1. Let ͉ϩ͘ which-way experiments, one usually invokes Heisenberg’sposition–momentum uncertainty relation. This has been ana- and ͉Ϫ͘ denote the state vectors corresponding to the two lyzed in great detail by, e.g., Wiseman et al.3 However, ways along which the object can pass through the interfer- Scully, Englert, and Walther4 pointed out that such an expla- ometer. After passing the first beam splitter, the density ma- nation need not always be possible, but that the entanglement trix in a representation with respect to the basis ͕͉ϩ͘,͉Ϫ͖͘ between the which-way marker and the interfering quantum object can always explain the loss of interference. Severalexperiments support this point of view.5–11 This entanglement need not always be perfect. In general, a measurement performed on the which-way marker yields only incomplete which-way knowledge. In order to quantify ϩ and wϪ that the object moves along one way or the other, respectively, fulfill Tr͕␳͖ϭw how much which-way information is available from such a measurement, one typically uses the ‘‘distinguishability,’’ D.
1. The magnitude of the difference between these prob- With incomplete which-way information stored, one obtains interference fringes with a reduced visibility, V, which is limited by the so-called duality relation which is obviously determined by the reflectivity of the first beam splitter. P quantifies how much which-way knowledgewe have. For Pϭ0, corresponding to a 50:50 beam splitter, This fundamental limit was recently discovered by Jaeger, we have no which-way knowledge, whereas for Pϭ1, we Shimony, and Vaidman,12 and independently by Englert.13 It know precisely which way the object takes.
can be regarded as a quantitative statement about wave– Without loss of generality, we assume that the second particle duality. In the special case, where full which-way beam splitter is a 50:50 beam splitter. Taking into account information is stored, Dϭ1, it implies that the interference the phase shift ␸ between the two interferometer arms, the fringes are lost completely, Vϭ0. The first experimental tests upper output beam corresponds to the state vector ͉u␸͘ of the duality relation have been performed recently.14,15 ϭ(͉ϩ͘ϩei␸͉Ϫ͘)/&. The intensity in this beam is Incomplete which-way information can alternatively be obtained without a which-way marker by setting up the in- u␸͉␳͉u␸͘ϭ 1 1ϩ2͉␳ terferometer such that the particle fluxes along the two ways differ. In this case, the which-way knowledge is expressed in Ϯϭ͉␳Ϯ͉ei␸0. The visibility of this interference pattern the form of the so-called ‘‘predictability,’’ P, which is lim- Am. J. Phys. 68 ͑11͒, November 2000
2000 American Association of Physics Teachers III. DUALITY RELATION
Let us now add a second quantum system ͑called which- way marker͒ to the interferometer. When an object is passingthrough the interferometer, a suitable interaction shall Fig. 1. Scheme of a typical two-beam interferometer. The incoming beam change the quantum state of the which-way marker depend- ͑left͒ is split into two beams, denoted ͉ϩ͘ and ͉Ϫ͘. After reflection from ing on the way the object took. This creates an entanglement mirrors, the phase of one of the beams is shifted by ␸. Next, the two beams between the which-way marker and the way of the object. A are recombined on a second beam splitter. Due to interference, the intensi- later measurement on the which-way marker can then reveal ties of the two outgoing beams vary as a function of the phase shift ␸.
which way the object took. In other words, which-way infor-mation is now stored in the which-way marker. For simplic-ity, we assume that the which-way marker does not sufferfrom decoherence25 ͑at least as long as we do not couple the where Imax and Imin denote the maximum and minimum in- marker to a macroscopic ‘‘needle’’͒.
tensities. The relation, Eq. ͑2͒, limiting visibility and predict- ability can easily be derived from Tr͕␳2͖ϭw2 tot denote the density matrix of the total system ͑ob- ject plus which-way marker͒ after the interaction ͑but before ϩ2͉␳Ϯ͉2ϭ͕1ϩP2ϩV2͖/2р1.
the phase shifter and the second beam splitter͒. Again, we We will now show that this inequality can alternatively be denote the pseudospin corresponding to the ways by ␴x , ␴y , obtained in the form of a Heisenberg–Robertson uncertainty z . And again, we choose the relative phase between states ͉ϩ͘ and ͉Ϫ͘ such that ͗ϩ͉Tr ͕␳ ͖͉Ϫ͘ ⌬A⌬Bу 1͉͓͗A,B͔͉͘, TrM denotes the trace over the which-way marker. Thus wereproduce the above results, in particular, which applies to each pair of Hermitian operators A and B,with the expectation values and standard deviations of opera- tors defined as ͗A͘ϭTr͕␳A͖ and ⌬Aϭͱ͗A2͘Ϫ͗A͘2, re- In order to read out the which-way information, we mea- sure an observable W of the which-way marker with eigen- In order to find suitable operators A and B, we investigate 1 , w 2 ,. ͖ and an orthonormal basis of eigenstates 1 , ͉ w 2 ,. ͖ . Let p ( Ϯ , w i) denote the joint probability i is found and that the object moves along way ͉Ϯ͘. If wi is found, the best guess one can make about the way is toopt for way ͉ϩ͘ if p(ϩ,wi)уp(Ϫ,wi), and for way ͉Ϫ͘ otherwise. This yields the ‘‘likelihood for guessing the way wϩϪwϪ . Obviously, ͗␴z reflects our which-way knowledge, whereas ͗␴ ͘ max͕p͑ϩ,wi ,p͑Ϫ,wi .
Without loss of generality, we choose the relative phase be- W can vary between 1/2 and 1, it is natural to scale this quantity by defining the ‘‘which-way knowledge’’ 26 tween states ͉ϩ͘ and ͉Ϫ͘ such that ␳Ϯ is real, i.e., ␸ ϭ 2LW 1ϭ ͚ p͑ϩ,wi p͑Ϫ,wi so that 0рKW 1. Obviously, KW depends on the choice of wave character and particle character of the ensemble, re- the measured observable W. In order to quantify how much spectively. The standard deviations of these observables, which-way information is actually stored, the arbitrariness ofthe read-out process can be eliminated by defining the are easily obtained, because ␴2ϭ␴2ϭ␴2ϭ1. Using the
2i ͚l jkl l , we can now evaluate the uncertainty relation, Eq. ͑7͒, for all possible pairs of the which is the maximum value of KW that is obtained for the best choice of W. The distinguishability is limited by the duality relation, Eq. ͑1͒, which has been derived in Refs. 12 1ϪV2ϭ⌬␴ ⌬␴ у͉͗␴ ͉͘ϭ and 13 without using the uncertainty relation.
ͱ1ϪP2ϭ⌬␴ ⌬␴ у͉͗␴ ͉͘ϭ We will now show that the duality relation—just as Eq.
͑2͒—can alternatively be derived in the form of a Heisenberg–Robertson uncertainty relation for some suitably chosen observables. For that purpose, let Equation ͑14͒ yields a trivial result, because standard devia-tions are non-negative by definition. However, Eqs. ͑12͒ and ͑13͒ are equivalent to the desired relation, Eq. ͑2͒. Hence, for the case without a which-way marker, Eq. ͑2͒ can be derivedin the form of an uncertainty relation for the components of denote which way to bet on if the eigenstate ͉w ͘ Am. J. Phys., Vol. 68, No. 11, November 2000 Second, we note that for the case without a which-way K ϭ ͚ ⑀ ͑͗ ϩ͉␳ ͉ ϩ͘Ϫ͗ Ϫ͉␳ ͉ Ϫ͒͘ marker, Eq. ͑2͒ is equivalent to the uncertainty relations for z , Eqs. ͑12͒ and ͑13͒. This equivalence can be read in both directions: In one direction, as discussed above, the uncertainty relation implies Eq. ͑2͒. In the other direction, Eq. ͑2͒ implies the uncertainty relation for these where we used ␴ ϭ͉ϩ͗͘ϩ͉Ϫ͉Ϫ͗͘Ϫ͉ the trace over the total system. Let us define the observable Third, we would like to draw attention to the fact that the result. This is somewhat surprising, because from Eq. ͑10͒ we concluded that ␴x represents the wave character, whereas ⑀ ϭ1 and ͓␴
z represents the particle character. Since we are investigat- x , W ⑀͔ ϭ ͓ ␴ y , W ⑀͔ ing the limit for the simultaneous presence of wave character z , W ⑀͔ ϭ 0. Inserting W ⑀ into Eq. ͑21͒, we obtain and particle character, one might have guessed that the un- Note that we are considering a joint observable of the total ever, this is not the case. Instead, ⌬␴y is employed in our system ͑object plus which-way marker͒ here, which is calculation. An intuitive interpretation of ␴y in terms of a clearly necessary to explore the correlations between the wave picture or a particle picture is not obvious.
which-way marker and the way taken by the object.
Next, we would like to mention that the observables whose uncertainty relations we evaluate in Eqs. ͑12͒ and ͑13͒ maximized. For simplicity, we will denote the corresponding depend on the density matrix, ␳. In the presentation in Sec.
observable defined by Eq. ͑22͒ by W ͑ II, this fact is somewhat hidden in our choice of the relative phase of states ͉ϩ͘ and ͉Ϫ͘, i.e., ␸ ϭ ␳ becomes more obvious, if we consider arbitrary values of ␸0 . In this case, we can define the observables It is easy to see that ␴zW0 is Hermitian and that (␴zW0)2 ϭ1, so that its standard deviation is
Additionally, let us consider the observable ␴ fulfills (␴yW0)2ϭ1. As it is also Hermitian, its expectation
x , ␴ y , and ␴ z in our above presen- tation. Obviously, these observables depend on ␳ via ␸0 . As the commutation relations of the ⌺’s and ␴’s are the same,Eq. ͑2͒ can be derived analogously. The situation is similar Using the commutator ͓(␴yW0),(␴zW0)͔ϭ2i␴x , we can now write down the corresponding uncertainty relation. In Finally, we will discuss whether either correlations ͑i.e., combination with Eqs. ͑15͒, ͑25͒, and ͑26͒, we obtain entanglement͒ or uncertainty relations are more closely con- nected to wave–particle duality. For that purpose, we will investigate all the explanations for the loss of interference This directly yields the duality relation, Eq. ͑1͒. Alterna- fringes, referenced in Sec. I. We will sort these explanations tively, the commutator ͓␴x ,(␴yW0)͔ϭ2i␴zW0 can be used into three categories, depending on whether they employ which again yields the duality relation.
͑3͒ correlations and some uncertainty relation.
To summarize, we have demonstrated here that in an ar- bitrary which-way scheme, the duality relation can be ex- The textbook explanations for Einstein’s recoiling slit in Ref.
pressed in the form of a Heisenberg–Robertson uncertainty 1 and Feynman’s light microscope in Ref. 2 are based on the relation for some suitably chosen observables.
Englert–Walther explanation4 as well as the derivations ofthe duality relation in Refs. 12 and 13 are based on the cor- IV. DISCUSSION
relations. Our derivation as well as the discussion of Wise-man et al.3 make use of both the correlations and some un- The above calculation reveals a new aspect of the connec- certainty relation. This is because these calculations involve tion between wave–particle duality and the uncertainty rela- the density matrix for the total system, consisting of the ob- tion. We would like to add a few comments concerning the ject plus the which-way marker. Consequently, the full quan- tum correlations between these subsystems are embodied in Let us first point out that the uncertainty relation used in our calculation is not the position–momentum uncertainty The above categorization reveals a crucial point: The ex- relation. This is obvious, because, for example, the observ- planations for the loss of interference fringes involving only ables considered here have only two eigenvalues, namely the uncertainty relation are ͑so far͒ limited to a few special Ϯ1, whereas position and momentum have a continuous schemes. In other words: There are several other schemes for which no such explanation is known, see, e.g., Refs. 4 and Am. J. Phys., Vol. 68, No. 11, November 2000 11. In the language of Ref. 3, the loss of interference in these 12G. Jaeger, A. Shimony, and L. Vaidman, ‘‘Two interferometric comple- schemes cannot be explained in terms of ‘‘classical momen- mentarities,’’ Phys. Rev. A 51, 54–67 ͑1995͒.
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Am. J. Phys., Vol. 68, No. 11, November 2000

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