On the philosophy of Cramer-Rao-Bhattacharya Inequalities in Quantum Statistics

On the philosoph y of Cram ´ er-Rao-Bhattac hary a Inequalities in Quan tum Statistics T o my r ever e d guru C. R. R ao fo r r eve aling the mystries of Chanc e K. R. P arthasara t h y Indian Statistical Institute Delhi Centre, 7, S. J. S. Sa nsan w al Marg, New Delhi - 110 016, India e-mail: krp@isid.ac.in Summary T o an y parametric family of states of a finite leve l quantum sys- tem w e asso ciate a space of Fisher maps and in tro duce the na tural notions of Cram ´ er-Rao-Bhatt a c harya tensor and Fisher infor ma t ion form. This leads us to an abstract Cram´ er-Rao-Bhatt a c harya low er b ound for the co v aria nce mat rix of an y finite num b er of un biased estimators of parameteric f unctions. A n um b er of illustrativ e examples is included. Modulo tec hnical assumptions of v arious kinds our metho ds can b e applied to infinite lev el quantum sy stems as w ell as pa r a metric families of classical probabilit y distributions o n Borel spaces. Key words : Finite lev el quan tum system, uncertain ty principle, generalized mea- suremen t, Cov ariance matrix of un biased estimators, F isher map, Fisher informa- tion form, Cram ´ er-Rao-Bha t tac harya tensor, Cram´ er-Rao- Bha ttac harya b ound. AMS Sub ject classification index: 81C20, 94A15. 1. Introdu ction The ev olutio n o f mo dern scien tific thought is strewn with sev eral examples ex- pressing the fo llo wing sen t iment: in an y effort to accomplish a task there can b e a certain limit to the efficiency of its p erformance. In the pr esen t context w e bring to attention t hree suc h famous examples whic h are based on the com bination of a deep conceptual approach and simple mathematical arg umen ts. Finally , w e shall fo cus on one of them, namely , limits to the efficiency of estimating an unkno wn parameter inv o lv ed in a family of states of a finite lev el quan tum system. Our first e xample is the celebrated uncertain t y pr inciple of Heisen b erg [8] in quan tum mec ha nics. F or an in teresting historical account of this great disco very in the philosoph y of science w e refer the reader to the essa y b y Jagdish Mehra 1 2 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics in [13]. If q and p denote the p osition and momen tum op erators of a quan tum mec hanical particle exe cuting motion o n the real line R t hey ob ey the comm utatio n relation q p − pq = i ℏ where ℏ = h/ 2 π , h b eing the Planc k’s c onstan t, and this implies the follo wing inequalit y . If ψ is the absolutely square inte grable w av e function describing the state of the system and V ar( X | ψ ) denotes the v ariance of the observ able in the state ψ then V ar( q | ψ )V ar ( p | ψ ) > ℏ 2 / 4 . (1.1) In particular, if the v aria nce of p in the state ψ is σ 2 then V ar( q | ψ ) > ℏ 2 / 4 σ 2 . In other w ords, this sets a limit to the a ccuracy with whic h the p osition q can b e measured in the state ψ . Such limits to accuracy ho ld for any ‘conjugate pair’ of observ ables in quantum theory . Our second example is t he famous Cram´ er-Rao inequalit y [4], [18] in the t heory of estimation of statistical parameters. F or an am using and insigh tful accoun t of the route by whic h this fundamen tal disco very w as made and ho w it came to b e recognized in the history of statistical science w e refer to [1 9]. Supp ose { p ( ω , θ ) } is a para metric f a mily of pro ba bilit y densit y functions with resp ect to a σ - finite measure in a Borel space (Ω , F ) , θ b eing a real para meter v arying in an op en in terv al ( a, b ) . Assume that the function I ( θ ) = Z Ω  ∂ ∂ θ log p ( ω , θ )  2 p ( ω , θ ) µ ( dω ) (1.2) is w ell-defined fo r all θ in ( a, b ) . On the basis of a sample p oint ω obtained from ex- p erimen t ev aluate a function T ( ω ) as an estimate of the pa rameter θ . The function T ( · ) on Ω is called an unbiase d estim a tor of θ if Z Ω T ( ω ) p ( ω , θ ) µ ( dω ) = θ ∀ θ ∈ ( a, b ) and, in suc h a case, its v ariance, denoted by V ( T | θ ) is defined b y V ( T | θ ) = Z ( T ( ω ) − θ ) 2 p ( ω , θ ) µ ( dω ) . Indeed, V ( T | θ ) is a measure of t he error in v o lved in estimating θ by T ( ω ) . The Cram ´ er-Rao inequality in its simplest form say s that V ( T | θ ) > I ( θ ) − 1 (1.3) where I ( θ ) is giv en by (1.2) and called the ‘Fisher information’ at θ . Th us (1 .3) sets a limit to the a ccuracy of estimating the unkno wn parameter θ from exp erimen ta l observ ation. It is a remark able fact tha t a sp ecial case of (1.3) implies the Heisen b erg un- certain ty principle (1.1) and m uc h more in emphasizing the profundity of Fisher information. Indeed, let ψ ∈ L 2 ( R ) b e a wa v e function so that k ψ k = 1 . By c hanging ψ t o a new w av e function e iαx ψ ( x ) , α ∈ R , if necessary , w e ma y assume, without loss of generality tha t the momen tum o p erator p satisfies the condition h ψ | p | ψ i = 0 and h ψ | q | ψ i = m, a real scalar. By Born’s in terpretat io n f = | ψ | 2 is K. R . P arth asarath y 3 the probability densit y function of the p osition observ able q in the state ψ . In tro- ducing the parametric family { f ( x − θ ) , θ ∈ R } of probability densities w e see that its Fisher information I ( θ ) is giv en b y I ( θ ) = Z R  f ′ ( x − θ ) f ( x − θ )  2 f ( x − θ ) dx = 4 Z R  Re ψ ′ ψ ( x )  2 | ψ ( x ) | 2 dx (1.4) and therefore indep enden t of θ . By Cram´ er-Rao inequality w e hav e V ar( q | ψ ) = Z ( x − m ) 2 f ( x ) d x > 1 I ( m ) . (1.5) On the other hand V ar( p | ψ ) = Z R x 2 | ( F ψ ) ( x ) | 2 dx where F is the unitar y F o urier tra nsform in L 2 ( R ) . Th us by (1.4 ) w e hav e V ar( p | ψ ) = h ψ | F † q 2 F | ψ i = k pψ k 2 = ℏ 2 Z | ψ ′ ( x ) | 2 dx = ℏ 2 Z     ψ ′ ψ ( x )     2 | ψ ( x ) | 2 dx > ℏ 2 Z      Re ψ ′ ψ  ( x )     2 | ψ ( x ) | 2 dx = ℏ 2 4 I ( m ) (1.6) whic h to gether with (1.5) implies (1.1). The more p o werful inequality (1.6 ) and its natural generalization for cov a riance matrices in L 2 ( R n ) are kno wn together as Stam’s uncertain t y principle. F or more inf o rmation along these lines and a ric h surv ey of info r ma t ion inequalities w e refer t o the pap er [5 ] b y A. Dem b o , T. M. Co ve r and J. A. Thomas. Our last illustrious example is of a differen t genre but again connected with the notion of information. It is Sha nno n’s noisy co ding theorem [20] whic h sets a limit to the a bilit y of communic ation through an information c hannel in the presence of noise. Again w e presen t the simplest v ersion of this strikingly b eautiful result in order to highlight the philosophical aspect and refer to [17] for more g eneral v ersions. Conside r an information ch annel whose input and o utput alphab ets a r e same and equal to the binary alphab et { 0 , 1 } whic h is also a field of tw o elemen ts with the op erations of additio n and m ultiplication modulo 2. If a n input letter x from this alphab et is t ransmitted through the c hannel assume that the output 4 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics letter is x or x + 1 with probabilit y 1 − p or p so that the probabilit y of error due to noise in tra nsmission is p. Suc h a c hannel is said to b e binary and symmetric. Assume that the transmission o f a sequence x 1 , x 2 , . . . , x n of n letters through this binary symmetric c hannel yields the output seque nce y 1 , y 2 , . . . , y n where y 1 − x 1 , y 2 − x 2 , . . . , y n − x n are indep endently and iden tically distributed Bernoulli random v ariables, each assum ing the v alues 0 and 1 with probabilit y q = 1 − p and p resp ectiv ely . Such a channel is called a memoryless binary symmetric c hannel. Denote the alphab et b y F 2 . By a c o de of size m, l e ngth n and err or pr ob ability not exceeding ε, where 0 < ε < 1 , we mean m pairs ( u j , E j ) , 1 6 j 6 m, u j ∈ F n 2 , E j ⊂ F n 2 , E 1 , E 2 , . . . , E m are pairwise disjoin t, satisfying the inequalities P (o utput sequence ∈ E j | input sequence = u j ) > 1 − ε ∀ j. Denote b y N ( n, p, ε ) the maximum p ossible size for co des of length n with error probabilit y not excee ding ε. Then lim n →∞ 1 n log 2 N ( n, p, ε ) = 1 + p log 2 p + q log 2 q ∀ 0 < ε < 1 , 0 < p < 1 / 2 . (1.7) If w e write H ( p ) = − p log 2 p − q log 2 q and call it the Shannon en tropy of the Bernoulli random v ariable with probability of success (error) p then (1.7) has the in terpretation that for large n, a mong the 2 n p ossible input sequences of length n roughly 2 n (1 − H ( p )) sequence s could b e tra nsmitted with error probability < ε and not more. F or this reason t he expression on the righ t hand side of (1.7 ) is called the Shannon capacit y of the binary symme tric c ha nnel w ith error proba bility p. A corresp onding generalization for memoryless and stationa r y quan tum c hannels describing their ‘capacity ’ to transmit classical alphab etic messages exists. F or a leisurely and self-contained exp o sition of suc h co ding theorems see [17]. The notio n of en trop y that arises in the brief discuss ion ab o v e can b e in tro duced for a la rge class of densit y functions and this, in turn, leads to some remark able connections with Fisher information and man y p o werful inf o rmation theoretic inequalities. Once again we refer t o the ve ry ric h surv ey article [5]. All the three examples describ ed ab o v e ha ve b een generalized in sev eral w ays , connections b et w een them and relations with other branche s of scienc e and en- gineering ha v e emerged a nd an enormous amount of literature has grow n around them. The last example has giv en birth to the sub j ect of quan tum information theory and co ding theorems for quan tum c hannels [14], [17 ]. The presen t essa y is dev oted to the second example but in the con t ext of parametric families of states of finite lev el quantum systems. Starting fro m the b o oks of Helstr¨ om [9], Holev o [10], and Ha yashi [7] there is quite some literature on the Cram ´ er Ra o b ounds for quan tum systems . By confining our selv es to finite lev el systems w e a void the t ech- nical difficulties of dealing with unbounded op erators and their v arying domains but we gain conceptual and a lg ebraic clarity . In Section 2 we give a brief accoun t of the quan tum probabilit y of finite level quan tum systems in a complex finite dimensional Hilb ert space including the no- tions of eve n ts, observ ables, states, generalized measuremen ts and comp osite sys- tems in the languag e of tensor pro ducts of Hilb ert spaces . Heisen b erg’s uncertain t y principle and an entropic uncertain t y principle are briefly describ ed. The notions K. R . P arth asarath y 5 of pa r a metric f a milies of states and un bia sed estimators of parametric functions along with their v ariances and cov ariances are in tro duced. Section 3 con tains the k ey notions, namely , Fisher maps, the Fisher information form and the Cram´ er-Rao-Bhatta c harya (CRB) tensor with respect t o a pa r a metric family of states o f a finite lev el quan tum system. The Cram´ er-Rao-Bhatta c harya (CRB) b ound is finally exp ressed in terms of the CRB tensor and the Fisher information fo rm. Sev eral illustrative examples ar e giv en. In the last section w e show how, b y using a dilation theorem of Naimar k, one can obtain a CRB b ound for the cov ariance matr ix of unb iased estimators of parametric functions based on generalized measuremen ts. 2. Preliminaries in the quantum probability and st a tistics of finite level sys t ems A finite lev el quan tum system is describ ed b y ‘states’ in a finite dimensional complex Hilb ert space. W e ch o ose and fix suc h a Hilb ert space H with scalar pro duct h u | v i whic h is linear in the v ariable v and an tilinear in u. A ty pical example obtains when H is t he n -dimensional complex vector space C n of column ve ctors and its dual is t he space of a ll ro w v ectors. In this case the scalar pro duct is expresse d as h u | v i = n X i =1 ¯ a i b i where u =     a 1 a 2 . . . a n     , v =     b 1 b 2 . . . b n     , a i , b i ∈ C ∀ i. Elemen ts of H are called ket v ectors, a t ypical elemen t in H b eing denoted b y | v i whereas any elemen t in the dual of H is called a br a v ector and a typic al bra vec tor is denoted b y h u | . The linear functional h u | ev alua t ed at a ket v ector | v i is the scalar pro duct h u | v i . If A is an op erator in H it is customary to write h u | Av i = h u | A | v i . The adjoint of A is denoted b y A † so that h u | A | v i = h A † u | v i = h u | Av i . In suc h a notation | u ih v | denotes the op erator satisfying ( | u ih v | ) | w i = h v | w i | u i ∀ | u i , | v i , | w i in H . The trace of an op erator A in H is denoted by T r A. In pa r ticularT r | u ih v | = h v | u i . Note that | u ih v | is a rank one op erator when | u i 6 = 0 , | v i 6 = 0 , and ( | u 1 ih v 1 | ) ( | u 2 ih v 2 | ) · · · ( | u k ih v k | ) = c | u 1 ih v k | where c = h v 1 | u 2 ih v 2 | u 3 i · · · h v k − 1 | u k i . Denote b y B ( H ) , P ( H ) , O ( H ) , S ( H ) respectiv ely the ∗ - algebra of all op erators on H with its usual (strong) top olog y , the orthomo dular lattice of all orthogona l 6 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics pro jection op erators on H , the real linear space of all hermitian op erators in H and the compact con v ex set of all nonnegativ e definite op erators of unit trace. W e ha ve P ( H ) ⊂ O ( H ) ⊂ B ( H ) and S ( H ) ⊂ O ( H ) ⊂ B ( H ) . If A, B ∈ O ( H ) w e say that A 6 B if B − A is nonnegative definite. Then O ( H ) is a partia lly ordered real linear space. A nonnegative definite hermitian op erator is simply called a p ositive op erator. The zero and iden tity op erators a re denoted resp ectiv ely by O amd I . Often, I is denoted b y 1 . F o r an y scalar λ the op erat o r λI is also denoted by λ. Thus , for A ∈ B ( H ) , λ ∈ C , A − λ stands for the op era t o r A − λI . F or any E ∈ P ( H ) , 0 6 E 6 1 and (1 − E ) ∈ P ( H ) . By a pro jection w e shall alwa ys mean an orthogonal pro jection operat o r i.e., an elemen t of P ( H ) . If E 1 , E 2 ∈ P ( H ) and E 1 6 E 2 then ( E 2 − E 1 ) ∈ P ( H ) . When a q uan tum system is described b y H w e sa y that the elemen t s of P ( H ) a re the events concerning the system, 0 is the nul l ev ent and 1 is the c ertain ev ent. If E 1 , E 2 ∈ P ( H ) and E 1 6 E 2 w e sa y that the ev en t E 1 implies the ev en t E 2 . If E ∈ P ( H ) then 1 − E is the ev ent ‘not E ’. If E 1 , E 2 ∈ P ( H ) t heir maximum E 1 ∨ E 2 and minim um E 1 ∧ E 2 are resp ective ly in terpreted as ‘ E 1 or E 2 ’ a nd ‘ E 1 and E 2 ’. If E 1 E 2 = 0 then E 1 ∨ E 2 = E 1 + E 2 . If E 1 and E 2 comm ute then E 1 ∧ E 2 = E 1 E 2 . The first basic difference b et w een classical probabilit y and quan tum probabilit y theory arises from the fact that for three ev en ts E i in P ( H ) , i = 1 , 2 , 3 one may not hav e E 1 ∧ ( E 2 ∨ E 3 ) = ( E 1 ∧ E 2 ) ∨ ( E 1 ∧ E 3 ) . Whenev er the E i ’s comm ute with eac h other the op erations ∧ and ∨ distribute with eac h other. An y hermitian op earator X in H is called a real-v alued or simply an o b s ervable ab out the system describ ed b y H . Th us O ( H ) is the real linear space of all real- v alued observ ables. If X , Y ∈ O ( H ) and X Y = Y X the n X Y is also an elemen t of O ( H ) . If X ∈ O ( H ) and σ ( X ) denotes the set o f all its eigenv alues then, by the sp ectral theorem, X admits a unique spectral resolution or represen tatio n X = X λ ∈ σ ( X ) λ E λ (2.1) where σ ( X ) ⊂ R is a finite set of cardinality not exceeding the dimension of H , 0 6 = E λ ∈ P ( H ) ∀ λ ∈ σ ( X ) a nd X λ ∈ σ ( X ) E λ = I , (2.2) E λ E λ ′ = δ λλ ′ E λ ∀ λ, λ ′ ∈ σ ( X ) . (2.3) This, at once, suggests the in t erpretat io n that the eigenpro jection E λ asso ciated with t he eigen v alue λ in (2.1) is the ev en t that the observ able X tak es t he v alue λ and σ ( X ) is the set of all v alues that X can take. If ϕ : σ ( X ) → R o r C is a real or complex-v a lued function then ϕ ( X ) = X λ ∈ σ ( X ) ϕ ( λ ) E λ (2.4) is the real or complex-v alued observ able whic h is the function ϕ of X. K. R . P arth asarath y 7 An y ele men t ρ ∈ S ( H ) is called a state of the quan tum sy stem described b y X . Suc h a state ρ is also called a density o p er ator. Clearly , ρ itself b ecomes an observ able. If E ∈ P ( H ) is an ev en t and ρ is a state then T r ρE is a quan tity in the unit in terv al [0 , 1] called the pr ob ability of the ev ent E in the state ρ. If E 1 , E 2 are t w o ev ents satisfying the relation E 1 E 2 = 0 then E 1 + E 2 is also a n ev ent and T r ρ ( E 1 + E 2 ) = T r ρE 1 + T r ρE 2 . Ho we v er, for tw o ev ents, E 1 , E 2 it is not necessary tha t T r ρ ( E 1 ∨ E 2 ) 6 T r ρE 1 + T r ρE 2 . In short, subadditivit y prop erty for probabilit y need not hold go o d. But this prop erty is retained whenev er E 1 and E 2 comm ute with each other. If ρ is a state and X is a n elemen t of O ( H ) with sp ectral resolution (2.1 ) t hen T r ρE λ is the probabilit y that X ta k es the v alue λ in the state ρ whenev er λ ∈ σ ( X ) . Th us the exp e ctation of X in the state ρ is giv en by X λ ∈ σ ( X ) λ T r ρE λ = T r ρ X λ ∈ σ ( X ) λE λ = T r ρX . More generally , the exp ectatio n of ϕ ( X ) defined by (2.4) is give n by T r ρϕ ( X ) . In particular, the varianc e of X in the state ρ, denoted b y V ar( X | ρ ) is giv en by V ar( X | ρ ) = T r ρX 2 − (T r ρX ) 2 = T r ρ ( X − m ) 2 where m = T r ρX is the exp ectation or me an of X in the state ρ. This sho ws, in particular, that V ar ( X | ρ ) v anishes if and only if the r estriction of X to the range of ρ is a scalar m ult iple of the iden tit y . The sp ectral theorem implies that t he extreme p oin ts of t he conv ex set S ( H ) are one dimensional pro jections of the fo rm | ψ ih ψ | where | ψ i is a unit vec tor in H . Here, the pr o j ection remains unaltered if | ψ i is replaced b y c | ψ i where c is a scalar of mo dulus unity . Extreme p oints of S ( H ) are called pur e states and a pure state is a one dimensional pro jection whic h, in turn, is determined b y a unit v ector in H mo dulo a scalar of mo dulus unity . By abuse of language an y determining unit vec tor itself is called a pure stat e. T h us whenev er w e say that a unit v ector | ψ i is a pure state w e mean the densit y o p erator | ψ ih ψ | . By spectral theorem an y state ρ can b e expres sed as P j p j | ψ j ih ψ j | w here p 1 , p 2 , . . . is a finite probabilit y distribution and {| ψ j i , j = 1 , 2 , . . . } is an orthono rmal set o f v ectors in H . If {| ψ j i} is an y set of unit v ectors and p j , j = 1 , 2 , . . . is a probabilit y distribution then P j p j | ψ j ih ψ j | is a state. If | ψ i is a pure state and X is a real-v a lued observ a ble then its v ariance V ar( X || ψ i ) in the pure state | ψ i is zero if and only if | ψ i is an eigen v ector for X . Thus, ev en in a pure state | ψ i , a n o bserv able need not hav e a degenerate distribution. This is a significant departure from classical probability . Hereafter, unless ot herwise explicitly men tioned, w e shall mean by an observ able a real-v alued o bserv able. Let X, Y b e tw o observ ables, ρ a state and let m = T r ρX , m ′ = T r ρY their resp ectiv e means. P ut e X = X − m, e Y = Y − m ′ and consider the nonnegative function f ( z ) = T r ρ ( e X + z e Y ) † ( e X + z e Y ) , z ∈ C . 8 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics Then the inequalit y inf z ∈ C f ( z ) > 0 implies (see [6], [16]) V ar( X | ρ )V ar( Y | ρ ) >  T r ρ 1 2 i ( e X e Y − e Y e X )  2 +  T r ρ 1 2 ( e X e Y − e Y e X )  2 (2.5) and th us puts a lo w er b ound on the pro duct of the v ariances of X and Y in a state ρ. The quan tum pro ba bilit y of finite level systems w e ha ve described here has a nat ur a l generalization when H is an infinite dimensional Hilb ert space. When H = L 2 ( R ) , X = q , Y = p are the w ell-kno wn p osition and momen tum o p erators satisfying the Heisen b erg commu tation relations q p − pq = i ℏ the inequalit y (2.5) yields the sp ecial form V ar( q   | ψ i )V ar( p   | ψ i ) > ℏ 2 4 ∀ | ψ i ∈ D where D is a dense domain in H where un b o unded o p erators like q p, pq etc. are w ell-defined. Th us (2.5) is at the heart of the Heisen b erg’s principle of uncertaint y . No w w e intro duce a notion which is more general than that of a n observ able. Indeed, it pla ys an imp ortan t r ole in the quantum vers ion of Shannon’s co ding theorems o f classical info rmation theory . Definition 2.1. A gener alize d m e a s ur emen t L of a finite level quantum s ystem with Hilb ert sp ac e H is a p air ( S, L ) wher e S is a finite set and L : S → B ( H ) is a map satisfying the c ondition: X s ∈ S L ( s ) † L ( s ) = I . (2.6) Suc h a generalized measuremen t L = ( S, L ) has the f ollo wing in terpretation. If the system is in the state ρ and the measuremen t L is p erformed then the ‘v alue’ s ∈ S is obtained with probability T r L ( s ) ρL ( s ) † and the system ‘collapses’ to a new state L ( s ) ρL ( s ) † T r L ( s ) ρL ( s ) † . (2.7) If, for example, the system is initially in the state ρ, a generalized measuremen t L 1 = ( S 1 , L 1 ) is p erformed and follow ed by another generalized measuremen t L 2 = ( S 2 , L 2 ) then the pro babilit y of obtaining the v alue s 1 ∈ S 1 is T r L 1 ( s 1 ) ρL 1 ( s 1 ) † and the conditional probabilit y of getting the v alue s 2 ∈ S 2 from L 2 giv en the v alue s 1 from L 1 is T r L 2 ( s 2 )  L 1 ( s 1 ) ρL 1 ( s 1 ) † T r L 1 ( s 1 ) ρL 1 ( s 1 ) †  L 2 ( s 2 ) † . Th us the probability of obtaining the v alue ( s 1 , s 2 ) fr o m L 1 follo w ed by L 2 is equal to p ( s 1 , s 2 ) = T r L 2 ( s 2 ) L 1 ( s 1 ) ρ L 1 ( s 1 ) † L 2 ( s 2 ) † K. R . P arth asarath y 9 and the final collapsed state is L 2 ( s 2 ) L 1 ( s 1 ) ρL 1 ( s 1 ) † L 2 ( s 2 ) † p ( s 1 , s 2 ) . More generally , if the measuremen ts L i = ( S i , L i ) , i = 1 , 2 , . . . , m are p erformed in success io n on a quan tum system with initial state ρ then the probabilit y p ( s 1 , s 2 , . . . , s m ) of getting the sequence s 1 , s 2 , . . . , s m of v alues s j ∈ S j ∀ j is giv en b y p ( s 1 , s 2 , . . . , s m ) = T r L m ( s m ) L m − 1 ( s m − 1 ) · · · L 1 ( s 1 ) ρL 1 ( s 1 ) † L 2 ( s 2 ) † · · · L m ( s m ) † and the final collapsed state is 1 p ( s 1 , s 2 , . . . , s m ) L m ( s m ) L m − 1 ( s m − 1 ) . . . L 1 ( s 1 ) ρL 1 ( s 1 ) † L 2 ( s 2 ) † . . . L m ( s m ) † . This at o nce suggests the pro duct rule for measuremen ts L i = ( S i , L i ) i = 1 , 2 as L = ( S 1 × S 2 , e L ) where e L ( s 1 , s 2 ) = L 2 ( s 2 ) L 1 ( s 1 ) , s 1 ∈ S 1 , s 2 ∈ S 2 . The measuremen t L stands for the measuremen t L 1 follo w ed b y the measuremen t L 2 . If L = ( S, L ) is a measuremen t with S ⊂ R o r C then its exp e ctation in the stat e ρ is giv en b y X s ∈ S s T r L ( s ) ρL ( s ) † = X s ∈ S s T r ρL ( s ) † L ( s ) . If S ⊂ R its v ariance V ar( L| ρ ) in the state ρ is giv en b y X s ∈ S s 2 T r ρ L ( s ) † L ( s ) − X s ∈ s s T r ρL ( s ) † L ( s ) ! 2 . When L ( s ) is a pro jection for ev ery s ∈ S then ( S, L ) is called a p r oje ctive or von Neumann me asur ement . If, in addition, S ⊂ R then the hermitian op erator P s ∈ S s L ( s ) is an observ able and our notion of generalized measuremen t reduces to measuring an observ able. It ma y b e of some in terest to form ulate and obtain an uncertain ty principle for a pair of t w o real-v a lued measuremen ts. F or a measuremen t with v alues in an abstract set S it is natur a l to r eplace the notion of v a r iance by its en t r o p y in a state ρ. Th us w e consider the quantit y H ( L| ρ ) = − X s ∈ S p ( s ) log 2 p ( s ) (2.8) where p ( s ) = T r ρ L ( s ) † L ( s ) and call it the entr opy o f the measuremen t L = ( S, L ) in the state ρ. With this definition one has the follo wing entropic uncertain t y principle. 10 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics Theorem 2.2 ([1 1], [1 2]) . L et L = ( S, L ) , M = ( T , M ) b e two gener alize d me a- sur eme n ts of a finite level quantum system in a Hilb ert sp ac e H . L et L ( s ) † L ( s ) = X ( s ) , M ( t ) † M ( t ) = Y ( t ) , s ∈ S, t ∈ T . Then for any state ρ the fol lo w ing holds: H ( L| ρ ) + H ( M| ρ ) > − 2 log 2 max s,t     X ( s ) 1 / 2 Y ( t ) 1 / 2     . (2.9) Remark It is imp ortant to not e that the righ t hand side in the inequalit y (2.9) is indep enden t of ρ. If X i , 1 6 i 6 k ar e k observ ables, ρ is a state in H and T r ρX i = m i define the scalar ν ij = 1 2 T r ρ { ( X i − m i )( X j − m j ) + ( X j − m j )( X i − m i ) } . (2.10) Then the real symmetric matrix ( ( ν ij )) of order k is called the c ov arianc e matrix of the observ ables X 1 , X 2 , . . . , X k in the state ρ and denoted by Co v  X 1 , X 2 , . . . , X k   ρ  . It is a p ositiv e semidefinite matrix and it is imp ortant to note the symmetrization in i, j in the righ t hand side o f (2.10). Without suc h a symmetrization ν ij could b e a complex scalar. Till now w e talk ed ab out a single quan tum system. Suppose w e ha ve a com- p osite quan tum system made out of sev eral simple system s A 1 , A 2 , . . . , A k with resp ectiv e Hilb ert spaces H A 1 , H A 2 , . . . , H A k . Then the Hilb ert space of the joint system A 1 A 2 . . . A k is the tensor pro duct H A 1 ...A k = H A 1 ⊗ H A 2 ⊗ · · · ⊗ H A k . This is the quan tum pr o babilistic analo g ue of cartesian pro duct of sample spaces in classical probability . It is clear that dim H A 1 ...A k = k Y i =1 dim H A i , dim indicating dimension. If ρ i is a state in H A i ∀ i then ρ 1 ⊗ · · · ⊗ ρ k is a state of the composite system A 1 A 2 . . . A k called the pr o duct state. If ρ is a s tate in H A 1 ...A k and we tak e its relativ e trace o ve r H A i 1 , H A i 2 , . . . , H A i ℓ then w e get the marginal s tate of the sy stem A r 1 , A r 2 . . . , A r m where { 1 , 2 , . . . , k } is the disjoint union { i 1 , i 2 , . . . , i ℓ } ∪ { r 1 , r 2 , . . . , r m } with ℓ + m = k . In this con text of comp o site quan tum systems there arises a new distinguishing feature of the sub ject with a remark able role in phys ics a s well as information theory . It is t he existence of a v ery ric h class of states in H A 1 A 2 ...A k whic h do not b elong to the conv ex hull o f all pro duct states. Suc h states are called entangle d states a nd they constitute a ric h resource in quan tum comm unication [14]. Till now w e restricted ourselv es to quan tum probabilit y . Now we describ e a few basic concepts in quan tum statistics dealing with a pa rametric fa mily of quan tum states o f a finite lev el system. Let Γ b e a p ar ameter sp ac e and let { ρ ( θ ) , θ ∈ Γ } b e a p ar ametric family of states in a Hilb ert space H . Supp ose X is an observ able, i.e., an elemen t of O ( H ) and T r ρ ( θ ) X = f ( θ ) , θ ∈ Γ , (2.11) K. R . P arth asarath y 11 where f is a real- v alued function on Γ . t hen we sa y t hat the o bserv able X is an unbiase d estimator of the p ar ametric function f on Γ . When the parametric family { ρ ( θ ) , θ ∈ Γ } is fixed w e write V ar( X | θ ) = V ar( X | ρ ( θ )) (2.12) If X 1 , X 2 , . . . , X m are m observ ables w e write Co v( X 1 , . . . , X m | θ ) = Cov( X 1 , . . . , X m | ρ ( θ )) . (2.13) A real-v alued f unction f on Γ is said to b e estimable with resp ect to { ρ ( θ ) , θ ∈ Γ } if there exists an observ able X ∈ O ( H ) suc h that T r ρ ( θ ) X = f ( θ ) ∀ θ ∈ Γ . Denote b y E (Γ) the real linear space o f all such estimable functions. An observ able X is said to b e b alanc e d with resp ect to the family { ρ ( θ ) , θ ∈ Γ } if T r ρ ( θ ) X = 0 ∀ θ ∈ Γ . Denote b y N the real linear space of all suc h balanced observ ables. F or an y f ∈ E (Γ) , an unbiase d estimator X of f write ν f ( θ ) = inf { V ar( X + Z | θ ) , Z ∈ N } . It is natural to lo ok for go o d lo wer b ounds for the function ν f ( θ ) . W e shall examine this problem in the nex t section and s tudy some examples. If f j , 1 6 j 6 m are estimable parametric functions we shall also lo ok for matrix lo wer b ounds for the p ositiv e semide finite matrices Co v( X 1 , . . . , X m | θ ) as eac h X i v aries ov er all un biased estimators of f i for each i = 1 , 2 , . . . , m. F or a more detailed in tro duction to quantum probabilit y theory we refer to [15], [16]. F or an initiation to estimation theory a nd testing h yp otheses in quan tum statistics w e refer to [7], [9], [10], References [7], [1 0 ], [14], [17 ] contain applications of the theory of generalized measuremen ts. 3. The Fishe r inform a tion form and the Cram ´ er-Rao-Bha tt achar y a tensor W e consider a fixed parametric family { ρ ( θ ) , θ ∈ Γ } o f states of a finite lev el quan tum system in a Hilb ert space H with parameter space Γ . As men tioned in the preceding section denote b y E (Γ) and N resp ectiv ely the r eal linear spaces o f estimable functions and balanced observ ables. Recall t ha t for a n y tw o unbias ed estimators X and Y of an elemen t f ∈ E (Γ) , the observ a ble X − Y is an elemen t of N . Definition 3.1. A ma p F : Γ → B ( H ) is c al le d a F isher map for the family { ρ ( θ ) , θ ∈ Γ } of states in H if the fol lowing two c onditions hold : (i) T r ρ ( θ ) F ( θ ) = 0 ∀ θ ∈ Γ , (ii) T r ρ ( θ )  F ( θ ) † X + X F ( θ )  = 0 ∀ θ ∈ Γ , X ∈ N . Denote by F the real linear space of all Fisher ma ps with respect to { ρ ( θ ) , θ ∈ Γ } and b y A (Γ ) the algebra of all r eal-v a lued functions on Γ . If a ∈ A (Γ) and F ∈ F 12 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics then aF define d b y ( aF )( θ ) = a ( θ ) F ( θ ) is also in F . In other w ords F is an A (Γ)-mo dule. F o r any t w o F isher maps F , G in F de fine I ( F , G )( θ ) = T r ρ ( θ ) 1 2  F ( θ ) † G ( θ ) + G ( θ ) † F ( θ )  = R e T r ρ ( θ ) F ( θ ) † G ( θ ) . (3.1) Then I is called the Fisher in f o rmation f o rm asso ciated with { ρ ( θ ) , θ ∈ Γ } . It ma y b e noted that, f or all F , F 1 , F 2 , G ∈ F and a ∈ A (Γ) , the follo wing hold: I ( F , G ) = I ( G, F ) , I ( aF , G ) = a I ( F , G ) , I ( F 1 + F 2 , G ) = I ( F 1 , G ) + I ( F 2 , G ) , I ( F , F ) > 0 . In particular, for an y F i , 1 6 i 6 n in F the matrix I n ( F 1 , F 2 , . . . , F n | θ ) = (( I ( F i , F j )( θ ))) , θ ∈ Γ , i, j ∈ { 1 , 2 , . . . , n } (3.2) is p o sitiv e semidefinite. It is called t he informa tion matrix a t θ corresp onding to the elemen ts F i , 1 6 i 6 n in F . If f ∈ E (Γ) , F ∈ F define λ ( f , F )( θ ) = T r ρ ( θ ) 1 2  F ( θ ) † X + X F ( θ )  , θ ∈ Γ (3.3) where X is an y unb iased estimator of f . Note that, in vie w of property (ii) in Definition 3.1 the r igh t hand side of (3.3) is indep enden t of the c hoice of the un biased estimator of f . Clearly , λ ( f , F ) is real linear in the v ariable f when F is fixed and A (Γ)-linear in the v ariable F when f is fixed. Th us λ ( · , · ) can b e view ed as an elemen t of E (Γ) ⊗ F . W e call λ ( · , · ) the C r am´ er-R ao-Bhattacharya tensor or simply the CRB-tensor a sso ciated with { ρ ( θ ) , θ ∈ Γ } . Let f i ∈ E (Γ ) , X i an unbiased estimator of f i for eac h 1 6 i 6 m a nd let F j , 1 6 j 6 n b e Fisher maps with resp ect to { ρ ( θ ) , θ ∈ Γ } . Define the m × m ma t r ix Λ mn ( θ ) = (( λ ij ( θ ))) , 1 6 i 6 m, 1 6 j 6 n, θ ∈ Γ (3.4) λ ij ( θ ) = λ ( f i , F j )( θ ) θ ∈ Γ , (3.5) λ b eing t he C RB-tensor. W e now introduce the family of p ositiv e semidefinite sesquiline ar forms indexed by θ ∈ Γ in the v ector space B ( H ) b y B θ ( X , Y ) = T r X † ρ ( θ ) Y , X , Y ∈ B ( H ) . (3.6) By prop ert y ( i) in Definition 3.1, equations (3.3) and (3 .5 ) w e ha v e λ ij ( θ ) = T r ρ ( θ ) 1 2  F j ( θ ) † ( X i − f i ( θ )) + ( X i − f i ( θ )) F j ( θ )  = Re B θ  X i − f i ( θ ) , F j ( θ ) †  . K. R . P arth asarath y 13 Multiplying b oth sides by the real scalars a i b j and adding o v er 1 6 i 6 m, 1 6 i 6 n, we obtain a ′ Λ mn ( θ ) b = Re B θ m X i =1 a i ( X i − f i ( θ )) , n X j =1 b j F j ( θ ) † ! (3.7) where Λ mn and B θ are given b y (3.4), (3.5) and (3.6) and a , b a re resp ectiv ely col- umn v ectors of length m, n with prime ′ indicating transp ose. No w an application of the Cauc h y- Sch w arz inequalit y to the righ t hand side of (3.7) implies ( a ′ Λ mn ( θ ) b ) 2 6 B θ n X i =1 a i ( X i − f i ( θ )) , m X i =1 a i ( X i − f i ( θ )) ! × B θ n X j =1 b j F j ( θ ) † , n X j =1 b j F j ( θ ) † ! = { a ′ Co v ( X 1 , X 2 , . . . , X m | θ ) a } { b ′ I n ( F 1 , F 2 , . . . , F n | θ ) b } . Dividing b oth sides of this inequalit y b y b ′ I n ( F 1 , F 2 , . . . , F n | θ ) b , fixing a and maximizing the left hand side ov er all b satisfying I n ( F 1 , F 2 , . . . , F n | θ ) b 6 = 0 w e obtain the matrix inequality : Λ mn ( θ ) I − n ( F 1 , F 2 , . . . , F n | θ )Λ mn ( θ ) ′ 6 Cov( X 1 , X 2 , . . . X m | θ ) , I − n denoting the generalized in vers e of I n ( F 1 , F 2 , . . . , F n | θ ) . In ot her w ords we ha v e pro ved the follo wing theorem Theorem 3.1 (Quantum Cram´ er-Rao-Bhatta chary a (CRB) inequ alit y) . L et { ρ ( θ ) , θ ∈ Γ } b e a p ar ametric family of states of a finite level quantum system in a Hilb ert sp a c e H , f i , 1 6 i 6 m estimable functions on Γ , X i an unbiase d estimator of f i for e ac h i and let F j , 1 6 j 6 n b e Fis h er maps with r e sp e ct t o { ρ ( θ ) , θ ∈ Γ } . The n the fol lowing ma trix ine q uali ty holds: Co v ( X 1 , X 2 , . . . , X m | θ ) > Λ mn ( θ ) I − n ( F 1 , F 2 , . . . , F n | θ ) Λ mn ( θ ) ′ ∀ θ ∈ Γ wher e Λ mn ( θ ) is the m × n matrix define d by (3.3) - (3.5) a n d I − n ( F 1 , F 2 , . . . , F n | θ ) is the g ener alize d i n verse of the Fisher information matrix I n ( F 1 , F 2 , . . . , F n | θ ) asso ciate d with F 1 , F 2 , . . . , F n . Pr o of. Immediate. Corollary 3.1. L et X i , 1 6 i 6 m, F j , 1 6 j 6 n b e as in Th e o r em 3. 1 . Then Λ mn ( θ ) I − n ( F 1 , F 2 , . . . , F n | θ )Λ mn ( θ ) ′ > Λ mn − 1 ( θ ) I − n − 1 ( F 1 , F 2 , . . . , F n − 1 | θ )Λ mn − 1 ( θ ) ′ , θ ∈ Γ for n > 2 . 14 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics Pr o of. This is immediate fro m the f act that b oth the sides of the inequalit y ab ov e are ar riv ed at by taking suprem um ov er certain sets in R n , the set for the left hand side b eing larger than the set for the right hand side. W e call the righ t hand side of the inequalit y in Theorem 3.1 the CRB low er b ound. Remark 1 Theorem 3 .1 and Corolla ry 3.1 imply the p ossibilit y o f improv ing the CRB low er b ound b y searc hing for a larger class of A (Γ ) -linearly indep enden t Fisher maps for a parametric f a mily of states. Remark 2 The CRB low er b ound in Theorem 3.1 has some natural in v ariance prop erties. If f i , 1 6 i 6 m are fixed a nd X i , F i ( θ ) , ρ ( θ ) are changed respectiv ely to U X i U † , U F i ( θ ) U † , U ρ ( θ ) U † b y a fixed unitary op erator U in H then the CRB lo wer b ound in Theorem 3.1 remains the same. If the Fisher maps F j are replaced b y G j ( θ ) = n X r =1 α j r ( θ ) F r ( θ ) , 1 6 j 6 n (3.8) where the matrix A ( θ ) = (( α r s ( θ ))) is in ve rtible for a ll θ then Λ mn ( θ ) I − n ( F 1 , . . . , F n | θ )Λ mn ( θ ) ′ = ˜ Λ mn ( θ ) I − n ( G 1 , . . . , G n | θ ) ˜ Λ mn ( θ ) ′ , the tilde o v er Λ mn on the righ t hand side indicating tha t G i ’s a re used in place of F i ’s. In other w ords the CRB b ound is in v ariant under A (Γ)- linear inv ertible transformations of the form (3.8). Example 3.1 Let Γ = ( a, b ) , H = C n and let ρ ( θ ) = diag ( p 1 ( θ ) , p 2 ( θ ) , . . . , p n ( θ )) , θ ∈ Γ b e states in C n with respect to the standard orthonormal basis, diag denoting diagonal matrix. An estimable function f on Γ has the form f ( θ ) = n X i =1 a i p i ( θ ) where a i are real scalars. An un biased estimator X for f is X = diag( a 1 , a 2 , . . . , a n ) . Note that p i ( θ ) > 0 and P i p i ( θ ) = 1 ∀ θ ∈ Γ . Assume that p i ( θ ) are differen tiable in θ a nd p i ( θ ) > 0 ∀ i, θ . Then F ( θ ) = diag  p ′ 1 ( θ ) p 1 ( θ ) , p ′ 2 ( θ ) p 2 ( θ ) , . . . , p ′ n ( θ ) p n ( θ )  yields a Fisher map with I ( F , F )( θ ) = n X i =1 p ′ i ( θ ) 2 p i ( θ ) K. R . P arth asarath y 15 and λ ( f , F ) = n X i =1 a i p ′ i ( θ ) = f ′ ( θ ) . Theorem 3.1 for the single observ able X and single Fisher map yields V ar ( Y | θ ) >  n P i =1 a i p ′ i ( θ )  2 n P i =1 p ′ i ( θ ) 2 p i ( θ ) ∀ θ ∈ ( a, b ) and an y un biased estimator Y of f . This is the Cram´ er-Rao inequalit y for finite sample spaces in classical mathematical statistics. Example 3.2 (Quantum v ersion of Barankin’s example [1], [21]). Let ρ ( θ ) b e an in ve rtible densit y op erato r for ev ery θ in Γ . F or an y γ ∈ Γ define F γ ( θ ) = ρ ( γ ) ρ ( θ ) − 1 − 1 . Then F γ is a Fisher map and for an y estimable function f ∈ E (Γ) we ha v e λ ( f , F γ )( θ ) = f ( γ ) − f ( θ ) . The Fisher information f o rm I satisfies I ( F γ 1 , F γ 2 )( θ ) = Re T r ρ ( γ 1 ) ρ ( θ ) − 1 ρ ( γ 2 ) − 1 If X is an un biased estimate of f ∈ E (Γ) one obtains as a sp ecial case the CRB b ound V ar( X | θ ) > ( f ( γ 1 ) − f ( θ ) , f ( γ 2 ) − f ( θ ) , . . . , f ( γ n ) − f ( θ )) I − n ( γ 1 , γ 2 , . . . , γ n , θ ) ( f ( γ 1 ) − f ( θ ) , . . . , f ( γ n ) − f ( θ )) ′ where I − n ( γ 1 , γ 2 , . . . , γ n θ ) is the g eneralized inv erse o f the informa t io n matrix  Re T r ρ ( γ i ) ρ ( θ ) − 1 ρ ( γ j ) − 1  for any γ 1 , γ 2 , . . . , γ n ∈ Γ . Example 3.3 (Quan tum Bhattach ary a b ound [2]). Let Γ ⊆ R d b e a connected op en set and let ρ ( θ ) , θ ∈ Γ b e a family of in v ertible states suc h that the correspondence θ → ρ ( θ ) is C m -smo oth. then ev ery estimable function f is also C m -smo oth. F or an y linear differen tial op erator D on Γ with C m -co efficien ts satisfying D 1 = 0 define F D ( θ ) = ( D ρ )( θ ) ρ ( θ ) − 1 where D is applied to eve ry matrix entry of ρ ( · ) o n the righ t hand side in some fixed orthonormal basis. The n F D is a Fisher map and the CRB tensor λ satisfies λ ( f , F D )( θ ) = ( D f )( θ ) ∀ f ∈ E (Γ) . If D 1 , D 2 are t wo linear differential op erators in Γ with C m -co efficien ts a nnihilat ing the constant function 1 the Fisher information satisfies I ( F D 1 , F D 2 )( θ ) = Re T r ( D 1 ρ )( θ ) ρ ( θ ) − 1 ( D 2 ρ )( θ ) , θ ∈ Γ . 16 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics If X is an un biased estimate of f and D i , 1 6 i 6 n are C m -differen tial op erators on Γ then the CRB inequalit y has the form V ar( X | θ ) > ( D 1 f , . . . , D n f )( θ ) I − n ( D 1 , D 2 , . . . , D n | θ )( D 1 f , . . . , D n f )( θ ) ′ where I − n ( D 1 , D 2 , . . . , D n | θ ) is the generalized inv erse of the p o sitiv e semidefinite matrix  Re T r ( D i ρ )( θ ) ρ ( θ ) − 1 ( D j ρ )( θ )  , i, j ∈ { 1 , 2 , . . . , n } . Example 3.4 Example 3.2 leads us to the fo llowing natural abstraction. Supp o se Γ is a d -dimensional C m -manifold and θ → ρ ( θ ) is a C m -smo oth parametrization of states in H as θ v aries in Γ . If L is a smo oth vec tor field on Γ then F L ( θ ) = ( Lρ )( θ ) ρ ( θ ) − 1 , θ ∈ Γ is a C m -smo oth Fisher map with resp ect to { ρ ( θ ) , θ ∈ Γ } under the assumption that ρ ( θ ) − 1 exists for ev ery θ . C m -smo oth Fisher maps constitute a C m (Γ)-mo dule and E (Γ) ⊂ C m (Γ) . The CRB tensor λ and the Fisher information form I satisfy the relations λ ( f , F L )( θ ) = ( Lf )( θ ) I ( F L , F M )( θ ) = Re T r ( Lρ )( θ ) ρ ( θ ) − 1 ( M ρ )( θ ) for any tw o v ector fields L, M . As a sp ecial case of the CRB inequalit y w e hav e for an y un biased estimator X o f f ∈ E (Γ) , V ar( X | θ ) > ( Lf )( θ ) 2 T r ρ ( θ ) − 1 ( Lρ )( θ ) 2 , θ ∈ Γ for any C m -smo oth vector field L. As a sp ecial case of t he example a b ov e, consider a connected Lie group Γ with Lie alg ebra G . L et ρ ( g ) = U g ρ 0 U † g , g ∈ Γ where ρ 0 is a fixed inv ertible state. Any elemen t L of G is lo ok ed up on as a left in v ariant v ector field on Γ . let U exp t L = exp t π ( L ) , t ∈ R , L ∈ G where L → π ( L ) is a represen ta t ion of G in H . Then the CRB inequalit y yields V ar( X | g ) > (( Lf ) ( g )) 2 T r ρ − 1 0 [ π ( L ) , ρ 0 ] 2 ∀ L ∈ G (3.9) where X is an un biased estimator of f . If L i , 1 6 i 6 d is a ba sis fo r G and the nonnegativ e definite matrix I d is defined b y I d =  Re T r ρ − 1 0 [ π ( L i ) , ρ 0 ] [ π ( L j ) , ρ 0 ]  , i, j ∈ { 1 , 2 , . . . , d } then a maximization o ver all elemen ts L in G on the rig ht hand side of (3.9) yields V ar( X | g ) > ( L 1 f , L 2 f , . . . , L d f )( g ) I − d ( L 1 f , L 2 f , . . . , L d f )( g ) ′ , I − d b eing the generalized in v erse of I d . Example 3.5 (adapted fro m [3]) . W e now consider a n example in whic h the dif - feren t states ρ ( θ ) may fa il to hav e an inv erse, indeed, their ranges need not b e the K. R . P arth asarath y 17 same. Let Γ ⊆ R d b e an o p en domain and let ρ ( θ ) , θ ∈ Γ o b ey the set of linear partial differen tial equations of the form ∂ ρ ∂ θ j = 1 2  L j ( θ ) ρ ( θ ) + ρ ( θ ) L j ( θ ) †  , 1 6 j 6 d (3 .10) where the op erators L j ( θ ) ∈ B ( H ) . T aking tr ace on b oth sides w e see that Re T r ρ ( θ ) L j ( θ ) = 0 , 1 6 j 6 d, θ ∈ Γ . If I m T r ρ ( θ ) L j ( θ ) = m j ( θ ) we can replace in (3.10) L j ( θ ) by L j ( θ ) − im j ( θ ) without alt ering the differen tial equations. He nce w e may assume, without loss of generalit y , that in (3.10) T r ρ ( θ ) L j ( θ ) = 0 , 1 6 j 6 d, θ ∈ Γ . (3.11) W e then sa y that the states ρ ( θ ) whic h ob ey (3.10) and (3.11) constitute a Liapunov family. A sp ecial case of suc h a Lia puno v family of states is obtained when d = 1 and ρ ( θ ) = p ( θ ) e 1 2 θ L ρ 0 e 1 2 θ L † , θ ∈ R where L is a fixed op erator in H , ρ 0 is a fixed state and p ( θ ) = n T r ρ 0 e 1 2 θ L † e 1 2 θ L o − 1 . Then ρ ′ ( θ ) = 1 2 (  p ′ ( θ ) p ( θ ) + L  ρ ( θ ) + ρ ( θ )  p ′ ( θ ) p ( θ ) + L  † ) . If ρ 0 = | ψ 0 ih ψ 0 | is a pure state then ev ery ρ ( θ ) is a pure state. Th us rank ρ ( θ ) = 1 ∀ θ ∈ R and we hav e a situation where { ρ ( θ ) } admits a ‘score op erator f unction’ with a classical part p ′ /p and a quan tum part L. Going bac k to the Liapunov family satisfying (3 .10) and (3.11) w e observ e that eac h of the maps θ → L j ( θ ) , 1 6 j 6 d is a F isher map. Inde ed, if X is a balanced observ able w e ha ve 0 = ∂ ∂ θ j (T r ρ ( θ ) X ) = 1 2 T r  L j ( θ ) ρ ( θ ) X + ρ ( θ ) L j ( θ ) † X  = 1 2 T r ρ ( θ )  L j ( θ ) † X + X L j ( θ )  . F or an y estimable function f λ ( f , L j )( θ ) = ∂ f ∂ θ j and the Fisher infor mation form I satisfies I ( L i , L j )( θ ) = Re T r ρ ( θ ) L i ( θ ) † L j ( θ ) . If w e write I d ( θ ) = (( I ( L i , L j )( θ ))) , i, j ∈ { 1 , 2 , . . . , d } 18 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics then the CRB inequalit y assumes the form V ar( X | θ ) > ( ∇ f )( θ ) I − d ( θ )( ∇ f )( θ ) ′ for any un biased estimator X of f , ∇ f b eing the gradien t v ector  ∂ f ∂ θ , ∂ f ∂ θ 2 , . . . , ∂ f ∂ θ d  . In the sp ecial case d = 1 introduced in the course of the discussion ab o v e the CRB b ound assumes the form V ar( X | θ ) > ( f ′ ( θ )) 2 T r ρ ( θ )  p ′ ( θ ) p ( θ ) + L  †  p ′ ( θ ) p ( θ ) + L  . If ρ ( θ ) , σ ( θ ) , θ ∈ Γ a r e Liapunov families of states in Hilb ert spaces H , K resp ectiv ely with co efficien ts L j ( θ ) , M j ( θ ) in the resp ectiv e differen tia l equations corresp onding to (3.10) then the tensor pro duct states ρ ( θ ) ⊗ σ ( θ ) , θ ∈ Γ constitute again a Liapuno v family with the co efficie n ts L j ( θ ) ⊗ 1 + 1 ⊗ M j ( θ ) , 1 6 j 6 d in the differen tial equations corresp onding to (3.10) and its Fisher information form satisfies I ( L i ⊗ 1 + 1 ⊗ M i , L j ⊗ 1 + 1 ⊗ M j ) ( θ ) = I ( L i , L j )( θ ) + I ( M i , M j )( θ ) . Eexample 3.6 Our la st example is the case when ρ ( θ ) is a mixture of t he form ρ ( θ ) = N X r =1 p r ( θ ) ρ r ( θ ) where { p r ( θ ) , 1 6 r 6 N } is a family of probabilit y distributions on the finite set { 1 , 2 , . . . , N } indexed b y θ ∈ Γ and for eac h fixed r, { ρ r ( θ ) , θ ∈ Γ } is a Liapunov family of states o b eying t he differen tial equations ∂ ρ r ∂ θ j = 1 2  L r j ( θ ) ρ r ( θ ) + ρ r ( θ ) L r j ( θ ) †  , 1 6 j 6 d, θ ∈ Γ and the conditions T r ρ r ( θ ) L r j ( θ ) = 0 ∀ θ ∈ Γ . Let now f i , 1 6 i 6 m b e estimable functions with resp ect to { ρ ( θ ) , θ ∈ Γ } and let X i b e an y un biased estimator of f i for each i. Differen tiating with resp ect to θ j the iden tit y T r ρ ( θ ) ( X i − f i ( θ )) = 0 w e get ∂ f i ∂ θ j = N X r =1 p r ( θ ) Re T r M r j ( θ ) ρ r ( θ )( X i − f i ( θ )) (3.12) where M r j ( θ ) = p r ( θ ) − 1 ∂ p r ∂ θ j + L r j ( θ ) . (3.13) K. R . P arth asarath y 19 Multiplying b oth sides of (3.12) b y real scalars a i b j and adding ov er i and j w e get a ′  ∂ f i ∂ θ j  b = N X r =1 p r ( θ ) T r d X j =1 b j M r j ( θ ) ! ρ r ( θ ) m X i =1 a i ( X i − f i ( θ )) ! . Applying Cauc h y-Sc hw arz inequalit y t o each tr a ce scalar pro duct on the right hand side follow ed by the same inequalit y to the scalar pro duct with resp ect to the probabilit y distribution p 1 ( θ ) , p 2 ( θ ) , . . . , p N ( θ ) w e obtain  a ′  ∂ f i ∂ θ j  b  2 6 ( N X r =1 p r ( θ ) T r d X j =1 b j M r j ( θ ) ! ρ r ( θ ) d X j =1 b j M r j ( θ ) ! †    a ′ Co v( X 1 , . . . , X m | θ ) a (3.14) Let Ψ r ( θ ) =  Re T r ρ r ( θ ) M r i ( θ ) † M r j ( θ )  , i, j ∈ { 1 , 2 , . . . , d } , Ψ( θ ) = N X r =1 p r ( θ )Ψ r ( θ ) . Then the v alidity of (3.14 ) for all a i , b j , 1 6 i 6 m, 1 6 j 6 d implies Co v( X 1 , X 2 , . . . , X m | θ ) >  ∂ f i ∂ θ j  Ψ − ( θ )  ∂ f i ∂ θ j  ′ , the sup er index - in Ψ indicating its generalized in vers e. 4. Estima tors based on gene r alize d measurements As in Section 3 w e consider a parametric family { ρ ( θ ) , θ ∈ Γ } of states of a finite lev el quan tum system in a Hilb ert space H and a real-v alued parametric function f on Γ . In order to estimate f w e now lo ok at a generalized measuremen t L = ( S, L ) as described in Definition 2.1. Choose a real-v alued function ϕ on S and if the outcome of L is s then ev aluate ϕ ( s ) and treat it a s an estimate of f ( θ ) . W e say that ( L , ϕ ) is an unbiase d estimator of f if X s ∈ S ϕ ( s ) T r ρ ( θ ) L ( s ) † L ( s ) = f ( θ ) ∀ θ ∈ Γ . (4.1) Indeed, it ma y b e recalled fro m Section 2 that T r ρ ( θ ) L ( s ) † L ( s ) is t he probability of the outcome s if the unkno wn par a meter is θ . Then the v ariance o f ( L , ϕ ) is giv en b y V ar( L , ϕ | θ ) = X s ∈ S ϕ ( s ) 2 T r ρ ( θ ) L ( s ) † L ( s ) − f ( θ ) 2 . (4.2) If w e write X = X s ∈ S ϕ ( s ) L ( s ) † L ( s ) (4.3) 20 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics Then X is an observ able and (4.1) sho ws that X is an un biased estimator of f whenev er ( L , ϕ ) is an un biased estimator of f . How ev er, V ar( X | θ ) need not b e the same as V ar( L , ϕ | θ ) . In (4.1) put T ( s ) = L ( s ) † L ( s ) , s ∈ S. Then T ( s ) > 0 and b y Definition 2.1, P s ∈ S T ( s ) = I . In other words { T ( s ) , s ∈ S } is a p o sitiv e op erator-v a lued distribution on S with total op erator mass I . By a w ell-kno wn theorem of Naima r k [10], [16] one can im b ed the Hilb ert space H isometrically in a larger Hilbert space b H = H ⊗ K with dim K < ∞ and construct m utua lly o rthogonal pro jection op erators on b H with the blo c k op erator form E ( s ) =  T ( s ) M ( s ) M ( s ) † N ( s )  , s ∈ S (4.4) satisfying the following: (i) P s ∈ S E ( s ) = I , (ii)  E ( s )  u 0  , s ∈ S, u ∈ H  spans b H . Suc h a dilatio n of T ( · ) in H to E ( · ) in b H is unique upto a natural unitary isomor- phism. No w w e go bac k to the unbiased estimator ( L , ϕ ) of f des crib ed in (4.1). Put b ρ ( θ ) =  ρ ( θ ) 0 0 0  , b X = X s ∈ S ϕ ( s ) E ( s ) . Then { b ρ ( θ ) , θ ∈ Γ } is a pa rametric fa mily of states in b H , b X is an observ able in b H and equations (4.1) and (4.4) imply that T r b ρ ( θ ) b X = f ( θ ) . F urthermore V ar( b X | θ ) = T r b ρ ( θ )( b X − f ( θ )) 2 = X s ∈ S ϕ ( s ) 2 T r ρ ( θ ) T ( s ) − f ( θ ) 2 = V ar( L , ϕ | θ ) . Th us b X is an unbiased estimator of f w ith resp ect t o { b ρ ( θ ) , θ ∈ Γ } with the same v ariance as the un bia sed estimator ( L , ϕ ) based on generalized measuremen t for the orig ina l family of states. If F is a Fisher map fo r { ρ ( θ ) , θ ∈ Γ } then b F defined b y b F ( θ ) =  F ( θ ) 0 0 0  , θ ∈ Γ is a Fisher map f o r { b ρ ( θ ) , θ ∈ Γ } in b H . If b I is the Fisher information fo rm for { b ρ ( θ ) , θ ∈ Γ } we ha ve b I ( b F 1 , b F 2 )( θ ) = I ( F 1 , F 2 )( θ ) . Th us from Theorem 3.1 we conclude the following theorem. K. R . P arth asarath y 21 Theorem 4.1. L et { ρ ( θ ) , θ ∈ Γ } b e a p ar ametric fami l y of states of a finite l e v el quantum system in a Hilb ert sp ac e H and let ( L , ϕ ) b e any unbiase d estimator of a r e al-va l ue d p ar am etric function f b ase d on a gener alize d me asur ement L and a r e al sc a lar function ϕ on the set o f val ues of the me asur ement. Supp ose F j , 1 6 j 6 n ar e Fisher map s f o r { ρ ( θ ) , θ ∈ Γ } . Th en V ar (( L , ϕ ) | θ ) > ( λ ( f , F 1 ) , λ ( f , F 2 ) , . . . , λ ( f , F n )) I − n ( λ ( f , F 1 ) , λ ( f , F 2 ) , . . . , λ ( f , F n )) ′ ( θ ) wher e λ is the CRB tensor and I − n is the gener alize d inverse of the information matrix I n = (( I ( F i , F j ) )) , i, j ∈ { 1 , 2 , . . . , n } . Pr o of. Immediate. W e shall briefly consider the case of estimating man y parametric functions f i ( θ ) , 1 6 i 6 m. In order to es timate them it a pp ears that sev eral gene ralized mea- suremen ts are to be made. Such measuremen ts ha ve to b e made in succes sion. As directed in Section 2 w e may treat them all as a single comp ound generalized measuremen t L = ( L, S ) . Let ( L , ϕ i ) b e an unbiased estimator of f i for eac h i . Th us the measuremen t L is carried out a nd if the o utcome is s ∈ S then ϕ i ( s ) is the estimate of f i ( θ ) . The probabilit y for the outcome s is T r ρ ( θ ) L ( s ) † L ( s ) . Th us the cov ariance matrix of the differen t estimators is given b y Co v ( L , ϕ 1 , ϕ 2 , . . . , ϕ m | θ ) = T r ρ ( θ ) X s ∈ S ϕ i ( s ) ϕ j ( s ) L ( s ) † L ( s ) − f i ( θ ) f j ( θ ) !! , i.j ∈ { 1 , 2 , . . . , m } . (4.5) As in the discussion preceding Theorem 4.1 w e can construct the Naimark dilation { E ( s ) , s ∈ S } fo r the p ositiv e op erato r -v a lued distribution { L ( s ) † L ( s ) , s ∈ S } in an enlarged Hilb ert space and view the co v ariance mat r ix (4.5) a s Co v  b X 1 , b X 2 , . . . , b X m | θ  for the observ ables b X i , = P s ϕ i ( s ) E ( s ) with resp ect to the states b ρ ( θ ) . This at once leads us to the CRB matrix inequalit y Co v ( L , ϕ 1 , ϕ 2 , . . . , ϕ m | θ ) > (( λ ( f i , F j )))  I − n ( F p , F q )  (( λ ( f i , F j ))) ′ ( θ ) , 1 6 i 6 m ; j, p, q ∈ { 1 , 2 , . . . , n } . for any set { F j , 1 6 j 6 n } of Fisher maps, λ b eing the CRB tensor, (( I n ( F i , F j ))) the Fisher information matrix with r esp ect to { F j , 1 6 j 6 n } and the sup er index - denoting generalized inv erse. 22 On the philosophy of Cram´ er-Rao-Bhattac harya Inequalities i n Quan tum Statistics A ckno w l e dge m ent The author thanks H. P artha sara th y f or sev eral useful com- men ts and also p o inting out the references [13] and [21] in the ph ysics a nd engi- neering literature. He thanks B. V. Rao f o r bringing his atten t io n to the v ery ric h surv ey article [5]. Reference s [1] E. W. Ba rankin, L o c al ly b est u nbiase d estimators, Ann. Math. Stat., 20, 47 7-501 (194 9). [2] A. 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