On the statistical meaning of complex numbers in ... - Springer Link

L3ETTERE AL NUOVO CIIVIENTO

v e t . 34, N. 7

12 Giugno 1982

On the Statistical Meaning of Complex Numbers in Quantum Mechanics. L. ACCARDI I s t i t u t o di M a t e m a t i c a dell' Universit& - Cagliari, I t a l i a I s t i t u t o di Cibernetica del C . N . R - Arco Felicc, N a p o l i , I t a l i a

A. FEDULLO Istituto di S c i e n z c dell' Tn/ormazione dell' Unive~'sith - Salerno, I t a l i a

(ricevuto iI 2 Novembre 1981)

S u m m a r y . - Bell's inequality is a necessary condition for the existence of a classical probabilistic model for a given set of correlation functions. This condition is not satisfied by the quantum-mechanical correlations of two-spin systems in a singlet state. We give necessary and sufficient conditions, on the transition probabilities, for the existence of a classical probabilistic model. We also give necessary and sufficient conditions for the existence of a complex (respectively real) Hilbert space model. Our results apply to individual-spin systems hence they need no (( locality >>assumption. When applied to the quantum-mechanical transition probabilities, they prove not only the necessity of a nonclassical probabilistic model, but also the necessity of using complex rather than real Ililbert spaces. S t a t e m e n t of the p r o b l e m . - The problem of understanding tile empirical basis of the quantum-mechanical formalism has been studied by many authors, both physicists and mathematicians (cf. the excellent survey (~) and the bibliography therein). Recently a new approach to this problem has been proposed (cf. (2-4)) in which one considers the conditional {i.e. transition) probabilities as the basic cmpirical data from which the mathematical model should be deduced. The main idea of this approach is to classify the probabilistic models, both Kolmogorovian and non-Kolmogorovian, according to statistical i n v a r i a n t s , which are expressed in terms of the transition probabilities.

(1) A_. S. WIGItTMAN: Hilbert's sixth problem: mathematical treatment o] the axioms o/ physics, in IVlathematieal Developments .Arising ]rom Hilbert Problems. Proceedings Symposia in Pure Mathematics,

Vol. 28 (Providence, 1976). L. A.CCARDI: Topics in quantum probability, to appear Rep. Phys. (3) L. ACC~RDI: Non-Kolmogorovian probabilities, in Rendiconti del Seminario Matematico dell'Uni(2)

versitd e del Politecnieo di Torino. (4) L. ACCARDI:Foundations o! quantum probability, invited address to the I I I Vilnius Con]erence, Probability and Mathematical Statistics, Vilnius, June 1981 (to appear).

161

162

L.

ACCARDI and A.

FEDULLO

I n t h e p r e s e n t w o r k t h e s t a t i s t i c a l i n v a r i a n t s f o r s o m e s i m p l e s y s t e m s are e x p l i c i t l y c o m p u t e d a n d i t is s h o w n t h a t t h e y allow u s to d i s t i n g u i s h a m o n g K o l m o g o r o v i a n , realHilbert-space and complex-Hilbert-space models. These statistical invariants depend o n l y o n t h e t r a n s i t i o n p r o b a b i l i t i e s w h i c h c a n b e c o n s i d e r e d as e m p i r i c a l d a t a , a n d i t is v e r y s i m p l e to p r o d u c e e x a m p l e s of q u a n t u m s y s t e m s s u c h t h a t t h e s t a t i s t i c a l i n v a r i a n t s a s s o c i a t e d to t h e i r t r a n s i t i o n p r o b a b i l i t i e s do n o t allow a n y K o l n m g o r o v i a n or r e a l - H i l b e r t - s p a c e m o d e l , b u t o b v i o u s l y allow a c o m p l e x - H i l b e r t - s p a c e m o d e l . T h e r e f o r e t h e m e t h o d of s t a t i s t i c a l i n v a r i a n t s allows u s to solve t h e p r o b l e m of 0 . . . . .

W e will s a y t h a t t h e t r a n s i t i o n p r o b a b i l i t i e s (1) a d m i t there exist

a K o l m o g o r o v i a n m o d e l , if

1) a p r o b a b i l i t y space (f2, 0, ~), 2) for e a c h o b s e r v a b l e A , B , C . . . . . (C~) . . . .

(~) (s) (~) 207 (*) 788

a measurable partition

of ~ 2 - - ( A a ) ,

(B~),

J . M . JAUaIX: Foundations of Quantum Mechanics (Reading, Mass., 1968). G. EMCH: H d v . Phys. Acta, 36, 739, 770 (1963). D. FINKELSTEIN, J. M. JAUCH, S. SCHIMINOVICH and D. SPEISER: J. 2llath. Phys. (N. Y.), S, (1962). D. FINKELSTEIN, J. M. JAUCH, S. SCHXMINOVIeH and D. SPEISER: J. Math. Phys. (N. Y.), 4, (1963). Q) E, C. G, STUECKELBERG: Helv. Phys. Acta, 33, 727 (1960); E. C. G. STUECKELBERG and M. GUENIN: Helv. Phys. Acta, 34, 621 (1961); E. C. G. STUECKELBERG, l~I. GUENIN, C. PIRON and. H. RUEGG: Helv. Phys. Acta, 34, 675 (1961); E. C. G. STUECKELBERG and M. GUENIN** Helv. Phys .Acta, 35, 673 (1962).

ON THE STATISTICAL MEANING OF COMPLEX NU:MB:ERS :ETC.

163

such t h a t , for each ~, [~, y . . . . (4)

P(A

= a~lB = be) = ~(A~ v~ Be) . . . . .

**(Be) W e will say t h a t t h e t r a n s i t i o n p r o b a b i l i t i e s (1) a d m i t a c o m p l e x (respectively, real) H i l b e r t space model, if t h e r e exist 1) a c o m p l e x (respectively real) H i l b e r t space of d i m e n s i o n n - - ~ ; 2) for each observable A, B, C .... , an o r t h m m r m a ] basis of J ~ such t h a t , for each ~., fi, y ....

(5)

P(A

(~0a), (~e), (Zy) .... :

-- aalB = be) = [(~a, Y~e)]2 . . . . .

I n this case we say t h a t the given o r t h o n o r m a l basis realize the corresponding transition probabilities. Thus, for a g i v e n set of t r a n s i t i o n p r o b a b i l i t i e s P ( A = a ~ l B = b~) . . . . . w h i c h we can consider as 0,

(7)

Z Fg,~,v = P ( A = a~lB = b~) ,

(s)

Z F~,.,v =/'(A =

(9)

EF~,,,v = P(B = b~lC = cv).

a.IC = c~),

o~

Proo]. Necessity. Let (Q, ~, f~), (A~), (Ba), (Cv) be a Kolmogorovian model for the given transition matrices. Then the symmetry condition (2) implies that (cf. (2)) 1

if(At,) = ff(B~) = ff(Cv) = -Ib,

Vo~, fi, ~ ,

hence 1

which implies (6) and (7) with F~,~,v = ~.ff(A~ t~ B a n Cv). (8) and (9) are obtained in a similar way. Conversely, let (F~,~,v) be n a numbers which satisfy (6)-(9). Then on the set D of all triples (a, fl, 7) (a, fl, Y = 1..... n) one can define the measure it by 1

if(a,

fl, 7) =

- F.,a,v n

and the partitions

A.=U(~,/~,7),

B~=U(~,/~,7),

c~=u(~,~,~)

and it is easy to check that they provide a Kolmogorovian model for tile given transition matrices. From now on we will limit our discussion to the case in which the three observables A, B, C take only two (arbitrary) values. The associated transition probability matrices will be denoted

1 P(A/B) =P=

(

p \ 1--p

= (cos2(~/2) pP)

\sin~(~/2)

1--q)=(c~ (10)

q \ 1--q

sil~-"(~/2)], cos2(~/2)/ sin2 (/3/2)t,

P(B/C) = (2 = (

P(C/A) = R =

r \ 1--r

q

\sin2(fl/2)

1 --

= (cos2@/2)

(

r r)

\sin2@/2)

cos~(fl/2)/ sin2(7/2) 1 , eos~(r/2)/

ON

TIlE

STATISTICAL

MEANING

OF

COMPLEX

NUMBERS

165

:ETC.

where, because of t h e a s s u m p t i o n (3), 0 < p, q, r < 1 and t h e r e f o r e t h e angles e, fi, y can always be chosen to satisfy (ll)

0 < ~,fi, y < z .

Theorem 2. The t r a n s i t i o n m a t r i c e s P , (2, R defined by (10) a d m i t a K o l m o g o r o v i a n model, if and only if

(12)

Ip-i-ff-lll 2 = cos ~ ( ~ / 2 ) ....

(17)

K~l(a), F2(b)}I 2 = sin 2 ( ~ / 2 ) ....

(where a~ denotes t h e angle b e t w e e n a and b), one easily verifies t h a t a spin m o d e l for P , Q, R exists if and o n l y if t h e r e exist t h r e e vectors a, b, c e S (2), such t h a t (18)

cos~ = eos~,

cosfl = c o s ~ ,

cosy =cosca.

Proposition 3. T h r e e v e c t o r s a, b, c e S (2) satisfying (18) exist if and only if (19)

cos 2a + cos 2 f l § cos ~ , - l < 2 c o s ~ c o s f l c o s y .

Proo]. T w o vectors a, b e S (2) satisfying the first of t h e equalities (18) always exist, and we can choose co-ordinates in R a so t h a t a = (1, 0, 0), b = (cos a, sin ~, 0). T h e existence of a c e S (~) satisfying (18) is t h e r e f o r e e q u i v a l e n t to t h e existence of a v e c t o r (el, c2, cs) e R s, satisfying 3

(20)

c,= cosy,

cl c o s ~ + c 2 s i n ~ =

cosfl,

~c~ = 1

ON T H E

167

S T A T I S T I C A L M E A N I N G OF COMPL"EX N U M B E R S .ETC.

a n d (19) is the necessary and sufficient c o n d i t i o n for this occurrence. P r o p o s i t i o n 4. i) If a triple of coplanar vectors satisfies (18), t h e n all the triples with this p r o p e r t y are m a d e of coplanar vectors, ii) If there is a triple of vectors a, b, c e S (z) which are not coplanar a n d satisfy (18), t h c n for a n y other triple a/, b', c'e S (2) satisfying (18) one has

(21)

ab = a ' b ~,

PToo].

~ = b' c ~,

~ = c' a' .

A n y three vectors a, b, c e S'(~) are such t h a t

(22)

a~ + ~ + ~ < 2 ~ .

(23)

(cf. (~), wX-689, 690) a n d the e q u a l i t y in (23) holds if a n d only if a, b, c are coplanar. If a, b, c satisfy (18), t h e n for each of the angles wc have only two possibilities: (24)

a~ = ~

or

2~--~,

b ~ = fl

or

2,~--fi,

~a = y

or

2n--r.

Because of (11), 2 n - - a , 2z~--fi, 2 ~ - - y > . u therefore, because of (23), the second possibility can take place at most for one of the angles. Let n o w a', b', c' e S re) be another triple of vectors satisfying (18). P u t 0~ = a~, 05 -- b~, 0a = ~ a n d 0~ the corresponding angles for a', b', c'. If (21) does n o t hold, t h e n for t h e reasoning above there is at most an i n d e x - - l e t us call it / - - s u c h t h a t 0'~ = 2~ - - 0~. D e n o t i n g ~, k the r e m a i n i n g indices, one has, from (23) a n d (22), (25)

2.~>O~t § 0~' + 0~1~ 2 n - - O i § O j §

2~--0i§

O i = 2n

T h u s a', b', c' m u s t be coplanar. B u t in this case from [0~ - - . ~j l'~~k ~' we deduce 2Jr~ (0~ § 0j § 0k)< 2zr, hence also a, b, c m u s t be coplanar. If a', b', c' are n o t coplanar the first i n e q u a l i t y in (25) is strict, a n d this contradicts the last i n e q u a l i t y . Hence for no i n d e x i we can have 0 i # O~i a n d this proves (21)9 C o m p l e x Hilbert-space models ]or P , (2, R . - Assume t h a t there exists a complexHilbert-space m o d e l for P, Q, R. Then, there are a Hilbert space ~ ~_ C2; three orthon o r m a l basis ( ~ ) ( ~ ) , (Z~) satisfying (5); and real n u m b e r s O~r B), e~v(B, C), Ova(C, A ) such t h a t

(26) Therefore using tile o r t h o g o n a l i t y relation

one i m m e d i a t e l y verifies the equality (27)

exp [-- i~ll(~y, A)] c,os ~2/2 : exp [i[~ll(A , B) _1_ Eli(B, C)]] cos 0~/2cos/~/2 -~ 9exp [i[e12(A, B) § e~(B, C)]] sin ~/2 sin/~/2.

(~3) F. ENRIQUEZ and U. AMALDI: Elemenli di geometria (Bologna, 1921).

168

L. ACCARDI and A. FEDULLO

L e m m a 5. L e t 41, 42, )'3 be p o s i t i v e real n u m b e r s . A necessary and sufficient condition for t h e existence of real n u m b e r s x, y, z, which solve the e q u a t i o n 41 exp [ix] § ~ exp [iy] = 43 exp [iz],

(28) is t h a t

--1~

(29)

41-(4] + ~)

~1

2414~

or e q u i v a l e n t l y , that

4~ + 4 ~ - 4 ~ --I~

(30)

i

~1.

241~3

Proof. - If a solution of (28) exists, t h e n

(31)

cos(x--y)

--

24142

and (29) is satisfied. Conversely, if (29) holds one can choose an a r b i t r a r y couple x, y such t h a t (31) is satisfied, and w i t h this choice b o t h sides of t h e e q u a l i t y (28) h a v e t h e s a m e modulus, t h u s a solution of (28) exists. Finally, the s o l v a b i l i t y of (28) is e q u i v a l e n t to t h e s o l v a b i l i t y of 41 exp [ix] - - 23 exp [iz] = - - 22 exp [iy]

(32) and, w r i t i n g finds (30).

condition

(29)--with

the

appropriate

coefficients--for cq.

(32)

one

Remark. - One easily verifies t h a t an e q u i v a l e n t condition for the existence of solutions of eq. (28) is t h a t the i n e q u a l i t y 41< 42 + 43 and those o b t a i n e d f r o m this b y circular p e r m u t a t i o n hold. T h i s f o r m of the condition, as we will show elsewhere, has a m o r e general c h a r a c t e r t h a n (29) or (30), b u t in t h e p r e s e n t p a p e r only this last t y p e of i n e q u a l i t i e s will be used. Corollary 6. T h e following e q u i v a l e n t inequalities are necessary conditions for t h e existence of a c o m p l e x - H i l b e r t - s p a c e m o d e l Ior the t r a n s i t i o n m a t r i c e s P , Q, R :

(33)

(34) (35)

cos ~

§ cos ~fl~- cos 2 F - l ~ 2 c o s ~ c o s f l c o s y , eosZ ~/2 + cos ~ fl/2 + cos ~ f l / 2 - I

--1~