completly positive maps in generalized quantum dynamics

8/8/2019 Completly Positive Maps in Generalized Quantum Dynamics

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Foundations o f Phy sics, Vol. 11, No s. 1/2, 1981

On Completely Positive Maps in Generalized QuantumDynamicsRalph F. S im mo ns , Jr 1,2 and Jam es L . Pa rk AReceived April 28, 1980

Several authors have hypothesized that completely positive ma ps shouM providethe means fo r generalizing quantu m dynamics. In a critical analysis o f thatproposal, we show that such maps are incompatible with the standard ph eno m.enological theory o f spin relaxation and that the theoretical argu ment whichhas been offered a s justification for the hypothesis is fallacious.

1. N O N U N IT A R Y M O T I O N I N Q U A N T U M S Y S T E M S A N DS U B S Y S T E M S

Recen t ly the re have been a numbe r o f pape rs a rgu ing tha t con ven t iona lquan tum dynamics needs to be gene ra l i zed so a s t o i nc lude nonun i t a ryevolu t ion of the densi ty op era tor . Som e autho rs (1-0~ bel ieve tha t the needfor a new dynamical pr inc ip le i s fundamenta l , s ince convent ional uni tarys ta te evolu t ion in a c losed quantum system is incompat ib le wi th the secondlaw o f t he rmod ynamics . O the rs (1-1~ advoca te the use o f ce r t a in non un i t a r ymapping s as e legant phen om enolo gica l descr ip t ions of s ta te evolu t ion insubsystems of com pos i te qu ant um systems which as a whole m ay s t il l berega rded as undergo ing o r thodox un i t a ry evo lu t ion .

In e i ther case it i s obvious tha t the des i red new theory o f mo t ion can notbe e rec t ed mere ly on the p remise tha t i t s maps be no t un i t a ry . Some newmathemat i ca l concep t mus t be added to quan tum theo ry to g ive de f in i t es t ruc tu re to a ny p ro posed genera l i za t ion o f the dynamica l l aws . One suchpro pos al tha t has rece ived some s tudy in depth ~13a~) involves the conc ept ofcomp le te posi tiv i ty . The pre sent n ote offers a cr i tica l analys is of tha ta p p ro a c h .1 Department of Physics, Washington State University, Pullman, Washington.Present address: Disc Memory Division, Hewlett Packard, Bo ise, Idaho.

470015-9018]81]0200-0047503.00/0 1981 Plenum Pubfishing Corporation


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48 Simmons and Park

2. M A T H E M A T I C A L F R A M E W O R KWith every quantum system there is associated a complex, separable,

complete inner product space, a Hilbert space ~#'. If ~ and ~ are Hilbertspaces associated with distinguishable systems A and B, then the directproduct space ~ @ ~ is associated with the composite system of A and Btogether.To every reproducible preparation of state for a quantum system therecorresponds a density operator p, which is a positive-semidefinite, self-adjoint, unit trace linear operator on ~ . All the density operators are ele-ments of y(d/d), the real normed linear space of all linear self-adjoint traceclass operators on ~ . Within the space J - ( ~ ) is a set called the positivecone ~+(d/d), which contains the positive-semidefinite operators on ~f~,and within the set ~'>(Jd) is a convex subset ~']+(oW) containing the elementsof ~/r+(~) with unit trace. Therefore ~ + ( ~ ) is the set of density operators,the mathematical representatives of quantum states.

Any proposed dynamical taw must involve only mappings of the convexset ~ + ( ~ ) into itself, and the formulation of such laws is complicated bythe fact that neither ~F+(~f~) nor ~]+(Jd) is a subspace of J ( ~ ) . The dyna-mical postulate of conventional quantum theory describes the motions ofa system by a one-parameter (time) unitary group {At} of linear transforma-tions of J ( J d ) into y(cs~), the unitary guaranteeing that no density operatorwill be transformed out of k]+(Jt~). Unfortunately the unitarity also con-strains the entropy functional S = --k Tr pin p to be a constant of themotion.

A natural generalization of the dynamical postulate is obtained byletting {At} be a one-parameter semigroup of linear transformations of3-(~t~) into J- (R) which need not be unitary. In this way irreversible motionsare included among the dynamical possibilities, since there may now bemaps At which have no inverse and under which S is not invariant.The Hille-Yoshida theorem (~5) can be used to obtain an equation ofmotion for p in terms of the infinitesimal generator L of the semigroup,

d p ( t ) ddt -- dt A~p(O) = LA~p(O) (t)with A,p(O ) = p( t ) ,

dp(t) /dt = Lp( t ) (2)The requirement that mappings {Ad generated by L transform each

element of ~ + ( ~ ) only into another element of ~~+(Jf) places severerestrictions on L. The conditions satisfied by an admissible L have beer/given by Kossakowski.aG~


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On Completely Positive Map s in Generalized Quantum Dynam ics 49A l i n e a r m a p p i n g A f r o m Y ( Y f ) i n t o . Y - ( W ) i s c a l l e d p o s i t i v e i f

A a ~ /'+ (~ g" ) w h e n e v e r a E y r + ( W ) . T h u s o n e a p p r o a c h t o g e n e r a l i z in gq u a n t u m d y n a m i c s c o u l d b e b a se d u p o n t h e h y p o t h e s is t h a t t h e e q u a t i o n o fm o t i o n ( 2 ) s h o u l d f e a t u r e a g e n e r a t o r L a s s o c i a t e d w i t h p o s i t i v e , t r a c e -p r e s e r v in g m a p s { A , } . H o w e v e r , t h is p r e s c r i p ti o n s e e m s r a t h e r b r o a d a n dinexp l i c i t , and some r i che r , more s t ruc tu red p remise i s c l e a r ly needed .

T h e p a r t i c u l a r a l t e r n a t i v e h y p o t h e s i s w i t h w h i c h t h e p r e s e n t a n a l y s i si s c o n c e r n e d i n v o l v e s t h e n o t i o n o f c o m p l e t e p o s it iv i t y . A l i n e a r t r a n s f o r -m a t i o n ~ f r o m 3 " ( ~ ) i n t o 3 - ( J r ) i s s a id to b e c o m p l e t e l y p o s it iv e i f t h et e n s o r p r o d u c t m a p @ I ~ o n ~ -'(Y ) @ M ( n ) i s pos i t i ve fo r a l l n , w he reM ( n ) i s t h e C * - a l g e b r a o f n n c o m p l e x m a t r ic e s .

3 . Q U A N T A L M O T I O N S D E S C R IB E D B Y C O M P L E T E L Y P O S IT I V EM A P S

C o m p l e t e p o s i ti v i ty is a s t r o n g e r r e s t r i c ti o n t h a n p o s i t i v it y a n d a s s u c hp e r m i t s o n e t o g i v e t h e e x p li c it f o r m f o r t h e g e n e r a t o r o f a c o m p l e t e ly p o si ti v ed y n a m i c a l s e m i g ro u p . G o r i n i e t a t. ~1~)h a v e d e r i v e d t h e n e c e s s a r y a n d s u f f ic i e n t c o n d i t i o n s f o r L t o b e t h e g e n e r a t o r o f a c o m p l e t e l y p o s i ti v e s e m i g r o u po f a n N - le v el s y st e m . A s a n e x a m p l e , t h e y f in d t h e g e n e r a t o r f o r a t w o - l e v e lsys t em , i .e ., a sys t em w hose a s soc i a t ed H i lb e r t space i s tw o-d im ens io na l .T h e g e n e r a t o r L g o f G o r i n i et al . m a y b e e x p r e ss e d in a m a t r i x r e p r e s e n t a t i o nb y u s i n g as th e b a si s fo r 3 - ' ( ~ ) t h e m a t r ic e s

{ul l i = 0, 1, 2, 3} ~:~ t~ -~ ~/ ~ t (3)w h e r e 1 i s t h e 2 x 2 i d e n t i t y m a t r i x a n d a i s t h e s t a n d a r d s e t o f P a u l i m a t r ic e s .T h e m a t r i x e l e m e n t L , j i s d e f i n e d b y

L i; = Tr(v~Lv~) (4)I n t h i s r e p r e s e n t a ti o n , L o h a s t h e f o r mo o o o )

L~ = a l - -Y l -h 3 h ,,a~ - -h2 h i - - 7 ~ /

w h e r e t h e h~ a r e r e la t e d t o t h e e n e r g y o p e r a t o r H t h r o u g h8H = ~/2 ~ h~v~, y~ >~O

i--1

(5 )

( 0

8 2 5 / I I / ~ / z - 4


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50 Simmons and Park

a n d3

at -- Tim , 0 + Z e,j~rnjhk (7)t c= l

the m~ b e i n g co n s t an t s s u b j ec t t o co n d i t i o n s wh i ch g u a ran t ee t h a t p(t) ispos i t ive .

In t h i s s am e b as i s t h e d en s i t y o p e ra t o r i s co n v en i en t l y ex p an d ed as

O = -Q~ o~,~ (8)i~O

s o t h a t t h e eq u a t i o n o f m o t i o n (2) h as t h e r ep res en t a t i o n-- ~ L . p j (9)dt

I f we in terpre t the two- level sys tem as a spin- magne t ic d ipo le imm ersedin a magnet ic f i e ld B, then the energy opera tor (6) assumes the fo rm

3H = ~/ 2 ~, h,v,, . . . . m " B (i0)wh ere m , t h e m ag n e t i c m o m en t o p e ra t o r , i s g i v en b y

m = h ~ ( 1 1 )b e i n g t h e g y ro m ag n e t i c r a t i o .

T h e q u a n t a l m e a n v a l u e o f t h e it h c o m p o n e n t o f m a t ti m e t is c o m p u t e du s i n g t h e s t a n d a r d t r a c e f o r m u l a

( m i } ( t ) = Tr[p( t ) mi] (12)a n d t h e e q u a t i o n o f m o t i o n f o r ( m , } m a y t h e r e f o re b e d e r i v e d f r o m

d ( m d t == Tr ( - ~ m , ) (1 3)by us ing (9) together wi th a speci f i c cho ice fo r L .

If Lg is ch ose n as the ge ner ator , then (5), (6) , (9) , an d (12) yield thee q u a t i o n s o f m o t i o n

d ( m i } ( t ) adt = ~j . l c= l eisl~h~((rn~}(t) - rnk) -- y i ((m ~}(t) -- mi ) (14)


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On Completely Positive Maps in Generalized Quantum Dynamics 51Gorini e t al . ~3~ claim that these equations, a direct consequence of theassumption of complete positivity, are the Bloch equations ~7) used in the

phenomenological theory of spin relaxation. Were that the case, the hypoth-esis of completely positive maps would indeed seem to be remarkably fertile,for it would constitute an elegant principle of motion from which at leastone well-known phenomenological law could be deduced.We find, however, that when the standard Bloch equations are writtenin this same notation, they assume the form

d ( m i ) ( t ) 3d r - - - - ~J ,k= l e ~ k h j ( m ~ ) ( t ) - V i ( ( m ~ ) ( t ) -- m~) (15)

The difference ( m ~ ) ( t ) - m i only appears in the relaxation term inBloch's equations but appears throughout in Eq. (14) based upon Lg.Therefore, since Gorini e t aL a31 have shown that Lo satisfies the sufficientand necessary conditions to be a completely positive generator, the evolutionas given by the Bloch equations is in fact n ot completely positive. (A moredetailed proof of this point is given in the Appendix.) Consequently thehypothesis that completely positive dynamical maps might be useful indescribing the quantal motion of subsystems is not supported by the com-parison with Bloch's theory.Neither does it appear that complete positivity will help in the search ~1-9~for a new fundamental nonunitary dynamics of closed systems, since thefinal state p(~) of the motion generated by L o is determined by rn~.In fact l i m ~ (m~) ( t ) = m~; and therefore L~ as expressed by (5)-(7) dependson the final state of the system. In our opinion, this feature is philosophicallyunacceptable in a fundamental theory. A basic dynamical equation shouldexhibit genuine predictive power, not merely serve as a phenomenologicalcatalog of states of the system. Surely, classical or quantum mechanicswould be far less impressive if the generator of time evolution, the Hamil-tonian, were itself a function of future states.

It is instructive in this connection to compare the completely positiveL~ with the generator derived by Park and Band, ~5~ who studied all admissibleL's for a two-level system in search of those which were independent of pand which generated energy-conserving, entropy-increasing motions. Eventhough complete positivity played no role in their arguments, the generatorthey found is the same as L,, provided the first column in (5) vanishes, anessential requirement i fL is to be state-independent. One does not obtain theBloch equations in this case either, and indeed one should hardly expectto derive the Bloch equations without analyzing the composite system ofspins plus lattice.


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52 Simmons and Park4. I S T H E R E A V A L I D P H Y S I C A L B A S I S F O R A C O M P L E T E L Y

P O S I T I V E D Y N A M I C S ?G iven the seeming i r re l evance o f comple t e pos i t i v i t y i n t he phys i ca l

s i t ua t i ons d i scussed above , i t i s appropr i a t e t o i nqu i re w he the r t he re ex i s ts o u n d p h y s i c a l a r g u m e n t s i n b e h a l f o f c o m p l e t e ly p o s i ti v e d y n a m i c a l m a p s .T h e o n l y p u b l i s h e d j u s t i f i c a t i o n s o f w h i c h w e a r e a w a r e a p p l y o n l y t o t h em o t i o n o f s u b s y s t e m s .

Kra us ~ls) dem ons t ra t e s t ha t a causa l s t a t e changes l ike t hose a s soc i a t edw i t h m e a s u r e m e n t i n t e r v e n ti o n in t h e o r t h o d o x q u a n t u m t h e o r y o f m e a s u re -m e n t a r e d e s c r i b ed b y c o m p l e t e ly p o s i t iv e m a p s , b u t o f c o u r se t h is t h e o r e mdoes no t suppor t comple t e pos i t i v i t y a s a ba s i s fo r gene ra l i z ed causa ld y n a m i c s .

G o r i n i e t al . a3) p r o v e t h a t i f t h e e v o l u t i o n o f t h e i s o l a te d s y s t e m Aplus B i s g iven by a un i t a ry g roup and the cons t i t uen t s A and B a re i n i t i a l l yu n c o r r e l a t e d , t h e n t h e s u b d y n a m i c s o f s y s te m A w i ll b e d e s c r i b e d b y acom ple t e ly pos i t i ve semigrou p . Th i s i n t e re s t i ng theo rem c l ea r ly de l inea t esa n i m p o r t a n t f a m i l y o f m o t i o n s w h i c h a r e c o r r e c tl y d e s c r ib a b l e b y c o m p l e t e lypos i t i ve maps . H ow eve r , s i nce t he re ex i s t fo r i so l a t ed compos i t e sys t emsm o t i o n s w h i c h a r e d e m o n s t r a b l y n o n u n i t a r y , ~1 w e m u s t c o n c l u d e t h a t t h et h e o r e m d o e s n o t p r o v i d e s t r o n g s u p p o r t f o r a g e n e r a l i z e d d y n a m i c s b a s e don comple t e pos i t i v i t y . Moreove r , g iven our e a r l i e r r e su l t t ha t t he B loche q u a t i o n s a r e n o t c o m p l e t e l y p o s i ti v e , w e n o w h a v e t h e c o r o l l a r y p r o p o -s i t i o n t h a t t h e B l o c h e q u a t i o n s c a n n o t b e d e r i v e d f r o m a n y m o d e l i n w h i c ht h e c o m p o s i t e s y s t e m o f s p i n s p l u s l a t t i c e i s a s s u m e d t o e v o l v e u n i t a r i l yf r o m a n u n c o r r e l a t e d i n i t i a l s t a t e .

L in db lad ' s i nves t i ga t ion ~1~1 o f com ple t e ly pos i t i ve m aps i nc ludes ap h y s i c a l a r g u m e n t f o r c o m p l e t e p o s i t iv i t y w h i c h i s b o t h m o r e g e n e r a la n d m o r e d e t a i l e d t h a n t h o s e d i s c u s s e d a b o v e . T w o p o s s i b l e i n t e r p r e t a t i o n sa r e a d m i t t e d f o r t h e c o m p l e t e ly p o s i ti v e s e m i g r o u p e v o l u t i o n o f a n o p e ns u b s y s te m . E i th e r t h e s y b s y s t e m A is i n t er a c t i n g w i t h a h e a t b a t h o r t h esys t em A is a subsys t em of t he i so l a t ed sys t em A p lus B and th e i so l a t eds y s t e m u n d e r g o e s u n i t a r y t i m e e v o l u t i o n . T h e a r g u m e n t u s e d b y L i n d b l a dt o c o n c l u d e t h a t A ~ m u s t b e c o m p l e t e ly p o s i ti v e r u n s a s f o ll o w s .

C o n s i d e r t w o q u a n t u m s y st em s A a n d B a n d a h e a t b a t h R . A s s u m e Ais i n t e r a c t in g w i t h R a n d t h a t t h e t i m e e v o l u t i o n o f A i s g iv e n in t h e H e i s en -b e r g p i c t u r e b y t h e f a m i l y o f m a p s

A A *~" ~ (~ A ) ---~ ~ ( ~ 4 ) (16)w h e r e ~ ( ' ~ A ) is t h e d u a l o f 3 -( o ~A ) , a n d c o n t a i n s t h e c o n t i n u o u s , l i n e a r,s e l f -ad jo in t ope ra to r s w h ich rep re sen t t he quan ta l obse rvab le s {QA} o f s y s t e m


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O n Comp letely Positive M ap s in GeneraIizedQuantum Dynamics 5 3A. [Here we must bear in m ind th at in th e Heisenberg pic ture PA is constant ,an d ea ch obse rvable QA is represented at different t imes by different elementsof ~ ( ~ ) . Hence the time dependence o f mea n va lues such as (QA) ----Tr(pAQA) is induced by mappings (16) in ~(~fA) which are dual tothe mappings A A.t~ in J-(O~A) conside red in previous sections.] Fu rth erassume tha t B is a c losed system, i ts dynam ics being g overned by the H amil-ton ian H a , and the t ime evo lu t ion o f B i s g iven by the conven t iona l un i ta rymapping , Umt = exp{itH~}. Le t Ha = 0 so that UB,~ = I s , the id entityo n ~ .

One no w asks i f the m ap A*, , can be extended to a posit ive ma pAt*: N (J f) --~ N(zgF), where ~ = ida @ ~ , such that B is unaffected.I t tu rns ou t tha t the map A t * = A *t @ Is i s pos it ive fo r any N- levelsystem B with II~ ......... 0 i f an d on ly i f A*# is completely positive. Hence weare led to bel ieve that the res tr ic tions on the d ynam ics of A interact ing witha heat ba th R are in par t due to the system B which is not in teract ing withei ther A or R. F urtherm ore, complete posi tiv i ty fol lows only i f B is an N-levelsystem with arb itrar y N ~ {0, t , 2,...} an d H e = 0.

This physical argument , careful ly s ta ted by Lindblad, is c lear ly basedon the requ i rement tha t the d ynamics o f a compos i te sys tem mus t be pos i tiveeven if the com posi te system is ma de up of two noninteract ing , uncorrela tedsystems, and there is cer ta inly no reason to quest ion th is reasonable require-ment . How ever , to requ ire tha t the dynam ica l map A] , , of system A can beexte nde d to a po sitive m ap A t* on N(o~A @ d/riB) goes be yo nd th e pos itivitycond ition. If we assum e tha t AA*,~ mu st be such t ha t i t c an be e xtended to apos it ive mapping on ~ ( ~ @ ~ ) , we can show tha t A*~ mus t sa t is fy aposi t iv i ty requirement in general d ifferent f rom complete posi t iv i ty . Toaccomplish th is , we s imply take system B to be any system which is not anN-level system and for which the Hamil tonian is not t r iv ia l , Ha v~ 0. F o rexample, take B to be a free particle with He = p2/2m. N o w ~ ( ~ ) is n o tthe C*-algebra of complex N N matr ices and hence the requirement tha tAA*,~ be co m ple tel y positive does no t follow . I n gene ral AA*~ m us t satisfysome other posi t iv i ty condit ion.

The in tro duct ion of system B is unnecessary and has no physical basis .Indeed, i f A a nd B are not in teract ing, then the s ta te of the composi te systemA plus B, i f in i t ial ly uncorrela ted, remains a lways in the uncorrela ted f ormPA @ P, an d this op era tor on ~gFA @ ~f~ is alway s positive if PA is positiveand if pB is positive. Therefore if the evolution of A maintains pA positiveand the ev olut ion of B m ainta ins pB posi tive , then pAB = P~ @ pB is alwaysposi t ive . Furthermore, s ince A and B are not in teract ing, the t ime evolut ionof A plus B is a lways given by the Heisenberg evolut ion operator At* =A** @ A*,~. W e require only that A*. , and A*, t be posi tive . They need notbe com pletely positive.


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5 4 S i m m o n s and Park

We conclude that the complete ly posit ive dynam ical map, thou ghmathematical ly in tr iguing and in terest ing as a def ini te proposal for gener-alizing dynamics, is a construct too restrictive to serve as the basis for newquanta l laws o f mot ion .

APPENDIX

In order for the complete ly posi t ive equat ions (14) of Gorini et al. aa)to be ide ntical to th e B loch equa tions (15), th e sym bols rn~" in the tw oequations would have to be unequal . Thus to compare (14) and (15) , wereplace m, in (14) b y n~ so tha t (14) bec om es

~od( m ~) ( t ) _ ~ e i~kh , ( ( rnk ) (t ) n~) - - 7 d ( m i ) ( t ) - - n~) (A1)d t J , k = lGo r in i et aI. (13) provide necessary and sufficient conditions for (A1) togenerate completely positive maps; in particular, condition (vi) in Ref. 13sta tes that

ni = 0 if 9'W373 = 0 (A2)Now , i f we in troduce the symb ol m~ throu gh

37~ mi - - - - ~ ~ijT~hjn~ + 7in~ (A3)

Eqs. (A l) becom e superficially identica l to the BIoch equation s (15). How -ever, the constan ts m~ in the Blo ch e quatio ns are the final values of mag neticmo m e n t c o mp o n e n ts .

By solv ing (A3) for {n~ } in term s of {mi} an d {h~}, we can sho w th at(n, } can not satisfy bo th (A1) and (A3). Using stan dar d algebraic proced ure,we invert (A3) to ob tain

nl o = [m1(717~73 + 71 hi 2) -- m2O(7273h3 + hlh~7 ~)+ ma(y27ah~ - - hlh~ya) ]

X ~ ~ihi ~ + 717~7 (A4)i = 1

W h en t he co nd itio n (A2) is inv oke d in the special case ) '1 = 0, ~,~ = V3 = 7(cf. Ref. 13, p. 824), (A4) becomesnl = - -m2 Th~ - - m~Ohlh2 + ma(Th2 - - hlh3)h2~ + h~2 (A5)


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On Completely Positive Maps in Generalized Qu antum Dynamics 55

Since this is not zero, the constants ni cannot be chosen so that (A1)becomes the Bloch equations while still satisfying the necessary and suffi-cient complete positivity conditi ons in Ref. 13. Therefore Bloch evolut ionis not complet ely positive.

R E F E R E N C E S1. J. Mehra and E. C. G. Sudarshan, Nuovo Cimento 11, 215 (1972).2. G. N. Hatsopoulos and E. P. Gyftopoulos, Found. Phys. 6, 15 (1976).3. G. N. Hatsopoulos and E. P. Gyftopoulos, Found. Phys. 6, 127 (1976).4. G. N. Hatsopoulos and E. P. Gyftopoulos, Found. Phys. 6, 439 (1976).5. G. N. Hatsopoulos and E. P. Gyftopoulos, Found. Phys. 6, 561 (1976).6. J. L. Park and W. Band, Found. Phys. 7, 813 (1977).7. W. Band and J. L. Park, Found. Phys. 8, 45 (1978).8. J. L. Park and W. Band, Found. Phys. 8, 239 (1978).9. J. L. Park and R. F. Simmons, Jr., in OM and New Questions in Physics, Cosmology,Philosophy, and Theoretical Biology: Essays in Honor of Wolfgang Yourgrau, A. vander Merwe, ed. (Plenum Press, New York, 1981), to be published.10. R. S. Ingarden and A. Kossakowski, Ann. Phys. 89, 451 (1975).11. E. B. Davies, Commun. Math. Phys. 39, 91 (1974).12. F. Haake, in Springer Tracts in Modern Physics, VoL 66 (Springer, Heidelberg, 1966).

13. V. Goriui, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).14. G. Lindblad, Commun. Math. Phys. 48, 119 (1976).15. K. Yoshida, Functional Analysis, 4th ed. (Berlin, 1974).16. A. Kossakowski, Bull. Acad. Polon. Sci. Math. 20, 1021 (1972).17. F. Bloch, Phys. Rev. 70, 460 (1946).18. K. Kraus, Ann. Phys. (N.Y.) 64, 311 (1971).


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Foundations o f Phys ics, Vol. 12, No. 4, 1982

Remarks on "On Completely Positive Maps inGeneralized Quantum Dynamics"G . A . R agg i o ~ an d H . P r i m as ~Received May 8, 1981The assertion by Simm ons and Par k that the dynam ical map associated with theBloc h equations o f nuclear ma gne tic resonance is not completely positiv e iswrong.

In a recent paper, Simmons and Park "~ claim that Bloch's equationdescribing the time dependence of the macroscopic nuclear polarizationunder the influence of an external magnetic field cannot be obtained from acompletely positive dynamical evolution law. This assertion is incorrect.Furthermore, Simmons and Park's corollary "that the Bloch equationscannot be derived from any model in which the composite system of spinsplus lattice is assumed to evolve unitarily from an uncorrelated initial state"is also wrong (for well-known heuristic derivations compare, e.g., Refs. 2 and3; for a derivation fulfilling the modern requirements of mathematical rigorcompare Ref. 4).

Since Simmons and Park use these false statements to make far-reaching conclusions about the irrelevance of complete positivity for thesystem-theoretic description of open quantum systems, and since Bloch'ssystem-theoretic description of nuclear magnetic resonance is a paradigm ofoutstanding significance for modern experimentalists and theoreticians, adirect proof (which is, of course, known) of the complete positivity of thedynamical map associated with Bloch's equation may be appropriate.The differential equation proposed by Bloch ~5~ for the time dependence

xLaboratoryfor Physical Chemistry,ETH-Zentrum, Zfirich, Switzerland.433

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4 3 4 R a g g i o a n d P r i m a s

o f t h e m a c r o s c o p i c n u c l e a r p o l a r i z a t i o n M (t ) = M l ( e ) e 1 + M z ( t ) e z + M 3 ( t ) e 3u n d e r t h e i n f l u e n c e o f a m a g n e t i c f ie l d B e 3 i s g i v e n b y

d M ( t ) M l ( t ) M 2 ( t ) M , ( t ) - - M o= y B M ( t ) e 3 e 1 e 2 e 3d t T 2 T 2 T 1

w h e r e ~ i s t h e g y r o m a g n e t i c r a t i o , T ~ is t h e l o n g i tu d in a l a n d T 2 is thet r a n s v e r s e r e l a x a t i o n t i m e f u lf il li n g 2 T 1 > / T 2 > 0 . W e c o n s i d e r t h e c a s e o fs p i n 1 / 2 , s o t h a t M ( t ) is g i v e n b y t h e e x p e c t a t i o n v a l u e o f 7S , w h e r e S =( S ~ , $ 2 , $ 3 ) a n d $ 1 , $ 2 , S 3 a re t he u su a l s p i n - l / 2 m a t ri ce s . As t r a i g h t f o r w a r d c o m p u t a t i o n s h o w s t h a t i n t h e H e i s e n b e r g r e p r e s e n t a t i o n

S ( t ) = e x p ( t L ) S ( 0 )t h e o p e r a t o r 7 S ( t ) s at is f ie s B l o c h ' s e q u a t i o n f o r t h e c h o i c e o f t h e g e n e r a t o r La s

3L ( A ) : i [ H , A ] _ + ~ { V j * A V j - J [ V j * V j , A ] + }j= l

w h e r e A i s an a r b i t r a r y 2 2 m a t r i x a n dH = - ? B S 3V l = a + S l + i a _ S 2 , V ~ = a + S 2 - i a _ S 1V , = { ( 2 /T 2 ) - ( 1 / T 1 ) } 1 / 2 S ~

a = { (y + 2 M o ) / 4 7 T ~ } ~/~ { ( y - 2 M o ) / 4 7 T ~ } I /zT h e r e s u l t s b y G o r i n i et at . ~6) a n d L i n d b l a d ~7) i m p l y a t o n c e t h a t { e x p ( t L ) It >~ 0} i s a sem igro up o f c o m p l e t e ly p o s i t i v e b o u n d e d l in e a r m a p s f r o m t h eb o u n d e d l i n e a r o p e r a t o r s o f a t w o - d i m e n s i o n a l c o m p l e x H i l b e r t s p a c e i n t oits elf . T h e i ne q u al it ie s 2 T I ~ > T z > 0 a n d t 2 M o / 7 1 < , 1 a r e n e c e s s a r y a n ds u f f ic i e n t f o r th e e x i s t e n c e o f L a s d e f i n e d a b o v e . H e n c e B l o c h ' s e q u a t i o n f o rs p in 1 / 2 c a n b e g e n e r a t ed b y a c o m p l e t e l y p o s i ti v e d y n a m i c a l m a p ( c o m p a r ea lso Emch and Var i l ly~8) ) .

I t s e e m s t h a t t h e m o r e p h i l o s o p h i c a l r e m a r k s b y S i m m o n s a n d P a r k a r ed u e t o a m i s u n d e r s t a n d i n g o f t h e r o l e o f s y s t e m - t h e o r e t i c d e s c r i p t i o n s o fo p e n q u a n t u m s y s t e m s . S u c h d e s c r i p t i o n s n e v e r a r e f u n d a m e n t a l , b u t t h e yh a v e a m o s t u s e f u l i n t e r m e d i a t e p o s i t i o n . O n t h e o n e h a n d , t h e y a l l o w t h ee x p e r i m e n t e r t o d e s i g n a n d t o r e f i n e e x p e r i m e n t s , w h i l e o n t h e o t h e r h a n dt h e y a r e t h e o r e t i c a l l y m e a n i n g f u l a n d c a n b e r i g o r o u s l y r e l a t e d t o t h e f i r s tp r in c i p le s o f t h e t h e o r y o f m a t t e r. B l o c h ' s e q u a t i o n s a r e c o m p a t i b l e w i th t h es e c on d l a w o f t h e r m o d y n a m i c s , a n d t h e y c a n b e d e r i v e d f r o m a l a r g e r c l o s e d


12/15

Remarks on "O n Completely Positive M aps in Generalized Quantum Dynamics" 435s y s t e m h a v i n g a n a u t o m o r p h i c d y n a m i c s . A s y s t e m - t h e o r e ti c d e s c r i p ti o n o fa n o p e n s y s t e m h a s t o b e c o n s i d e re d a s p h e n o m e n o l o g i c a l ; t he r e q u i re m e n tt h a t i t s h o u l d b e d e r i v a bl e f r o m t h e fu n d a m e n t a l a u t o m o r p h i c d y n a m i c s o f ac l o se d s y s t e m i m p l ie s t h a t t h e d y n a m i c a l m a p o f a n o p e n s y s t e m h a s t o bec o m p l e t e l y p o s i t iv e . F a r f r o m b e i n g i r r e l e v a n t i n p h y s i c a l s i tu a t i o n s , t h ec o n c e p t o f c o m p l e t e p o s i ti v i t y i s o n e o f t h e g r e a t n e w a c h i e v e m e n t s o fm o d e r n q u a n t u m m e c h a n i c s .

REFERENCE S

1. R. F. Sim m ons and J. L. Park, Found. Phys. 11, 47 (1981).2, R. K. Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953).3. A. Abragam, The Principles o f Nuclear Magnetism (Clarendon Press, Oxford, 1961).4. J. V. Pul~, Commun. Math. Phys. 38, 241 (1974).5. F. Bloch, Phys. Rev. 70, 460 (1946).6. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).7. G. Lindblad, Commun. Math, Phys. 48, 119 (1976).8. G. G. Emch and J. C. Varilly, Letters" in M ath. Phys. 3, 113 (1979).


13/15

Foundations o f Phys ics, tlol. 12, No. 4, 1982

Another Look at Complete Positivityin Generalized Quantum Dynamics:Reply to Raggio and PrimasRalp h F. Sim mo ns, Jr. 1 and Ja mes L. Park -~Received November 2, 1981In this rejoinder to a critique by Raggio and P rima s of our paper, " OnCompletely P ositive Maps in Generalized Quan tum D ynam ics," we acknowledgethat, contrary to our original assertion, the Blo eh equations are indeedcompletely positive. W e then explain briefly w hy this modification o f ouranalysis does not alter its main conclusions.

As a byproduct of a continuing search for possible generalizations ofquantum dynamics that might accommodate in particular the entropy-increasing motions mandated by thermodynamics but forbidden by unitarymechanics, the present authors published last year an analysis ~) ofcompletely positive maps. That paper has generated correspondence bothpraising and condemning its content. Unfortunately our remarks apparentlystruck some readers as a somewhat inflammatory deprecation ofphenomenological studies based on the theory of such maps, even though weexplicitly noted that the elegant work ~2) of Gorini et aL "delineates animportant family of motions which are correctly describable by completelypositive maps." In fact our intent was never to comment adversely orotherwise on the methodology of such fields as nuclear magnetic resonance,but rather to assess the theoretical significance of complete positivity as apossible basis for a generalized quantum thermodynamics.

One section of our paper dealt with the interpretation of a theorem ofGorini et al. that provides necessary and sufficient conditions for completely

Disc Memory Division, Hewtett Packard, Boise, Idaho.2 Departme nt o f Physics, W ashington State University, Pullman, Washington.

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4 3 8 S i m m o n s , J r. a n d P a r k

positive maps when the system of interest has a two-dimensional Hilbertspace. We proved that these conditions yielded an equation of motiondiffering from the Bloch equations and then deduced that the latter must notbe completely positive. In their recent critique of our article, Raggio andPrimas ) objected to this statement and demonstrated that the mathematicalform of the Bloch equations does in fact satisfy the definition of completepositivity; it is then only natural for them to conclude that our entireanalysis consequently collapses. The situation is actually a bit more subtle.

First let us acknowledge that Raggio and Primas are correct on thetechnical point that the Bloch equations are completely positive. In statingthe opposite we misinterpreted our own mathematical analysis. In the contextof our investigation the problem had been this: Is the concept of completepositivity sufficiently powerful to determine the new laws of motion forgeneral quantum thermodynamics? We were accordingly quite excited by theprospect seemingly offered by the aforementioned theorem of Gorini e t a l . ,which we had thought of as a derivation of the Bloch equations from theprinciple of complete positivity. Further consideration revealed problemswith this view, fully discussed in the Appendix of our paper, which containsthe mathematics we misinterpreted as establishing that Bloch's equationswere not completely positive. The conclusion we should have drawn fromthis analysis was simply that the principle of complete positivity is notsufficient to determine Bloch's equations. Therefore our claim thatcomparison of the work of Gorini e t a l . with Bloch's theory does notsupport".., the hypothesis that completely positive dynamical maps might beuseful in describing the quantal motion of subsystems..." remains valid in thecontext in which we stated it.

If the conditions required by complete positivity were necessary andsufficient to derive the Bloch equations, then complete positivity would offeran excellent foundation for a new, non-Hamiltonian dynamical postulate ofthe type that we and others ") are seeking. Unfortunately, the Blochequations emerge from complete positivity only when a special additionalcondition (m0 h = 0 ) is imposed. In fact it is known that completepositivity also includes ordinary Hamiltonian motion as a special case. Thus,for us, complete positivity is essentially a neutral criterion, favoring neitherentropy-conserving nor entropy-increasing forms of mechanics.Though we regret the misstatement about the Bloch equations and arepleased to clarify the matter, nevertheless our final conclusion is in essenceno different: Despite its formal beauty and apparent phenomenologicalutility, the completely positive map is not an adequate construct for theformulation of fundamental dynamical principles.


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Com ple te Pos i t iv lty in Gene ra l ized Quan tum D ynam ics : Rep ly to Ragg io and Pr lmas 439

REFERENCE S

I . R. F. Simmons, J r . and J . L. Park , F o u n d . P h y s . 11, 47 (1981).2 . V . Gor in i , A . Kossakow sk i , and E , C . G . Suda rshan , J . M a t h . P h y s . 17, 821 (1976).3 . G . A . Ragg io and H. Pr imas , F o u n d . P h y s . , th is issue , preceding paper .4 . Go N. H a tsopou los and E . P . Gyf topou los , F o u n d . P h y s . 6, 15, 127, 439 , 561 (1976).

Prit~ted n Belgium

completly positive maps in generalized quantum dynamics

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