Vistas in Astronomy, Vol. 37, pp. 287-290, 1993 Printed in Great Britain. All rights reserved.

0083-6656/93 $24.00 @ 1993 Pergamon Press L~d

MECHANISMS OF WAVE-FUNCTION COLLAPSE AND ENTROPY OF DETECTORS

Tstmehiro Kobayashi Institute of Physics, University of Tsukuba, Ibaraki 305, Japan

INTRODUCTION

We have r e c e n t l y shown t h a t the t r a n s i t i o n of a d e n s i t y ma t r ix from a pure to a mixed s t a t e can be d e r i v e d in d e n s i t y m a t r i c e s wi th the i n t e r n a l t r a c e r e p r e s e n t e d by the p a r t i a l t r a c e o p e r a t i o n s f o r d e t e c t o r v a r i a b l e s d e l i b e r a t e l y i gno red in measurement p r o c e s s e s [1 ,2 ] . We have a l s o shown t h a t t h e r e are two dlfferent mechanisms for realizing the transition in the density matrix [2,3,4]. In the mechanism called the quantum collapse the transition is directly derived in the macroscopic limit, while it is derived from the thermal fluctuation of the initial detector variables in the statistical collapse. An explicit example including both mechanisms was presented in a simple model [5]. We would llke to show that the change of entropies of detector states in the trasltion is a quite interesting quantity to see the difference between the two mechanisms [6]. In thls paper we shall investigate a typical double-silt measurement, in which the incident wave of the object is split into two branch waves, ~A and ~B'

corresponding to the two different pathes. An apparatus is put only on the path A.

Let us consider the measurement process described by the Interatlon between an extremely relativistic particle [object) and a detector schematlzed as a one dimensional array of scatterers having two different states, ~0 and ~1" We

take a simple Hamlltonlan H N = Ho+HIN where H 0 = cp and HIN = ea~=l~(x-na)al (n)

Here p and x, respectively, denote the momentum and position operators of the particle, na(n=l...N) are the positions of the scatterers for the slte-length a,

e stands for a parameter with the energy dimension and ~i (n) is the Paull spin-

matrix for the nth scatterer. We can evaluate S N as S N = EN s(n), where the n=l

S-matrix for the nth scatterer S (n) is written by S (n} ea (n) = exp[-i~-~ I ] = cos(~)

(n) ea sin(~-~c ). Note that the probability for the transition of one scatterer is

2 ea given by ~msln (]-~).

288 ~ Kobayashi

MODELS FORQUANTUM COLLAPSE

Let us consider the apparatus, of which lnltlal state ls described by

yN ~N ~(n) After the interaction with the object the flnal apparatus state Is i=~n=l~0 •

evaluated In terms of S t as ¥~ ~ sN¥~ - K N (n) (n) = _ = n=l [ 1 ~ - ~ 0 -1~-~ 1 ] . In t h i s p r o c e s s

the o f f - d i a g o n a l terms of the d e n s i t y ma t r i x wl th t he i n t e r n a l t r a c e a r e

N = (l~-e) N. Then the off-dlagonal terms vanish In the evaluated as ( ~ e f f ) o f f macroscopic l i m i t for s /O. Quantum collapse does happen.

MODEL FOR STATISTICAL COLLAPSE

Let us study the case where the initial apparatus state Is written In terms of the superposltlon of the two states (~0 and ~i) for each scatterer, that

Is, ~N IN l,ex .l~(n) .(n) (n) .(n) I = n=I~22 t P~ 0 )~0 +exp(lal )~I ]'

(n) and (n) are taken to be real for simplicity. For the final state where a 0 a 1

evaluated as = S ~I' the off-dlagonal term In the density matrix wlth the

trace Is given by (~eff)off = g ,c÷Iscos,an)-a n),~l where c = 11-~, internal

s = J-~, and the coefficients and the elements for the object are neglected. In general the off-dlagonal term does not disappear In the limit.

(I) Random phase limit N ~n) (n)

The energy of the appa ra tu s s t a t e g iven by 4I does not depend on a and a 1 .

C o n s i d e r i n g t h a t In quantum measurement p r o c e s s e s the measurement must r e p e a t e d l y c a r r i e d out one by one over a l a r g e ensemble of the o b j e c t p a r t i c l e s to o b t a i n a c o r r e c t p r o b a b i l i t y d i s t r i b u t i o n of t he measured q u a n t i t y , we cannot expec t t h a t the phases a r e the same over a l l the r e p e t i t i o n of the measurements . Then those

phases g e n e r a l l y have the £-dependense such as a~n)(e)- and a~n)(£)," where

deno tes t he e th mesurement In the expe r imen t . In t h l s Idea the d e n s i t y m a t r i x i t s e l f has the e-dependence a r i s i n g from the ambigu i ty of the i n i t i a l s t a t e . The d e n s i t y m a t r i x f o r the measurement over the ensemble must be d e f i n e d by the average over ~ as

where L stands for the total number of the measurements. Here we study the extreme case where all phase parameters are completely random In the repetition.

N = The average over ~ derives the result <(~eff)off>L ' '~cj N In the large L l i m i t .

N Then we have llmN_~®<(~eff)off~ L = 0 for c = ~1-~<1. The transition from the pure

state to the mixed state ls realized.

(II) Fluctuation of the number of active scatterers In apparatus

The maximum of the off-dlagonal terms is glven in the case wlth cos2(a~n)(£) - "

a~n)(~)) = 1 for all n = 1, 2 ..... N and ~ = 1, 2 ..... L. Actually we have U

Wave-Func~onCoUapse 289

N = e liON, a l(n)(~)-a~n)(~) = 2~m (m=O or an integer) (aeff)off where Omea/~c and

are taken. It is trivial that the internal trace operation and the average over the phases do not derive the collapse. We, however, have also to take account of the fluctuation of the active scatterer number. In every repetition of the measurement process the number of the scatterers (N) actively working in every leasurement has some thermal fluctuations at non-zero temperatures. In general such a fluctuation may be described by the Oaussian distribution for the large N

limit as PN = exp[-(N-No )2/2~2]/(2~6) where N o = <S>, the mean value of N, is a

macroscopic number being propotlonal to N and the dispersion 62 may be proportional to N O .

N The statistical average for Oeff can be represented by the average in

terms of the distribution in the large L limit as

1 ImN-~®~(O~ f )o f f~L'l imL_~o (I/L)E~= l(P~f)of f~exp (-6202/2110No ) •

Taking account of 62 ~ N O at finite temperatures, we have the disappearance of

the off-dlagonal terms in the macroscopic limit. The transition of the density matrix is realized via the statistical average for the thermal fluctuation.

EUTROPIES OF DETECTOR STATES

We shall investigate the following Shannon-type entropy s(~ (n)) : -

EN (n)[21oglc~n)[ 2 for the nth scatterer state represented by ~(n) (n).,.(n) I=I ci = Co ~0

(n)~(n) (n) 2+icOn) 2 ÷Cl ~1 with c 0 [ [ = 1. Note that the entropy above defined is that

for the mixed state described by the two states ~0,1" Now we introduce the

entropy of the detector state defined by the sum of those entropies over all

scatterers as S(¥ N) ffi E N sly(n)). Let us calculate this entropy for the n=l " processes discussed in the last section.

(i) Quantum collapse In the process for the quantum collapse the entropies of the initial and final

respectively, are given as S(¥~) ffi 0 and S(¥~) ffi -N[l-e)log detector states,

(l-e)+elog~]. We easily see that the entropy of the final state (S(¥~)) is

positive and proportional to N for O<e<l. Note that the difference between the

two entropies aboved obtained, i,e. ASq = S(~)-S(¥~),__ linearly diverges in the

macroscopic limit N~.

(2) Random phase limit

For the simplicity we put a~n)(~) = al(~ ) and a~n)(£) = aO(£ ) for all n. The

entropies of the Initlal and flnal states for the ~th measurement are,

respectlvely, evaluated as S(~(~)) = Nlog2 and S(~(~)) IS~ ~I~0~ ~og~l~o ~ ~ J. 1

290 Z Kobayashi

~01og~ O] where ~ o = l / 2 + c s x s l n ( a l ( £ ) - a O ( E ) ) . The maxlmum of S ( ~ ( E ) ) i s g iven by

Nlog2, which c o i n c i d e s wi th S ( ~ ( £ ) ) . We t h e r e f o r e see t h a t the d i f f e r e n c e

between the above two e n t r o p i e s I s n e g a t i v e . I t i s impor t an t t h a t 15Ss(g) ] -~

i n d i c a t e s t h e p r o d u c t i o n of some macroscopic q u a n t i t i e s in t h e s e measurement p r o c e s s e s .

(3) F l u c t u a t i o n of a c t i v e s c a t t e r e r number In the case of the s t a t i s t i c a l c o l l a p s e induced by the f l u c t u a t i o n of the a c t i v e

( n ) , ^ , (n) s c a t t e r e r number, t h a t I s , in the case wi th a I ~ - a 0 (E) = 2~m (m=O or an

I n t e g e r ) f o r a l l of n and e, we have 5Ss(£)ffiO. This f a c t i n d i c a t e s t h a t we have

no macroscopic q u a n t i t y to be produced in the d e t e c t o r in t h i s measurement p r o c e s s .

r4ESOSCOPIC CHANGE OF ENTROPY

We would l l k e to i n v e s t i g a t e l i t t l e d i f f e r e n t macroscopic l i m i t which l s d e f i n e d by N-~ and N~ = f i n i t e . Let us put N~=r<~. In t h i s l i m i t we d e r i v e 5Sq~rlogN in (1) . The en t ropy d i f f e r e n c e l o g a r l s m i c a l l y d i v e r g e s in t he l i m i t ,

wh i l e I t l i n e a r l y d i v e r g e s In the l l m l t d e f i n e d on ly by N-~. In (2) 5S s i s

5 S s ( g ) ~ - 4 r s i n 2 ( a l ( £ ) - a O ( £ ) ) , which does not d i v e r g e even in t he o b t a i n e d as

macroscopic l i m i t . I t I s i n t e r e s t i n g t h a t in bo th ca ses the change of t he e n t r o p y i s not enough to r e a l i z e the thermodynamical o r d e r of e n t r o p y which i s g e n e r a l d i v e r g e s l i n e a l y wi th the increment of N. The d i f f e r e n c e of t h i s l l m l t from the l i m i t d e f i n e d on ly by N-*~ i s c l e a r l y seen in the c o l l a p s e . That i s , the o f f - d l a g o n a l terms does not van i sh in t h i s l i m i t as

r N r limN_~ ' N~fr<®fllmN_~(1-~-~) =exp (-~) fO.

The b e h a v i o r of the c o l l a p s e c r u c i a l l y depends on the magnitude of r , t h a t I s , In t he case r f f l n l t e but s a t i s f a c t o r i l y l a r g e r than 2(r>>2) the c o l l a p s e i s e f f e c t i v e l y comple ted , whereas in the case where r i s comparable wi th 2 the c o l l a p s e Is not completed and we can expec t to see the Incomple te c o l l a p s e . I f we p rov ide v e r y smal l s i z e a p p a r a t u s or ve ry low d e n s i t y ones such t h a t Nx~(--pV~) ~0(1) i s r e a l i z e d , we can d i r e c t l y see t h i s e f f e c t [6] .

REFERENCES

1. T. Kobayashl and K. Ohmomo, Phys. Rev. A41(1990) 5798. 2. T. Kobayashi , the P roceed ings of the Symposium on the Founda t ions of Modern

Phys i c s 1990, eds . P. Lah t l and P. M l t t e l s t a e d t , P170. 3. T. Kobayashi , Talk p r e s e n t e d a t the Second I n t e r n a t i o n a l Wigner sysposlum,

Gos la r in @ermany, 1991; p r e p r l n t of U n i v e r s i t y of Tsukuba, UTHEP-220 (1991). 4. T. Kobayashi , Nuovo Cimento 107B (1992) 657. 5. T. Kobayashl , (Founda t ions of Phys ics L e t t e r s v o l . 5 No.3 in p r e s s (1992)) . 6. T. Kobayashl , (Phys. r ev . A1 in p r e s s (1992)) .