Interference and distinguishability in quantum mechanics

Physica B 151 (1988) 309-313 North-Holland, Amsterdam

INTERFERENCE AND DISTINGUISHABILITY IN QUANTUM MECHANICS J. UFFINK and J. H I L G E V O O R D History and Foundations of Science, Physical Laboratory, P.O. Box 80.000, 3508 TA Utrecht, The Netherlands

Quantitative measures are introduced for the indistinguishability U of two quantum states in a given measurement, and the amount of interference 1 observable in this measurement. It is shown that these notions obey an inequality U _-__I, which can be seen as an exact formulation of Bohr's claim that one cannot distinguish between two possible paths of a particle while maintaining an interference phenomenon. This formulation is applied to a neutron interferometer experiment of Badurek et al. It is shown that the formulation is stronger than an argument based on an uncertainty relation for phase and photon number as considered by these authors.

A recent experiment in neutron interferometry [1] can be seen as a realisation of the double-slit thought experiment discussed by Einstein and Bohr. In this famous discussion, Bohr argued that one cannot distinguish between two possible paths of a particle while preserving an interference phenomenon. To reach this conclusion Bohr applied the uncertainty relation for position and momentum in a somewhat informal way. However, it has been shown that the Heisenberg formulation of the uncertainty relation by itself is not strong enough to justify Bohr's claim [2]. The discussion has been revived in the light of the new neutron experiments. In ref. 1. an argument is presented supporting Bohr's claim, based on an uncertainty relation for the phase and photon number of the electromagnetic field. The validity of this explanation was subsequently disputed [3], because of the dubious theoretical status of the phase-number uncertainty relation. This raises the question whether it is possible to give a direct quantitative formulation of Bohr's claim, without resource to the uncertainty relations. Work in this direction has been done by Wootters and Zurek [4]. Here we propose an alternative formulation that seems particularly apt for the interferometer experiments. The main obstacle for a direct formulation of Bohr's claim is the problem of choosing a quantitative measure for the extent to which the two paths are distinguishable in a given experiment.

Let ~1 and ~2 denote orthogonal quantum states that represent possible paths of a particle. A measurement performed on this particle may be described by a complete set of orthogonal projection operators { D k } , where k denotes a possible outcome of the measurement. The hypotheses that the particle travelled either one of the paths then provide two probability distributions, viz. Pk = ( ~ l l D k l ~ , ) ,

(1) qk = (~b21D~l~b2) • The two paths may be said to be discriminated if, from the observed outcome of the measurement, one can decide between these two hypotheses. Thus, the problem of distinguising between two paths can be seen as a special case of the general classical problem of discriminating between two statistical hypotheses. A solution for this problem depends, of course, on the distributions p and q, but also on the observed outcome. However, independently of the latter, one can indicate whether a discriminative answer is likely. There are two extreme cases. (i): Pkqk = 0 for all k, i.e. every outcome that has a positive probability according to one hypothesis is impossible according to the other. In this case a single observation will suffice for complete discrimination. (ii): p~ = q~ for all k. In this case no number of observations can

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J. Uffink and J. Hilgevoord / Interference and distinguishability

310

discriminate between p and q. In all other cases an incomplete discrimination is to be expected. One is tempted to define a "degree of indistinguishability", reflecting the expected lack of discrimination between p and q per observation. For this purpose we choose:

U(p, q) = ~ (pkqk) '/2 .

(2)

k

The significance of this expression in statistical theory has been studied by Bhattacharyya [5], Rao [6] and Wootters [7]. Loosely speaking, a value of U(p, q) close to unity indicates that even when one of the distributions is " t r u e " , a typical outcome of the experiment will not allow one to infer accurately which of them is true. As an illustration we compare the expression proposed above with a more familiar statistical approach to the discrimination problem, the theory of significance testing [8]. A significance test for two hypotheses is a procedure by which one hypothesis is either accepted or rejected in favor of the alternative hypothesis. In this approach the "distinguishability" of the two hypotheses is judged by the so-called errors of the first and second kind, 1 - a and 1 - / 3 ; i.e., respectively, the probability of rejecting the first hypothesis when it is true, and the probability of accepting it when its alternative is true. A " N e y m a n - P e a r s o n test" is designed to minimize these two errors, and always obeys ( 1 - a ) + (1 - / 3 ) N 1. One may show that for all N e y m a n Pearson (N.P.) tests of the distributions p and q:

U{Dk}(tP~, qh) = ~ ( ( q'l[Dkl~l) ( qhlDk[tP2) ) ~/2 . k

(4)

Thus, even when it is assumed that the particle did travel one of the paths, the outcome of the measurement will only enable us to determine this path if U is small. One may easily show that for a given tpl and ~b2 expression (4) is nonincreasing when the projections are resolved into one-dimensional projections, as is intuitively reasonable. Maximal distinguishability is reached when the set { Dk} includes [g~1) ( ~1 [ or [if/2)( ~b2[. We now turn to the notion of interference, The state of a particle emerging from an interferometer may be regarded as a normalized superposition of g~l and ~b2, ~b = cj g5 ÷ c2qh, with levi=+ Ic212= 1 The probability that the measurement {Dk} yields an outcome k is then

P, = Icll~p~ + ICzl2q~ + i k , where Pk and qk are given by (1), and i k = 2 Re c~c2( ~bllDkl~,: )

Ik = I ( + , l D k l ~ 2 ) l

•

We define the interference power of tp1 and ~b2 for this measurement as

k -

~)(1-/3)---

(7)

k

U 2 ) 1/2 ,

(3) (1-

(6)

is the so-called interference term. The maximum value of i k for all choices of c l and c 2 is

I{Dk}(~bl, ~b2)= ~ Ik = ~ I<~llOkl~2)l • (1 - a) + (1 - / 3 ) ~ 1 - (1

(5)

¼U 2 .

Thus, a large degree of indistinguishability implies that the sum of the two kinds of error in a N.P. test is close to its upperbound 1, whereas a small value of U ( p , q) implies that any N.P. test of p and q is " g o o d " in the sense that the product of the two errors is small. In agreement with (2) we define the degree of indistinguishability of the quantum states ~b~ and q'2 for the m e a s u r e m e n t {Dk} as

Let us see how this relates to other familiar notions of interference strength. In many experiments a variable relative phase shift X between the two paths is introduced. In that case it is convenient to put O2(X)= eiX~b2• Let us further take c 1 = c 2 = 2 -1/2. Pk then oscillates as a function of X:

Pk(X) = l ( p k + qk) + COS )¢l($11Dklqh)l • A well-known measure for the amount of interference in this situation is the Michelson fringe visibility

J. Uffink and J. Hilgevoord / Interference and distinguishability gk = (Pmax - P m i n ) / ( & a x

+ Pmin) ,

where emax and emitl denote adjacent maximal and minimal values of Pk(X). This gives

311

are again coherently split and pairwise superposed, so that the neutrons emerge in two final beams. The emerging neutron state 4,0 may then be written as a superposition of orthogonal parts corresponding to the two paths,

I{Dk} = ~ Vk ½(Pk + qk) "

4,0 =2 -1,2(4,10 + e,X4,0) ,

Thus, the interference power is just the average visibility over all possible outcomes, weighted by the mean probability ½(Pk + qk)" For given g'l and ~02, Iwk } is non-decreasing when the projections D k are resolved into one-dimensional projections, so that a more resolving measurement will in general show more interference. Now, let us compare the interference power (7) with the degree of indistinguishability (4). By the Schwartz inequality one finds

where X is the variable phase shift. In the ideal case each path contributes with equal amplitude to both emerging beams,

k

0 0 + 4,1a) 4,0 = 2 -1/2 (4,1A 0

4'2 = 2

-1/2

0

(11)

+ 4,2 )

and 0 0 4,m=--4,2B,

(12)

(8)

Thus, the appearance of a pronounced interference phenomenon in an experiment is incompatible with the requirement that the interfering states are distinguishable in that experiment. This can be regarded as a quantitative expression of Bohr's claim. Equality in (8) occurs when all projections D~ are one-dimensional. In relation (8) the degree of indistinguishability and the interference power are compared for one and the same experiment. What if the interference is observed in one measurement, {Dk}, and one attempts to distinguish the two states by means of another measurement, described by the set {Dr}? Actually this will not improve the situation, as long as the two measurements are compatible. In that case, the product of two projections, Dka = DkDt, is also an orthogonal projection operator, and Dk./ is a resolution of {Dk} as well as of {Dl}, so that

U{o,} ~ U{ok,,} >=/{ok.z} --- l{ok).

0

(4,2

60 A = 602A, U{Dk}(~IYl' ~ 2 ) ~ l{Dk}(~lll' ~//2) "

(10)

(9)

We now apply the ideas discussed above to the neutron interferometer. In the interferometer an incident neutron beam is coherently split in two partial beams. A phase shift between these beams may be produced by placing a piece of material in one of them. Next, the two beams

where the second index labels the emerging beams. Thus, for X = 0, all neutrons are found in beam A, whereas for a phase shift X = "rr, all neutrons are found in beam B. This conforms to maximal interference in the intensity as a function of the phase shift. In the experiment of ref. 1 the incident beam is prepared with spin polarized along the positive z-axis. One of the partial beams is led through a magnetic coil which reverses the spin of the passing neutrons. In the emerging beams spin analysers and detectors are placed, and the timedependent intensity is recorded. When the spin analysers are turned into the y-direction a periodical interference pattern in the intensity is observed as a function of the phase shift. The question is now whether one can infer, without disturbing the interference, which path each neutron travelled through the interferometer by means of an extra measurement on the spin flipping coil. The idea behind this is that the interaction of a neutron with the magnetic field involves the exchange of a photon. Hence, if a measurement of the photon number of this field is made, one might hope to detect whether or not such an interaction has taken place. Let the spin flipper be placed in path 2. The combined final state $ of neutron and magnetic field may be taken as

312

J. Uffink and J. Hilgevoord / Interference and distinguishability

qt = 2-'/2(I/tl +

e'X62)

(13)

with

Ds=+,,z 1(1÷) + I-))(C÷l + C-I), Ds=-,,2 = ½(I + ) - l -

qq = 6°~:,

))(C+[- C-I),

(14) we obtain from (11) and (12)

where ~ is the final state of the magnetic field when no interaction occurs, and 02 is the combined final state of neutron and magnetic field after their interaction. Introducing the neutron eigenstates for the z-component of the spin, I---), and photon number states In), we may write

4,o = I+)(+1,¢,°),

(15)

= ~] I n ) ( n I ~ ) .

(16)

n

The spin flipper is assumed to be efficient, i.e. it fully reverses the spin of all passing neutrons. Further, we assume that the interaction does not alter the spatial wave function of the neutron. 02 may then be written as ~2 = ~2~2,

(17)

lwx,,} = ~ C ~ l n ) C n + l l ~ )

•

(21)

Let us now ask whether the passage of a neutron through the spin flipper, and the associated photon exchange can be detected by a measurement of the photon number. The two states to be distinguished are (14) and (17). For the measurement of the photon number, represented by the projections D, = [n)Cn[, the degree of indistinguishability (4) is

u{o°}(O,,O2)=~lC~ln)Cn+ll~)l

1/2 .

Thus, the extent to which one can determine whether a photon exchange has occurred depends on the extent to which two consecutive photon numbers are present in the spectrum [(n[~)[ 2. It is obvious from (21) and (22) that

with

6") >62 = e'°'l-)C + 16 ),

(18)

~2 = E e-i~"ln + 1 ) ( n ] ~ ) ,

(19)

n

where w is the frequency of the magnetic field. Suppose now that a measurement is made of the intensity in the emerging neutron beams with spin analyzed in the y-direction at all positions x along the beams at a fixed time. The corresponding projection operators are denoted by D~,~. It is easy to see that this procedure and the actual procedure, where a time-dependent observation at a fixed position is made, are equivalent in a quasi-monochromatic approximation for the neutron wave packet. The interference power of the states (14) and (17) for this measurement is /~D~,,~ = I(~lg2)l ~ f dx 1(4,°1Dx,,14~2)1,

(20)

where the integral is to be performed over both emerging beams. Using the fact that the projections D~,~ can be factorized, D~,, = DxD ,, with

(22)

n

(23)

q")'

in agreement with (9). This shows that the conditions under which a measurement on the spin flipper allows the distinction of the two paths exclude the conditions under which interference occurs. Finally, we compare the formulation of Bohr's claim given here with an approach employing an uncertainty relation for the phase and photon number. A satisfactory description of the phase of an electromagnetic field in quantum theory is that in terms of the exponential phase operator e i~ defined by Lrvy Leblond [9], eir=Z

I n ) ( n + 11. tt

Note that the expectation value of this operator appears in (21). An appropriate definition for the phase uncertainty as discussed in ref. 9 is

2 = x-I(

leJ'l

)l-I(01e)l

2,

(24)

where [(01~)l 2 is the probability to find zero

J. Uffink and J. Hilgevoord / Interference and distinguishability

p h o t o n s m the m a g n e t i c field. U n d e r e x p e r i m e n tal c o n d i t i o n s this t e r m m a y be neglected. T h e r e fore, in o r d e r to have a n a p p r e c i a b l e interference p o w e r , it is necessary for the m a g n e t i c field that

a 6 ¢ 1.

(25)

We can n o w e m p l o y the u n c e r t a i n t y r e l a t i o n for p h a s e a n d p h o t o n n u m b e r given in ref. 9 (An)2[(A~b)2 +

-

½1<01>121 2 -I<0l

An >> ½.

(26)

(27)

H o w e v e r , this c o n c l u s i o n by itself is n o t suffici e n t to exclude the d i s t i n g u i s h a b i l i t y of the two paths. F o r e x a m p l e , if the state of the p h o t o n field is such that I

~

the p h o t o n e x c h a n g e m a y be d e t e c t e d with complete c e r t a i n t y from a m e a s u r e m e n t of the p h o t o n n u m b e r , w i t h o u t violating (27). H o w ever, the fact that such a state is i n c o n s i s t e n t with c o n d i t i o n (25) follows directly f r o m r e l a t i o n (23). We c o n c l u d e that the f o r m u l a t i o n of B o h r ' s claim c o n s i d e r e d a b o v e is s t r o n g e r t h a n o n e based o n an u n c e r t a i n t y r e l a t i o n of the H e i s e n berg type.

References

>lz],

w h e r e (An) 2 = ( n 2) - ( n ) 2 is the s t a n d a r d deviation of the p h o t o n n u m b e r . It t h e n follows from c o n d i t i o n (25) that

Il z = 18 2 n,n 0

3/3

[no_nil.>1

[1] G. Badurek, H. Rauch and J. Summhammer, Phys. Rev. Lett. 51 (1983) 1015. [2} J.B.M. Uffink and J. Hilgevoord, Found. Phys. 15 (1985) 925. [3] C. Dewdney, A. Garuccio, Ph. Gueret, A. Kyprianides and J.P. Vigier, Found. Phys. 15 (1985) 1035. [41 W.K. Wootters and W. Zurek Phys. Rev. D 19 (1979) 473. [5] A. Bhattacharyya, Bull. Calcutta Math. Soc. 35 (1943) 99. [6] C.R. Rao, Bull. Calcutta Math. Soc. 37 (1945) 81. [7] W.K. Wootters, Phys. Rev. D 23 (1981) 357. [8] M.G. Kendall and A. Stuart, The Advanced Theory of Statistics, vol. 2 (Griffin, London, 1946). [9] J.M. Lrvy-Leblond, Ann. Phys. 101 (1976) 319.

DISCUSSION (Q) J.P. Vigier: I do not doubt your calculations but I feel they would have been rejected by Bohr who claimed particles do not travel through one or the other slit, as a result of the complementarity concept and the complete character of the quantum formalism. (A) J. Uffink: In the formalism I presented it is not assumed that the particle did travel one way or the other in an interference experiment. The point is that in the conditions in which interference is observed you cannot distinguish by statistical means between two hypotheses. Thus even if you would assume the particle travelled one path, you could not test such an hypothesis accurately against its alternative. (Q) H. Rauch: How would you comment on the formulation of the phase-particle uncertainty relation given by Nietto and Carruthers? (A) J. Uffink: The exponential phase operator of LrvyLeblond is a combination of the cosine and sine operators of Carruthers and Nietto, which has more desirable prop° erties, e.g. symmetry under a shift of origin of the phase. Further, since both formulations use the standard deviation An for the photon number, the counter example

mentioned in my talk shows that both formulations are not sufficient to establish Bohr's claim. (Q) J. Summhammer: Is there a special reason why you choose as a representation for indistinguishability qv"~-~xPr?(For instance, for reasons of the error interval of the binomial distribution.) (A) J. Uffink: The square root comes in naturally in a statistical consideration of the problem, see e.g. the paper of Wootters. (C) I. Bialynicki-Birula: In connection with Dr. Rauch's remark I would like to point out that the uncertainty relations for N and ~b can be formulated in terms of information theory concepts (the so-called entropic uncertainty relation). In this formulation the difficulties encountered in the traditional formulation (namely, the nonexistence of a Hermitian phase operator) do not appear. (A) J. Uffink: I agree with this. However, the entropic formulation is also insufficient for the considered purpose.