Stochastic processes and continual measurements in quantum

STOCHASTIC PROCESSES A N D CONTINUAL M E A S U R E M E N T S IN Q U A N T U M MI~CHANICS

Alberto Barchielli Dipartimento di Fisica dell'Universit~ di Milano. Istituto Nazionale di Fisica Nucleare, Via Calorie,

i.

Sezione di Milano.

16 - 20133 Milano - Italy.

Continual measurements in quantum mechanics and operation valued stochastic processes.

In the last twenty years a very flexible formulation of quantum mechanics

(QM) has been developed,

starting from a suitable generalization of

the notion of observable and of the Von Neumann reduction postulate /i,2/. A central point in this formulation is the notion of instrument /2,3/ which contains both the probabilities for the measured quantity and the way the state of the system changes under measurement. Let £

be a Hilbert space.

operators on £ , ~(T(~))

by T ( ~ )

(~(B(~)))

Denote by B ( ~ ) the algebra of bounded

the space of trace-class operators on ~

the space of bounded operators on T ( ~ )

I4B(£) we denote the identity operator on ~

is the probability measure of a GSP with values in ~ 't~,;~L); then, by theorems 1 and 2 at pgs.348-350 of ref.4, this measure

satisfies

the continuity

last member of eq.(2.9)

condition

at pg.309 of ref.4.

goes to zero and ~(...)

{ (see also the footnotes at pgs.57 and 3 5 O o f 3) Using definitions

then property iii) follows from Theor.2.1.

the

is strongly continuous in

ref.4).

(2.1) and (2.3), we can write

g-.-t

Thus,

..

19

4) The composition law (2.5) can be proved by a changement of integration variables in the double integral defining ~ ( t 3 t t 2 ; ¢ 2 ) ~ ( t 2 , t l ; ~ l using eq.(l.2). Now, let ~(...) ~& T(~)such

satisfy property i)-iv). For any Y-~0, Y~ B ( ~ ) , p ~ 0 ,

that~ ~0 , define

L~,e(f)=/< y,}/L ~,;o)e>. ~ L ~, ~ ( ~ )

) and by

is a positive definite continuous functional in

,~.,o,

~)~{',,t,) with

L~,D(0)=I. By Theor.2 at pg.350 of ref.4 there exists a unique probability measure?~,~(N)t on ( ¢ ' , ~ [ ~ ) ) such that

Then we set

so that eqs.(2.10) and (2.11) become •

t,e

'

where Fy, p(N) is a finite, positive, ~--additive measure on /~ (t,,~,). Now, %* it is easy to show that by property iii)

Therefore eq. (2.12) holds for any positive Y and ~ ; side of eq.(2.12) vanishes, one has Ff,~(.)=0.

Itrll=~

lly~=~

when the left hand

We have also

IIYII=~

lie/l,.

lYt111 As t h e

left

hand s i d e o f

(2.12)

is

linear

in

Y, we can e x t e n d b y l i n e a r i t y

F ~(N) to a positive bounded linear functional on B ( ~ ) .

Now, F

~($')=

= and, therefore, it is a positive normal functional on B ( k ) ( ~&(...)~ & T(~v)). But, for any Y>0, F~,,(N)< F~ ~(~/ ) and, therefore, also F

",,f--(N) is

normal and can be identified with an element of T(M,)

20

(ref.8,

pgs.50-51).

By linearity,

~j%~(N) can be extended to all ~ ~ T ( ~ ) ;

in this way we define an operator ~ ( . . . ) £ ~ ( T ( ~ ) ) . Therefore'

where

VY~

B(~),~

/F;% ~(...)~(T(L))

~ T(£),

we have

is a positive operator valued measure with value

space

(~ ',~ ~ ~ ); normalization follows from property i). Starting %12tLZ(n/T ~(T(kO{~ from the operator ~ ( . . . ) , 0 ~ )), that en3oies the same proper'

ties

as

~(...)~ne

valued measure

constructs

I::}" (...).

in the

same

way

the positive

operator

Then it is easy to show that

o which implies (by the uniqg,eness of the measure determined by a characteristic functional)

that

~

t~

(...)=~(...),~,~;

therefore ~(t2,tl;

N) is CP.

Finally, starting from

one can show that eq.(l.2) ref.4,

pg.313,

cylinder

sets.

holds when N and M are cylinder

it is shown how to construct Using that construction

a general

sets. Now in

set starting

and the ~-additivity

from

of the three

measures in eq.(l.2), one obtains that this equation holds for general sets N and M. Remark 2.1.

In the reconstruction

(...) in

has been used, so that for a characteristic

~

of the OVSP only the weak continuity of operator weak and

trong continuity turn out to be equivalent.

3.

Construction of a class of OVSP's. In ref.3 it is proved that for any CP instrument

exists an Hilbert space ~ sure E(N)~ B ( ~

,a state O- in T ( ~ ) ,

~(N)

in ~

there

a projection valued mea-

) and a unitary operator U on ~ s u c h

that

21

We call

{~

, 6- , E(o), U l a (projection valued) dilation of the CP

instrument ). C o n s i d e r ~ n o w two CP instruments /~i and /~2; up to technicalities the defines a new instrument (ref.2, theor.4.2.2). Let composition ]~2@ f~ {~ i,6-i, Ei(,), uilJ be a dilation of "Oi, i = 1,2; then, from eq.(3.1) we have

where UI-~ U l O I 2 , U2=-U2~I 1. Therefore, a dilation of /~2 0]~'1 is given by

{~,Let~now f~,s~ ~,~ w~J ~ , ~L, E~,, U(t~,tl)1

(E~= projection valued measure on ~ ( [ , , ~%) ) b~ ar'dila?ionr~f an OVSP J ~ (~2,tl;.). By eqs.(l.2) and (3.2), we are brought to require the following compatibility conditions among dilations referring to different time intervals (tl