Wave-Particle Duality || Unsharp Particle—Wave Duality in Double-Slit Experiments

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  • CHAPTER 10




    The quantum mechanical double-slit experiment was discussed from an epistemo-logical point of view in the Bohr-Einstein debate. (I) According to Bohr the particle-wave duality comes from the fact that the measuring instruments for the measurement of a particle (the path) and for the measurement of the wave (the interference pattern) are mutually exclusive. This was demonstrated by Bohr for many Gedanken experiments by means of the uncertainty relation between momentum p and position x. The same uncertainty relation can, however, also be used to show that in spite of the strict particle-wave duality, the path (momentum p) and the interference pattern (position x) can at least be measured simultaneously in an approximate sense. This surprising result was first demonstrated by Wootters and Zurek(2) within a quantum mechanical reformulation of one of the Gedanken experiments discussed by Bohr and Einstein. (I)

    A more rigorous treatment of unsharp quantum mechanical measurements must make use of the general theory of unsharp observables in quantum me-chanics, which replaces the sharp observables, which can be described by projector-valued measures, by effect-valued measures which correspond to un-sharp observables (see Busch(3. If in the double-slit experiment the observables of the path and the interference pattern are replaced by unsharp observables of this kind, unsharp joint measurements of the path (momentum p) and the interference pattern (position x) turn out to be possible. Unsharp measurements of this kind must not be confused with inaccurate ones since the unsharpness expresses the

    PETER MITTELSTAEDT Institut fur Theoretische Physik, Universitiit zu KOln, 0-5000 KOin 41, Germany.

    Wave-Particle Duality, edited by Franco Selleri. Plenum Press, New York, 1992.


    F. Selleri (ed.), Wave-Particle Duality Plenum Press, New York 1992


    objective indeterminacy of the path and of the interference pattern, but not the observer's ignorance about these properties.

    Several attempts have been made to realize unsharp joint measurements experimentally. A photon split-beam experiment has been performed using a modified Mach-Zehnder interferometer. (4) This experiment provides simul-taneous unsharp wave and particle knowledge in full agreement with the theoreti-cal predictions. More sophisticated experiments of this kind were proposed, but have not yet been realized. (5) Other kinds of experiments which were interpreted in the sense of unsharp joint measurements are neutron interference experiments, which were performed by several authors. (6--8) However, these experiments use a slightly different experimental setup which does not allow interpreting them as unsharp measurements, but rather as experiments with "unsharp preparation." This conceptual distinction can also be realized experimentally by extended experiments which are briefly discussed here.


    As an illustration of the Bohr-Einstein debate we consider the photon split-beam experiment shown in Figure 1. In the experimental setup, a Mach-Zehnder interferometer, the incoming photon-state 1'1') is split by the first beam splitter (BS\

    ,-/ y L

    I cBI 1 CA 2 L_~

    (85)2 ~ I I I C~f-- /' L ~


    (85)1 hp>


    FIGURE 1. Photon split-beam experiment. Beam splitters (BS)\ and (BS)2 with transparencies J3 = 112, 'Y = 112, two fully reflecting mirrors M\ and M2, a phase shifter, and two photon counters C\ and C2


    into two components described by the orthonormal states IB) and I-,B), respec-tively. The two parts of the split beam are reflected at the two mirrors M, and M2 and recombined with a phase difference 8 at the second beam splitter (BS)2' In this experiment there are two mutually exclusive measuring arrangements: If the photon counters C" C2 are in the position (q, q), one observes which way (B or -,B) the photon came, and if the counters are in the position (q, q), one observes the interference pattern, i.e., the intensities which depend on the phase difference 8.

    This experiment can be described completely within the framework of the two-dimensional Hilbert space 'JC2 The observables are then given by projection operators P(B) with eigenstates IB) and I-,B) (for the path) and P(A) with eigenstates IA) and I-,A) (for the interference pattern). The state I'P) of the incoming photon can be decomposed in the B-basis as I'P) = 1I'\I2(IB) + exp (i8)1-,B. Hence, the probabilities for the waysB and -,B are given by p('P, B) = 1('PIB)12 = 112 andp('P,-,B) = 1('PI-,B)12 = 112, res~ctively. The interference observable is given by P(A) with eigenstates IA) = ttV2(IB) + I-,B and I-,A) = ttV2(IB) - I-.B. Hence, the probabilities for A and -,A are given by p('P, A) = 112(1 + cos 8) and p('P, -,A) = 112(1 - cos 8), respectively. Obviously the observables P(B) and P(A) have a nonvanishing commutator [P(B), P(AL = 1I2{IB)( -,BI - I-,B)(BI} *' 0 and can thus not be measured simultaneously.

    Pure and mixed state operators of 'JC2 can be represented by means of a three-dimensional unit sphere, the Poincare sphere r;p (Figure 2). By using the Pauli operators {TI ,


    P( B) (particle) --..,-.--..,.-.~

    FIGURE 2. States and operators of the two-dimensional Hilbert space represented by the Poincare sphere. Initial preparation P['P], phase shift 8, path observable P(B), interference observable P(A). Mixed states W(Z) and WL('P, A) lie in the interior of the sphere.

    Mixed states are described by (positive trace one) operators W(Z) and are given by W(Z) = 1I2(Zi


    observable P(A) = 1I2(Aj U"j + TI) in the sense of von Neumann and Liiders consists of two steps (I) and (II)

    Step (I), the objectification, transforms the preparation P['P] into the Liiders mixture

    WL('P, A) = p('P, A)P(A) + p('P, -,A)P(-,A)

    whereas step (II) merely reduces the observer's subjective ignorance by reading the final result P(A), say, of the measuring process. For the above-mentioned preparation I'P) = 1I2(1B) + exp(i8)I-,B) one obtains either the mixture

    W L ( 'P, B) = 1I2P(B) + 1/2P( -,B)

    by measuring the path or the mixture

    WL('P, A) = 112(1 + cos'P)P(A) + 112(1 - cos'P)P(-,A)

    by measuring the interference pattern. In the Poincare sphere the mixed states WL('P, B) and WL('P, A.) are obtained

    geometrically by orthogonal projections of the vector q, onto the radius vectors ii and A, respectively. For a given angle 8 S.t. 8 = arccos.ffA the probabilities for P(A) and P( -,A) are p('P, A) = 112(1 + cos 8) and p('P, -,A) = 112(1 - cos 8), respectively (Figure 2).


    In contrast to the sharp first kind measurement described above, the final result of an unsharp measurement is not a projection operator P E C!P('/1e) but an effect operator E E ~('/1e). Effects are selfadjoint operators E, the spectrum Ell. of which lies within the interval [0,1]. In the two-dimensional Hilbert space '/1e2 effects are given by operators of the kind

    For the sake of simplicity we will restrict the discussion here to the special case a = 1. These special effect operators

    correspond to points in the interior of the Poincare sphere, and are formally


    equivalent to mixture operators W. To any effect there exists a counter effect E which is given by

    E = 11 - E = 112(11 - Xp)

    The spectral decomposition of an effect E(i..) of this kind reads

    E = 1I2(~(j + 11) = 112(1 + 1~I)p(A) + 112(1 - 1~1)p( - A)

    with projection operators

    The eigenvalues have the interpretation

    E+ = 112(1 + I~I) > 1!2-reality degree of property P(~) E- = 112(1 - I~I) < 1I2-unsharpness of property P(~)

    where we have assumed that E+ > 112 and E- < 112. By means of these considerations we are now prepared to describe the

    unsharp measuring process. An unsharp measurement of the observable P(A) = 112(1 + Aio) leads to the effect

    E(A,AA) = 1I2(AAAi



    FIGURE 3. Unsharp measuring process P(A) wiIh prepaIation P['Pj. The vector C lies in the plane spanned by Ihe vector


    and with eigenvalues E(A,AA) and E(B,AB), respectively. 1\vo arbitrary observ-abIes Ml and M2 are called coexistent, if the ranges on the M; are contained in the range of one joint observable M. For the unsharp observables E(A,AA) and E(B ,AB) this means that these two effects are coexistent iff

    If in particular Xjj = 0, this condition reads Ai + A~ ,.,; 1. The special observables P(A)-(of the interference pattern) and P(B)-(of

    the path) fulfill the relationXjj = O. Moreover, we consider here the limiting case of maximal coexistence Ai + A~ = 1. The corresponding effects E(A, AA) and E(B, AB) can then be estimated jointly by the ideal Liiders measurements of an observable

    P(C) = 1I2(C;0'; + ~) with Iq = 1 which fulfills the two conditions CiA; = AA and C;B; = AB. The observables P(C) with CX = AA lie on a circle CA and those with Cjj = AB on a circle CB' From the geometry of the Poincare sphere (see Figure 4) it is obvious that for values AA and AB with Ai + Ah = 1 there is exactly oneobservableP(C) for which both conditions are fulfilled. However, since C is neither in the plane spanned by qi and X nor in the

    P( 8) (particle) -~~--

    FIGURE 4. Poincare sphere descrip-tion of an unsharp joint measurement of the observables P(A) and P(A) by means of an intermediate observable P(C). Effects E(A, AA) and E(B, AB); preparations P[


    plane spanned by q; and jj, the joint measurement of A and B does not correspond to minimal demolishing measurement of both effects E(A, AA) and E(B, AB).


    In the case of a sharp measurement the observer knows either the eigenvalue of the path observable or the eigenvalue of the interference observable. Unsharp measurements provide unsharp knowledge about the path and the interference pattern. It is useful to describe these different situations by means of information theory. For a preparation W = P[ q>] the deficiency of information about the result of a sharp measurement of the observable P(A) can be expressed by the formula

    H(q>, A) = -p(q>, A) Inp)q> , A) - p(q>, ...,A)lnp(q>, ...,A) .;;; In2

    where p(q>, A) andp(q>, ...,A) are the probabilities of obtaining P(A) and P( ...,A) as measuring results. Conversely the (normalized) knowledge about the wave prop-erty P(A) of a system with preparation P['P] is given by W = 1 - H('P, A)/ln2.

    Here we consider preparations W = P[ 'P] which correspond to a pure state Iq and observables which are not degenerate. Then the deficiency of information H('P, A) about the result of an ideal first kind measurement of P(A) agrees with the von Neumann entropy of the Liiders mixture

    WL(q>, A) = p('P, A)P(A) + p('P, ...,A)P( ...,A)

    which is obtained in step (I)-the objectification-of the measuring process, i.e., we have

    where we have denoted the entropy of the mixture WL by H(WL). Hence, the deficiency of information H('P, A) about the property A an be interpreted as the observer's subjective ignorance about A in the Liiders-Gemenge which is de-scribed by the mixture WL('P, A).

    The result of an unsharp measurement of P(A) with the unsharpness 112(1 - AA) is given by the effect E(A,AA)' The gain of information, which is achieved by measuring the effect E(A,AA) on a system with preparation 'P should be determined by its minimal demolishing measurement, i.e., by the measurement of the projection operator P(C), which in addition to C'A = AA also fulfills the condition C;,(q; X A)j = O. The deficiency of information H('P, C) about the property C in the state q> is then given by

    H(q>, C) = -p(q>, C)lnp('P, C) - p('P, ...,C)lnp('P, ...,C)


    with (see Figure 3)





    In a similar way for the reality degree E+(B, AB) of the path observable P(B) one obtains

    E+(B, AB) = 112(1 + AB) = 112(1 + 11 - 2AI)

    An experimental arrangement with the transparency 'Y of the second mirror and the visibility S('Y) provides some information about the photon's particle and wave character. The corresponding formula for the deficiencies of information H('Y. A) and H('Y. B) about further measurements of P(A) and P(B) in an eigenstate IC('Y of P(C('Y are then given(2,4) by

    H('Y. A) = -p('Y. A)lnp('Y. A) - p('Y. -,A)lnp('Y. -,A) H('Y. B) = -p('Y. B)lnp('Y. B) - p('Y. -,B)lnp('Y, -,B)

    where p('Y. A) and p('Y, B) are the probabilities of measuring P(A) and P(B), respectively. with the preparation P(C('Y. These probabilities read

    p('Y, A) = 112(1 + 2Y'Y(1 - 'Y = 112(1 + AA) = E+(A, AA) p('Y. B) = 112(1 + 11 - 2'Y1) = 112(1 + AB) = E+(B. AB)

    and agree with the reality degrees of P(A) and P(B). For the deficiencies of information one thus obtains

    H('Y. A) = -112(1 + S('Y In 112(1 + S('Y - 112(1 - S('Y In 112(1 - S('Y H('Y. B) = -'Yln'Y - (1 - 'Y)ln(1 - 'Y)

    Conversely, one can express the wave character and the particle character of a system in the state IC('Y by the (normalized) knowledge W('Y) = 1 - H('Y. A)/ln2 about the wave property and the knowledge P('Y) = 1 - H('Y, B)/ln2 about the particle property. The particle property (P('Y) and the wave property W('Y) are plotted against the visibility S( 'Y) in Figure 5.

    For the realization of the described experiment. 18 different transparency values 'Y for the beam recombiner (BSh were used. The preparation 1'P(8 = 1/2(1B) + exp{i8)1-,B depends on the phase 8 which was varied continuously. The intensities

    which are measured in the counters C1 and C2, respectively, are then given by


    o~~--~-.--.-~r--r--r-~--~~ ..;

    U'l ci

    0.0 --+ S(y')


    FIGURE 5. Particle property P('y) and wave property W(y) are plotted against the visibility S(y).

    As an illustration of the experimental results, we consider two alternative situa-tions which correspond to almost sharp measurements of the particle and the wave property (Figure 6; for further details see Mittelstaedt et al. (4.

    1. 1'1 = 0.492: here we have W('YI) = 99.98% wave property and P('YI) = 0.02% particle property. The measured visibility agrees with the theoretical value S('YI) = 0.992 and corresponds to a very significant interference pattern.

    2. 1'2 = 0.994: here we have only W('Y2) = 1.8% wave property and P('Y2) = 98.2% particle property. Even in this extreme case the measured visibility S('Y2) = 0.145 which agrees with the theoretical value is still sufficient for a significant interference pattern.

    These results demonstrate in a very convincing way that the measurement of the observable C('Y) provides an un sharp simultaneous knowledge of the photon'S particle property as well as of its wave property. In accordance with previous theoretical expectations, (2) it follows that even if the path of the photon is known to a degree of 98.2% its wave character still manifests itself in a su,rprisingly significant interference pattern.

    It should be mentioned that the present photon split-beam experiment is not yet a proper realization of the theory of the unsharp measuring process. It has been pointed out by Busch(5) that only the extended experiment of Figure 7 with four outcomes (in the counters C I' C 2 and D I' D 2) would provide a complete description

  • 182

    = 0.492 y il 'c " .d . ?;o 'iii c .'!! .f;

    I,B) open I 8) open


    y = 0.994

    ~ hB}open , I B) open

    ~ ,


    ! .

    ~ '. :' .' .' "

    i\ ! : ..

    : :


    , , , ,

    , . , ,

    .' \! II


    r~ n " " '. " "

    i\ " il ' ,

    1\ 1 i . , , .

    : !


    .. , .


    ' . , ,

    : : , .

    \! .'

    .' " .' "

    " V ..



    ~ i\ .' .' n ., " " ., . ,

    " .. :! i: ' ' : : i i ' . , . II ' ' ! ! ' , : : ' ,

    ! , , . '

    ' , , , , ' , . . , l: ' ' , ' I' , ' .. .. , , , : ' . , , . , . , ' : ! i ' , , , : ! ~ , . I' : : , , " \i I: " if :: " :: V v V 8Tt Ii

    o '-____________ J -r---r-.,---.---r-.,---.--o 4Tt 8Tt


    FIGURE 6. Experimental interference pat-tern for two transparencies. Upper part: -y = 0.492, i.e., 0.02% particle property, 99.98% wave property, and a visibility S = 0.992. Lower part: -y = 0.994, i.e., 98.2% particle property, 1.8% wave property, and a visibility S = 0.145.

    of the unsharp measuring process. If the transparencies K and A of the additional mirrors are different, the counters Cl' Cz) and (Dp D z) can be considered as two distinct instruments for measuring un sharply the path P(B) and the interference observable P(A), respectively. Moreover, for K *' A the entire measuring instru-ment is statistically complete, i.e., the statistics of the four outcomes also provide complete information about the initial preparation 1'P(l3, &) of the system which depends on the transparency 13 of the first beam splitter and of the phase &.


    I 'P >

    FIGURE 7. Extended version of the photon split-beam experiment of Figure 1 with two additional beam splitters (transparencies K and A) and two additional counters Dl and D 2 .


    Another attempt to realize unsharp joint measurements of wave and particle properties are neutron interference experiments which were performed and dis-cussed by Rauch and Summhammer, (6) Zeilinger, (7) Greenberger and YaSin, (8) and Rauch. (9) These experiments use a slightly different experimental setup (Figure 8). The incident beam is again split into beams Band -,B but in one of the beams, say B, an absorber is introduced which partially absorbs this beam. As an absorber of this kind, one could use another partly transparent mirror with transparency IX < 1. In this way the intensity of the path B can be arbitrarily reduced, whereas the -,B intensity remains unchanged.

    Since in this experimental setup the normalization of the incident beam is destroyed by absorption, for the theoretical discussion we replace the absorbing mirror by a beam splitter with variable transparency 13. The incoming beam can then be decomposed as

    If with this preparation the observable P(A) of the interference pattern is meas-ured, one obtains the probabilities

  • 184

    [""------I I I I I I I



    I I I I I I I


    Measuring instrument r-----------,

    I (BSh L ________ -1

    I - ... -~_-... -_1 Ii 1-_...1..-___ -1, I hB)



    arrangement of Figure 8: the preparation and the measuring instrument. In the neutron experiments the initial state l


    r: = p(cp, -'C('Y = p(cp(~), -,A)

    one obtains

    Tr(WL(cp(~), A)P(B = p+1I2(1 + cos 8) + p-1I2(1 - cos 8)

    and these expectation values are obviously different. The two cases of unsharp preparation and unsharp measurement could also be distinguished by the extended split-beam experiment proposed by Busch. (5)


    1. N. BOHR, in: Albert Einstein: Philosopher-Scientist (P. A. SCHILPP, ed.), pp. 119-155, Open Court, La Salle, Ill. (1949).

    2. K. WOOTTERS and W. H. ZUREK, Phys. Rev. D 19, 473-484 (1979). 3. P. BUSCH, Phys. Rev. D 33, 2253-2261 (1986). 4. P. MITTELSTAEDT, A. PRIEUR, and R. SCHIEDER, Found. Phys. 17, 891-903 (1987). 5. P. BUSCH, Found. Phys. 17, 905-937 (1987). 6. H. RAUCH and K. SUMMHAMMER, Phys. Lett. A 104, 44-46 (1984). 7. A. ZEILINGER, Physica B 137,235-244 (1986). 8. D. M. GREENBERGER and A. YASIN, Phys. Lett. A 128, 391-394 (1988). 9. H. RAUCH, in: Proceedings of the 3rd International Symposium "Foundations of Quantum

    Mechanics," Physical Society of Japan, Tokyo (1989).