Mechanisms of wave-function collapse and entropy of detectors

0083-6656/93 $24.00

Vistas in Astronomy, Vol. 37, pp. 287-290, 1993

@ 1993 Pergamon Press L~d

Printed in GreatBritain. Allrights reserved.

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 t h e t r a n s i t i o n o f a d e n s i t y m a t r i x from a pure t o 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 w i t h t h e 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 t h e 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 g n o r e d i n 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 a r e 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 A.

to the two different pathes.

An apparatus is put only on the path

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

matrix

for

for a parameter with the energy dimension and ~i (n) is the Paull spinthe

nth

scatterer.

We can evaluate SN as SN = EN s(n), where the n=l

S-matrix for the nth scatterer S (n) is written by S (n} (n)

ea sin(~-~c).

ea

=

(n)

exp[-i~-~ I

] = cos(~)

Note that the probability for the transition of one scatterer is

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

~ Kobayashi

288

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

yN ~N ~ ( n ) i=~n=l~0 •

In terms of S t as ¥~~ = sN¥~-_ = KnN = l [ 1 ~ - ~ 0(n) -1~-~ 1(n) ].

evaluated

the

After the interaction with the object the flnal apparatus state Is

off-diagonal

terms

of

the

density

matrix

wlth

In this process internal trace are

the

= (l~-e) N. Then the off-dlagonal terms vanish In the evaluated as ( ~N eff)off 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~22t P~ 0 )~0 +exp(lal )~I ]' a 0(n)

where

evaluated internal

and

a 1(n)

as

=

are taken to be real for simplicity. S ~I'

For the final state

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-~,

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

In

(I) Random phase limit

The

energy

N ~n) (n) o f t h e a p p a r a t u s s t a t e g i v e n by 4 I 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 t h e 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 o f t h e o b j e c t p a r t i c l e s t o 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 o f t h e measured q u a n t i t y , we cannot e x p e c t t h a t t h e p h a s e s a r e t h e same over a l l t h e r e p e t i t i o n o f t h e measurements. Then t h o s e phases

generally

have

the

£-dependense

such

as a~n)(e)- and a~n)(£)," where

d e n o t e s t h e e t h mesurement In t h e e x p e r i m e n t . In t h l s I d e a t h e d e n s i t y m a t r i x i t s e l f has t h e e-dependence a r i s i n g from t h e a m b i g u i t y o f t h e 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 t h e measurement over t h e ensemble must be d e f i n e d by t h e a v e r a g e 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 ''~cjN 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

a~n)(~)) U

=

of 1

the for

off-dlagonal terms is glven in the case wlth cos2(a~n)(£) - " all n = 1, 2 ..... N and ~ = 1, 2 ..... L.

Actually we have

Wave-Func~onCoUapse

289

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

macroscopic number being proportional to N O .

propotlonal

to

N

and

the

dispersion

62

may

be

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 )off~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)) : ENI=I ci(n)[21oglc~n)[ 2 for the nth scatterer state represented by ~ ( n ) = Co(n).,.(n) ~0 ÷Cl(n)~(n) ~1 with c0(n) [2+icOn) [2 = 1. Note that the entropy above defined is that for

the

entropy

mixed of

scatterers

state

the

detector

state

by

the two states ~0,1"

Now we introduce the

defined by the sum of those entropies over all

ffi EN sly(n)). Let n=l " processes discussed in the last section. In

as

described

S(¥N)

us

calculate

this entropy for the

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

detector states, respectively, are given as S(¥~) ffi0 and S(¥~) ffi-N[l-e)log (l-e)+elog~]. positive two

and

We

easily

see

that

the

entropy of the final state (S(¥~)) is proportional to N for O
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.

entropies of the Initlal and flnal states for the respectlvely, evaluated as S(~(~)) = Nlog2 and S(~(~)) J.

1

~th

measurement

The are,

IS~ ~I~0~ ~og~l~o ~~

290

Z Kobayashi

~01og~ O] Nlog2,

where which

between

the

~o=l/2+csxsln(al(£)-aO(E)). coincides

above

indicates the processes.

two

production

with

S(~(£)).

The maxlmum o f S ( ~ ( E ) )

We t h e r e f o r e s e e t h a t t h e d i f f e r e n c e

entropies Is negative. of some

i s g i v e n by

It is important that

macroscopic quantities

15Ss(g)]-~

i n t h e s e measurement

(3) F l u c t u a t i o n o f a c t i v e s c a t t e r e r number t h e c a s e of t h e s t a t i s t i c a l c o l l a p s e induced by t h e f l u c t u a t i o n o f t h e a c t i v e (n),^, (n) scatterer number, t h a t I s , i n t h e c a s e w i t h a I ~ - a 0 (E) = 2~m (m=O o r an In

Integer)

f o r a l l of n and e, we have 5Ss(£)ffiO.

no m a c r o s c o p i c process.

quantity

to

be

produced

in

This f a c t i n d i c a t e s

t h a t we have

t h e d e t e c t o r in t h i s measurement

r4ESOSCOPIC CHANGE OF ENTROPY We would l l k e t o 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 m a c r o s c o p i c 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<~. I n t h i s l i m i t we d e r i v e 5Sq~rlogN in ( 1 ) . The e n t r o p y 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 h e l i m i t , while

It

obtained

linearly as

diverges

In

the

l l m l t d e f i n e d o n l y by N-~.

5Ss(g)~-4rsin2(al(£)-aO(£)),

which

does

not

I n (2) 5S s i s

d i v e r g e even i n t h e

macroscopic limit. It Is interesting t h a t in b o t h c a s e s t h e change o f t h e e n t r o p y i s not enough t o r e a l i z e t h e thermodynamical o r d e r o f 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 w i t h t h e i n c r e m e n t o f N. The d i f f e r e n c e o f t h i s l l m l t from t h e l i m i t d e f i n e d o n l y by N-*~ i s c l e a r l y seen i n t h e c o l l a p s e . That i s , t h e o f f - d l a g o n a l terms does not v a n i s h i n 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 o f t h e c o l l a p s e c r u c i a l l y depends on t h e magnitude o f r , t h a t I s , In the case rfflnlte but satisfactorily l a r g e r than 2(r>>2) t h e c o l l a p s e i s effectively c o m p l e t e d , whereas in t h e c a s e where r i s comparable w i t h 2 t h e c o l l a p s e I s not completed and we can e x p e c t t o see t h e I n c o m p l e t e c o l l a p s e . If we p r o v i d e v e r y s m a l l s i z e a p p a r a t u s or v e r y 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, t h e P r o c e e d i n g s o f t h e Symposium on t h e F o u n d a t i o n s of Modern P h y s i c s 1990, e d s . P. L a h t l and P. M l t t e l s t a e d t , P170. 3. T. K o b a y a s h i , Talk p r e s e n t e d a t t h e Second I n t e r n a t i o n a l Wigner sysposlum, G o s l a r in @ermany, 1991; p r e p r l n t of U n i v e r s i t y o f Tsukuba, UTHEP-220 (1991). 4. T. Kobayashi, Nuovo Cimento 107B (1992) 657. 5. T. Kobayashl, ( F o u n d a t i o n s o f P h y s i c s L e t t e r s v o l . 5 No.3 in p r e s s ( 1 9 9 2 ) ) . 6. T. Kobayashl, (Phys. r e v . A1 in p r e s s (1992)).