Elementos que deben contener las propuestas didácticas

Since I(A,B,T,) is triv ia lly zero for classical sy stem s (as p e r re q u ire m e n t (ii)), only th e case of q u a n tu m sy stem s will be co n sid ered in th is section. In p a rtic u la r, it will be show n th a t

I(A,B,T) = 0 IF F [0 Vj,k (10)

th u s c h a ra c te ris in g c o m p a tib ility for q u a n tu m o b se rv ab le s (see [9] for f u r th e r d isc u ssio n of th is condition). Also, th e in e q u a lity a p p e a rin g in e q u a tio n (9) for c o m p le m e n ta ry o b se rv a b le s w ill be im p ro v e d , a n d g en eralized to th e re la tio n

I(A,B,<L) > % ---\--- r , ^ 1 + su p I I

(ID

w h ere A ,B a re n o n -d eg en erate observables of a n N -level q u a n tu m system for w h ich [A/, B k] * Ö V j , k = 1 ,2 ,..., N, a n d ej, fk a re b a sis vectors for

A j(H ), B k(H ) resp ectiv ely . T h is r e s u lt m ay be com pared w ith e q u atio n s (13), (5), a n d (24) of references [2], [3] an d [4] respectively.

To prove e q u a tio n (10), it is sufficient to show th a t J (E ,F ,rE) = 1 if a n d only i f [E , F] W <£ = 0. Now, d e fin in g th e p ro jectio n G as th e su m of th e m u tu a lly orth o g o n al projections E a F, E aF ' E 'aF, E a F ', it follows for a d e n s ity o p e ra to r on H w ith diag o n al form W = Si Wi I \|/j > < \|/j I (Wi>0Vi)y t h a t eq u atio n (5) m ay be re w ritte n as

J(E,F,t) = wi < yi I d y . > . i

T h u s if J(E ,F ,'E ) = 1 th e n <\|/; I G \ji> = 1 for each i, i.e., I \|/j> = G I \|/j>, a n d so W<z = G W<e . B u t F F G = F aF= F E G , an d hence [E ,F ] W<£ = [F, F ]GW<e = 0. C onversely, suppose [E ,F ] W ^ = 0. H ence, [E , F ] I\j/;> =0 for each i, i.e., F F I \j/;> = F F I a n d so F F I \|/> e E(H) rF (H ) = (E a F)(H). T hus, u sin g (E a F) E = E a F an d (F a F )F ' - 0 , one h as

(£a i) I y/> = (ixfutf I y/.> + a - if) I ¥,>]

=

vp +

a-if) 11// >

=

I Vi> + d/i) if' I y.>

=

I y> .

F inally, since [F, F] = - [F ,F (| = - [ F ' F] = [ F 'F 1 , it follows th a t

d l\)/i> = (^ + i / /

I y. > =

I y. > ,

a n d hence t h a t J(E , F, T.) = 1 a s req u ired .

To prove e q u a tio n (11), note th a t since th e Aj(H ), Bk (H) are d istin c t one­ d im e n sio n al su b sp ac es of H in th is case, th e ir in te rs e c tio n m u s t be zero­ d im e n sio n al, i.e., A j a B k = 0 . Also, A jH ) n B ^ H ) can a t m o st be one-

/ A

d im en sio n al, in w hich case one m u st h av e Aj(H ) £ B k (H), i.e. A j an d B k m u s t be orth o g o n al an d so com m ute. B u t [Äj, Bk] * Ö by supposition,

A A ^ A A ^ A A ^

a n d hence (A j a B )(H) is zero -d im en sio n al, i.e. A j aB =0. S im ilarly , A .

aB k = Ö, a n d th u s , a s in th e (less g e n e ra l) case of c o m p le m e n ta ry o b serv ab les, e q u atio n (8) holds for A an d B. Now, if A j v Bk projects the vector \jf e H onto th e u n it direction \\rjk e (Aj (H), th e n

(y,ÄjV) = (y r A . v B k y) \(ef y/jf)

12 ;

(y .£ kv> = Ufk, v jk) i 2 ;

a n d n o tin g t h a t e j , fk, \f/jk lie in th e sam e p lan e, (Aj vBk) (H) , it follows also t h a t

\(ef ¥ j k )

| 2 +

S u b s titu tio n of th e above in to e q u atio n (8), w ith W w ritte n in diagonal form , yields

trrJ + tr KA.B.'E) > s u p ----— — --- . (12)

j.k 1 + 1 Q I

F in a lly , n o tin g sup {t r[W^Äj]f t r[ W ^ B k ] } ^ N - 1 > e q u a tio n (11) m ay be o b ta in e d from e q u atio n (12). F o r th e p a rtic u la r case w here A an d B are co m p lem en tary , eq u atio n (11) becom es

I(A,B,-e) >---, (13)

2V + / N th u s im proving th e low er b ound of e q u atio n (9).

4.

An example - two spin-172 particles

We co n sid er h e re th e well kno w n E PR -B ohm th o u g h t e x p erim e n t for an ensem ble of p a ire d spin-1/2 p a rticle s described by th e sin g let s ta te [10,11], a n d c a lc u la te th e m e a s u re of in c o m p a tib ility b e tw e e n m e m b e rs of a g e n eral class of observables for th is system .

F o r (u n it) d irectio n s m i , m 2 le t S ( m i , m 2 ) be th e p ro p o sitio n w hich is v erified if a n d only i f p a rtic le s 1 an d 2 a re m e a su re d to h av e spin “u p ” in d ire c tio n s m i , m 2 resp ec tiv e ly . T h u s S ( m i , m 2 ) is re p re s e n te d by th e p ro jectio n

S ( m v m2) : = ( m ^ m .

w h ere I m i , m 2) = I m i) ® I m2), an d Im ) d enotes th e s ta te of a spin-1/2 p a rtic le w ith sp in “u p ” in d irection m . T he discrete observables A, B are d efin ed as th e p ro p o sitio n s £ (01,02), £(61,62) resp ectiv ely , for a r b itr a r y directions a \, (12, 61, 62 .

In th e sp ecial cases 61 = ± a \ , 62 = ± ci2> one h a s [A ,B ] = 0, a n d hence I(A,B, £) = 0 from e q u atio n (10). In all o th e r cases, A a B = A 'a B = A a B'

= 0, so th a t

K A , B t T) = 1 -J{A,B,<Z) = tr[W £ (A v £ ) ] . (14) H ere, is ta k e n to be th e sin g let s ta te i y/s)(Ys I > w here

1%) = ^ • [ l o 1. - a 1> - ' - “ i»0 !)) G5)

/V A

to w ith in a n a rb itr a r y p h a se factor, a n d A w B is th e projection onto th e tw o -d im en sio n al su b sp ace sp a n n ed by lai,<22), ‘ 61,62). W ritin g

Ä w ß - l a 1,a 2)(a 1,a 2 l + I u){u \ u) 1 (u I , (16a) w h e re

\ u ) : = I a i.,a 2) “ ^61,62 1 ^ 1,a2) I 6 ^ 6 ^ (16b) a n d satisfies (u 101,02) = 0, th e m ea su re of in co m p atib ility of A a n d B m ay be ev alu ated by su b stitu tio n of (15), (16a), (16b) into eq u atio n (14). Several p ag es of sp in o r a lg e b ra th e n yield

KA.B.'E) = | ( 1 + K /L ) , (17a) w h e re

K : = (1 + a r 62)(l + - (1 + 1 + 6r 62) (17b) L : = 4 - (1 + a 1.61)(l + a 2.62) . (17c) M axim um in co m p atib ility , I(A ,B ,T ) = 1, is achieved by th e case 61 = 02, 62 = a i ; conversely, I( A ,B ,(E) = 0 for th e case a i = a2, 61 = 62 . F o r th is la tte r case, A a n d B m ay b o th be a ssig n e d th e a p rio ri v a lu e of 0, as for th e sin g let s ta te b o th p a rtic le s can n e v er have spin “u p ” in th e sam e direction.

F in a lly , for th e case a \ = 61, 02 * ± b2, one o b ta in s th e sim ple r e s u lt

KA,B,<E) = 1/2.

5.

Discussion

T he proposed m ea su re of in co m p atib ility (eq u atio n (7)) m ay be applied to b o th c la ss ic a l a n d q u a n tu m sy ste m s, a n d s a tis fie s a n in e q u a lity for c o m p le m e n ta ry o b s e rv a b le s (e q u a tio n s (9), (13)) a n a lo g o u s to th e H e is e n b e r g in e q u a lity fo r c o n ju g a te o b s e rv a b le s . A lso, a sim p le c h a ra c te ris a tio n of com patible d iscrete o b serv ab les for q u a n tu m system s h a s b een given (equation (1 0 )).

F u r th e r w ork is in p ro g ress co n cern in g th e in te r p r e ta tio n of non-zero n u m e r ic a l v a lu e s of / ( A ,# ,^ :) . F o r th e case w h e re A a n d B a re p ropositions, as in th e exam ple of §3, th e q u a n tity 1/2 x 7(A,ß,*E) ap p ea rs to be a m e a su re of th e e rro r w ith w hich th e “q u a n tu m ” jo in t d istrib u tio n of A

a n d B m ay be m odelled by a classical jo in t d istrib u tio n .

T he “n o n -cla ssic al” b e h a v io u r of q u a n tu m jo in t p ro b a b ilitie s h a s been show n [7 ,8] to le a d to th e v io la tio n of c e rta in in e q u a litie s of classical p ro b ab ility , such as th e well know n Bell in e q u a lity [9]. T he re s u lts of th is L e t t e r f u r t h e r d e m o n s tr a te t h a t t h is n o n -c la s s ic a l b e h a v io u r, as ex em p lified by th e n o n -triv ia l low er b o u n d of e q u a tio n (6), lead s to th e existence of incom patible observables for q u a n tu m system s.