OPERATOR PROBABILITY THEORY

OPERATOR PROBABILITY THEORY Sta Gudder Departmet of Mathematics Uiversity of Dever Dever, Colorado 80208 sta.gudder@sm.du.edu Abstract This article presets a overview of some topics i operator probability theory. We do ot strive for geerality ad oly simple methods are employed. To give the reader a flavor of the subject we cocetrate o the two most importat topics, the law of large umbers ad the cetral limit theorem. 1 Itroductio This article surveys various aspects of operator probability theory. This type of work has also bee called ocommutative probability or quatum probability theory. The mai applicatios are i quatum mechaics, statistical mechaics ad quatum field theory. Sice the framework deals with Hilbert space operators it is also of iterest to operator theorists. The article is directed toward mathematicias who are ot experts i this field but who wat to lear somethig about it. For this reaso we shall ot strive for great geerality ad shall cocetrate o the two importat topics, the law of large umbers ad the cetral limit theorem. A lot of work has bee devoted to probability theory o operator algebras such as C -algebras ad vo Neuma algebras 1, 2, 3, 5, 6, 7, 8, 10]. However, to avoid various techicalities we shall oly cosider the full algebra BH) of bouded operators o a Hilbert space H. Also, we shall ot discuss importat but techically difficult recet research such as ocommutative measure ad ergodic theory 2, 4, 5] ad free probability theory 1

2, 11]. Strictly speakig, the results we shall preset are ot ew. However, we thik that our methods are simpler ad preset a clearer picture. Sice some of our defiitios are ot stadard we preset some ew versios of kow results. 2 Notatio ad Defiitios Let H be a complex Hilbert space ad let BH) be the set of bouded liear operators o H. We deote the set of bouded self-adjoit operators o H by SH) ad the set of positive trace class operators with uit trace by DH). The trace of a trace class operator T is deoted by trt ). We thik of BH) as a set of ocommutative) complex-valued radom variables ad DH) as a set of states probability measures). If A SH) with spectral measure P A the P ρ A ) = tr ρp A ) ] is iterpreted as the probability that A has a value i the Borel set for the state ρ. It is the atural to call the real probability measure tr ρp A ) ] the distributio of A i the state ρ. It follows that the expectatio E ρ A) ofa i the state ρ is give by E ρ A) = λtr ρp A dλ) ] = trρa) A arbitrary A BH) has the uique represetatio A = A 1 + ia 2 for A 1,A 2 SH) give by A 1 =A + A )/2, A 2 =A A )/2i. We the write A 1 = ReA) ad A 2 =ImA). The A geerates two distributios P ρ A 1 ) ad P ρ A 2 ). It is atural to defie the ρ- expectatio of A by E ρ A) =E ρ A 1 )+ie ρ A 2 )=trρa 1 + ia 2 )] = trρa) We ca ow defie other probabilistic cocepts i the usual way. The ρ- momets of A are trρa ), =0, 1,..., ad the ρ-variace of A is Var ρ A) =E ρ A Eρ A)I) 2] = E ρ A 2 ) E ρ A) 2 Notice that Var ρ A) may be complex ad we have E ρ A ) = trρa )=trρa) ]=trρa) =E ρ A) 2

ad hece, Var ρ A )=Var ρ A). For A BH) we write A =A A) 1/2 SH). The ρ-absolute variace of A is Var ρ A) =E ρ A Eρ A)I 2] = E ρ A E ρ A)I) A E ρ A)I)] = E ρ A A + E ρ A) 2 I E ρ A)A E ρ A)A ] = E ρ A A) E ρ A) 2 = E ρ A 2 ) E ρ A) 2 Sice Var ρ A) 0 we coclude that E ρ A) 2 E ρ A 2 )or trρa) 2 tr ρ A 2) 2.1) Of course, if A SH), the Var ρ A) = Var ρ A). For x H with x = 1, we deote the oe-dimesioal projectio oto the spa of x by P x. Of course, P x DH) ad if ρ = P x we have E ρ A) = Ax, x ad Var ρ A) = A 2 x, x Ax, x 2 = Ax 2 Ax, x 2 Elemets of DH) that have the form P x are called pure states. It is well kow that ay state is a covex combiatio of pure states, that is, if ρ DH) the ρ = λ i P xi 2.2) where λ i > 0 with λ i = 1. I this case, x i is a eigevector of ρ with correspodig eigevalue λ i. Our first result shows that Var ρ A) = 0 oly uder rare circumstaces. Lemma 2.1. a) Var ρ A) =0for every ρ DH) if ad oly if A = λi for some λ C. b) Suppose ρ has the form 2.2). The Var ρ A) =0if ad oly if Ax i = E ρ A)x i, i =1, 2,... c) Suppose ρ = P x is a pure state. The Var ρ A) =0if ad oly if Ax = E ρ A)x. Proof. a) If A = λi the λ = E ρ A) so Var ρ A) = 0. Coversely, if Var ρ A) = 0 for all ρ DH) the A E ρ A)I = 0. It follows that A = E ρ A)I. b) If Ax i = E ρ A)x i, i =1, 2,..., the A Eρ A)I 2 x i,x i = A Eρ A)I) x i 2 =0 3

Hece, Var ρ A) =tr ρ A E ρ A)I 2] = λ i A Eρ A)I 2 x i,x i = 0 2.3) Coversely, if Var ρ A) = 0, the as i 2.3) we have λi A Eρ A)I 2 x i,x i =0 It follows that A E ρ A)I) x i 2 = A E ρ A) 2 x i x i =0 for i =1, 2,... Hece, Ax i = E ρ A)x i, i =1, 2,... c) This follows directly from b). If we write ρ i the form 2.2), the {x i } forms a orthoormal set ot ecessarily a basis) i H. The rage of ρ is Raρ) =spa {x i }. We say that ρ is faithful if trρa) = 0 for A 0 implies A =0. Lemma 2.2. The followig statemets are equivalet a) ρ is faithful. b) Raρ) =H. c) {x i } forms a orthoormal basis for H. d) ρ is ivertible. Proof. a) b) If Raρ) H, the there exists a x H with x =1 such that x x i, i =1, 2,... But the P x 0 ad P x 0, but trρp x )=0. Hece, ρ is ot faithful. b) c) If Raρ) =H the spa {x i } = H so that {x i } is a orthoormal basis. c) d) If {x i } forms a orthoormal basis, the all the eigevalues of ρ are positive. It follows that ρ is ivertible. d) a) Suppose ρ is ivertible. If A 0 ad trρa) = 0, the trρ 1/2 Aρ 1/2 ) = trρa) = 0. Sice ρ 1/2 Aρ 1/2 0, we coclude that ρ 1/2 Aρ 1/2 = 0. Sice ρ 1/2 is ivertible, it follows that A = 0. Hece, ρ is faithful. It follows from Lemmas 2.1 ad 2.2 that if ρ is faithful the Var ρ A) =0 if ad oly if A = E ρ A)I. 3 Types of Covergece Various types of covergece have bee discussed i the literature 2, 4, 8, 10] ad we shall ow cosider four of them. I the followig defiitios A 4

BH) will be a sequece of operators ad ρ DH) will be a fixed state. A sequece A coverges to A almost uiformly ρ] if for ay ε>0there exists a projectio P such that tr ρi P )] <εad lim A A)P = 0. Projectios are thought of as ocommutative) evets ad the trace coditio is equivalet to trρp ) > 1 ε. The defiitio says that for small ε the evet P is likely to occur ad A coverges to A uiformly o the rage of P. A sequece A coverges to A almost surely ρ] if lim E ρ A A 2) = lim tr ρ A A 2) =0 We frequetly write A A a.s. ρ] for almost sure covergece. We say that A coverges i ρ-mea to A if lim E ρa A) = lim tr ρa A)]=0 ad that A coverges i ρ-probability to A if for ay ε>0 lim P ρ A A ε) =0 I this defiitio, P ρ A A ε) is shorthad for P ρ A A ε, )). A fifth type of probabilistic covergece will be discussed i Sectio 5. We say that a sequece A coverges to A strogly o K H if lim A A)x = 0 for all x K. If K = H the we just say that A coverges strogly to A. We ow compare these various types of covergece. Lemma 3.1. If A A a.s. ρ] the A A i ρ-mea. Proof. Applyig Equatio 2.1) gives ad the result follows. E ρ A A) 2 E ρ A A 2) Theorem 3.2. a) A A a.s. ρ] if ad oly if A A strogly o Raρ). b) If ρ is faithful, the A A a.s. ρ] if ad oly if A A strogly. 5

Proof. a) Assumig ρ has the form 2.2) we have tr ρ A A 2) = λ j A A 2 x j,x j = λ j A A)x j 2 j j Hece, if A A a.s. ρ] the A x j Ax j for every j so A x Ax for all x i a dese subset of Raρ). It follows that A A strogly o Raρ). Coversely, suppose that A A strogly o Raρ). Let A,A : Raρ) H deote the restrictios of A ad A to Raρ). By the uiform boudedess theorem there is a M>0 such that A A 2 M for all. Let ε>0 be give. The there exists a iteger N such that i=n+1 λ i < ε 2M Moreover, there is a iteger K such that K implies Hece, K implies tr ρ A A 2) = N A A )x i 2 < ε 2 N λ i A A )x i 2 + < ε 2 + M i=n+1 λ i <ε i=n+1 Hece, A A a.s. ρ]. Part b) follows from Lemma 2.2. λ i A A )x i 2 It follows from Theorem 3.2 that if A A ad B B a.s. ρ] ad α, β C, the αca + βdb αca + βdb a.s. ρ]. The ext result is a versio of Egoroff s theorem. Theorem 3.3. If A A a.s. ρ], the A A almost uiformly ρ]. Proof. Sice A A a.s. ρ], A x Ax for every x Raρ). Let ρ have the form 2.2) ad let Q be the projectio oto spa {x 1,...,x }. The 6

Q = P xi ad Q = P xi is the projectio oto Raρ). The lettig Q = I Q we have tr ρi Q )] = tr ρq + Q Q ) ] =trq Q )] ) ) =tr ρ P xi =tr λ i P xi = i=+1 i=+1 λ i i=+1 Give ε>0 there exists a such that tr ρi Q )] <ε. Lettig P = Q we have that tr ρi P )] <ε. Let ε > 0. Now there exists a iteger N such that m N implies A m A)x i < ε, i =1,...,. The for ay x H with x 1wehave Px 1 ad Px = c i x i where ci 2 = Px 2 1. Hece, c i 1, i =1,...,. Thus, for m N we have A m A)Px = A m A) c i x i ci A m A)x i < ε ci ε Therefore, m N implies that A m A)P ε so A A almost uiformly ρ]. The coverse of Theorem 3.3 does ot hold. However, it is show i 5] that if A are uiformly bouded the A A uiformly ρ] implies that A A a.s. ρ]. 4 Law of Large Numbers We first preset some stadard probabilistic results. Lemma 4.1. Markov) If A 0 ad a>0, the P ρ A a) E ρa) a Proof. By the spectral theorem we have A = 0 λp A dλ) a λp A dλ) a 7 a P A dλ) =ap A a, ))

Takig expectatios gives E ρ A) ae ρ P A a, )) ] = ap ρ A a) Corollary4.2. If A A a.s. ρ], the A A is ρ-probability. Proof. By Lemma 4.1 ad Equatio 2.1) we have P ρ A A ε) E ρ A A ) ε The result ow follows 1 ε Eρ A A 2)] 1/2 Corollary4.3. Chebyshev) If A BH) with µ = E ρ A), σ = Var ρ A), the for ay k>0 we have P ρ A µi k) σ2 k 2 Proof. By Lemma 4.1 we have P ρ A µi 2 k 2) E ρ A µi 2 ) k 2 = σ2 k 2 But P ρ A µi k) =P ρ A µi 2 k 2). The followig results are called oe-sided Chebyshev iequalities. Corollary4.4. If A BH) is self-adjoit, µ = E ρ A), σ 2 =Var ρ A) ad a>0, the P ρ A µ + a) σ2 σ 2 + a, P 2 ρ A µ a) σ2 σ 2 + a 2 Proof. First assume that µ =0. Forb>0wehave P ρ A a) =P ρ A + bi a + b) =P ρ A + bi) 2 a + b) 2] Applyig Lemma 4.1 gives P ρ A a) E ρ A + bi) 2 ] a 2 + b 2 = σ2 + b 2 a + b) 2 8

Lettig b = σ 2 /a we obtai P ρ A a) σ2 4.1) σ 2 + a 2 Now suppose that µ 0. Sice A µi ad µi A have mea 0 ad variace σ 2, by 4.1) we have P ρ A µi a) The result ow follows σ2 σ 2 + a, P ρµi A a) σ2 2 σ 2 + a 2 For A, B BH) we defie the ρ-correlatio coefficiet by Cor ρ A, B) =E ρ A E ρ A)I) B E ρ B)I)] = E ρ A B) E ρ A )E ρ B) Notice that Cor ρ A, A) = Var ρ A). We say that A ad B are ucorrelated if Cor ρ A, B) = 0. Of course, A ad B are ucorrelated if ad oly if E ρ A B)=E ρ A )E ρ B). Also, A ad B are ucorrelated if ad oly if B ad A are ucorrelated. It is easy to check that Cor ρ A + ai,b + bi) = Cor ρ A, B) 4.2) for all a,b C. It follows from 4.2) that if A ad B are ρ-ucorrelated the A + ai ad B + bi are ρ-ucorrelated. It is clear that Var ρ λa) =λ 2 Var ρ A) ad that I geeral Var ρ λa) = λ 2 Var ρ A) Var ρ A + B) Var ρ A)+ Var ρ B) but additivity does hold i the ucorrelated case. Lemma 4.5. a) For A, B BH) we have Var ρ A + B) = Var ρ A)+ Var ρ B) + 2Re Cor ρ A, B) b) If A ad B are ucorrelated, the Var ρ A+B) = Var ρ A)+ Var ρ B). 9

Proof. a) The result follows from Var ρ A + B) =trρa + B) A + B)] tr ρa + B)] 2 = trρa A) + trρb B) + trρa B) + trρb A) trρa) + trρb)] trρa ) + trρb )] = Var ρ A)+ Var ρ B) + Cor ρ A, B)+Cor ρ A, B) = Var ρ A)+ Var ρ B) + 2Re Cor ρ A, B) b) this follows directly from a). The ext result gives the core of the proof of the law of large umbers. Theorem 4.6. Let A i BH) be mutually ρ-ucorrelated with a commo mea µ = E ρ A i ), i =1, 2,...IfS = 1 A i, the for ay λ C we have tr ρ S λi 2] = 1 2 Var ρ A i )+ µ λ 2 Proof. First assume that µ = 0. The tr ρ S λi 2] =tr ρ SS + λ 2 )] I λs λs =trρss )+ λ 2 λtrρs ) λ trρs ) = 1 trρa 2 i A i )+ 1 trρa 2 i A j )+ λ 2 = 1 trρa 2 i A i )+ 1 2 = 1 Var 2 ρ A i )+ λ 2 i j=1 i j=1 trρa i )trρa j )+ λ 2 If µ 0, the by Equatio 4.2), A i µi are mutually ucorrelated with mea 0 ad S = 1 A i µi) =S µi 10

Hece, by our precedig work tr ρ S λi 2] =tr ρ S +µ λ)i 2] = 1 Var 2 ρ A i µi)+ µ λ 2 = 1 Var 2 ρ A i )+ µ λ 2 The law of large umbers says that uder certai coditios, if A i are mutually ρ-ucorrelated with commo mea µ, the their average S = A i coverges to the mea operator µi. We ow preset several versios 1 of this law. Theorem 4.7. Strog law of large umbers) Let A i be mutually ρ-ucorrelated with commo mea µ = E ρ A i ), i = 1, 2,..., ad suppose there exist real umbers M > 0, 0 < r < 2 such that Var ρ A i ) M r for all. If S = 1 A i, the S λi a.s. ρ] if ad oly if λ = µ. Proof. Sice lim 1 2 applyig Theorem 4.6 gives Var ρ A i ) lim M r 2 =0 lim tr ρ S λi 2] = µ λ 2 Hece, lim E ρ S λi 2) = 0 if ad oly if λ = µ. A simple coditio that implies the variace property of Theorem 4.7 is the uiform boudedess coditio Var ρ A i ) M for all i. Applyig Corollary 4.2 ad Theorem 4.7 we obtai the followig result. Corollary4.8. Weak Law of large umbers) Uder the same assumptios as i Theorem 4.7 we have that S µi i ρ-probability. 11

We ca eve relax the commo mea coditio i Theorem 4.7 to obtai the followig stroger versio. Theorem 4.9. Let A i be mutually ρ-ucorrelated, let µ i = E ρ A i ), i = 1, 2,..., ad suppose there exist real umbers M > 0, 0 < r < 2 such that Var ρ A i ) M r.ifs = 1 A i the S 1 ) µ i I 0a.s. ρ] Proof. Let A i = A i µ i I ad S = 1 A i. The A i are mutually ρ- ucorrelated ad E ρ A i)=0,i =1, 2,... As i the proof of Theorem 4.6 we have tr ρ S 2) = 1 Var 2 ρ A i) M r 2 Hece, lim tr ρ S 2) = 0. But S = 1 A i µ i I)= 1 A i 1 µ i I = S 1 ) µ i I Sice S 0a.s. ρ], the result follows 5 Cetral Limit Theorem We say that A, B BH) are idepedet i the state ρ if trρa 1 B m1 A r B mr ) = trρa 1+ r )trρb m 1+ +m r ) for all 1,..., r,m 1,...,m r N. For example, if A BH 1 ), ρ 1 DH 1 ), B BH 2 ), ρ 2 DH 2 ) the A I BH 1 H 2 ) ad I B BH 1 H 2 ) are idepedet i the state ρ 1 ρ 2. I this case the operators commute ad it is ot surprisig that they are idepedet. However, there are simple examples of ocommutig idepedet operators. For example, suppose 12

Ax = αx, Bx = βx x = 1. The A ad B are idepedet i the pure state P x. Ideed, A 1 B m1 A r B r x, x = α 1+ + r β m 1+ +m r = A 1+ + r x, x B m 1+ +m r x, x The momet geeratig fuctio of A BH) relative to ρ DH) is the fuctio M ρ,a : R R give by M ρ,a t) =E ρ e ta ) the termiology comes from the fact that d dt M ρ,at) t=0 = E ρ A ) which is the -the momet. Lemma 5.1. If A ad B are idepedet i the state ρ, the M ρ,a+b = M ρ,a M ρ,b. Proof. Sice A ad B are idepedet we have Hece, E ρ A + B) ]= k=0 E ] ρ e ta+b) = E ρ I + ta + B)+ t ] 2 2! A + B)2 + ) E ρ A k )E ρ B k ) k =1+t E ρ A)+E ρ B)] + + t! k k=0 ] = 1+tE ρ A)+ + t! E ρa )+ ] 1+tE ρ B)+ + t! E ρb )+ = E ρ e ta )E ρ e tb ) We coclude that M ρ,a+b t) =M ρ,a t)m ρ,b t). ) E ρ A k )E ρ B k )+ 13

We say that a sequece {A i } is idepedet i the state ρ if A i+1 is idepedet of A 1 + +A i for all i N. Notice that if {A i } is idepedet, the by Lemma 5.1 we have M ρ,a1 + +A t) =E ρ e ta 1 + +A 1 ) ] = E ] ρ e ta 1 + +A 1 ) E ρ e ta ) = = E ρ e ta 1 )E ρ e ta 2 ) E ρ e ta ) = M ρ,a1 t) M ρ,a t) 5.1) We say that A ad B are ideticallydistributed i the state ρ if E ρ A )= E ρ B ) for all N. A sequece {A i } coverges i distributio relative to ρ if lim M ρ,a t) = M ρ t) for all t R, where M ρ t) is the momet geeratig fuctio of a classical radom variable. We do ot require that M ρ t) = M ρ,a t) for a A BH). It is well kow that the momet geeratig fuctio for the classical ormal distributio with zero mea ad variace oe is Mt) =e t2 /2. The cetral limit theorem says that if {A i } is a idepedet, idetically distributed sequece the suitably ormalized averages of the {A i } coverge i distributio to the ormal distributio. Theorem 5.2. Let {A i } be idepedet, idetically distributed i the state ρ with commo mea E ρ A i )=µ ad commo variace Var ρ A i )=σ 2.If ) Ai µi T = σ the lim M ρ,t t) =e t2 /2 for all t R. Proof. First assume that µ = 0 ad σ 2 = 1 ad let T = 1 Equatio 5.1) we have M ρ,t t) = M ρ,a1 t/ ) ] Let Lt) =lm ρ,a1 t). Note that L0) = 0 ad L 0) = M ρ,a 1 0) M ρ,a1 0) = µ =0 A i. The by 5.2) L 0) = M ρ,a 1 0)M ρ,a 1 0) M ρ,a 1 0) 2 M ρ,a1 0) = E ρ A 2 1) = 1 5.3) 14

We wat to prove that lim M ρ,t t) =e t2 /2 5.4) Applyig 5.2), Equatio 5.4) is equivalet to lim Mρ,A1 t/ ) ] = e t 2 /2 5.5) ad Equatio 5.5) is equivalet to lim Lt/ )= t2 2 5.6) To verify Equatio 5.6) we apply 5.3) ad L Hospital s rule to obtai Lt/ ) L t/ ] )t lim = lim = lim L t/ ] ) t2 = t2 1 2 1/2 2 2 Now suppose that µ ad σ 2 are arbitrary ad let A i = A i µi. The {A σ i } is a idepedet, idetically distributed sequece with E ρ A i) = 0,Var ρ A i)=1. The lettig T = 1 A i, by Equatio 5.4) we have lim M ρ,t t) =e t2 /2. However, ) Ai µi T = σ σa i + µi µi = σ = 1 A i = T Hece, lim M ρ,t t) =e t2 /2. The ext result gives a versio of the usual cetral limit theorem. Corollary5.3. If {A i } is a idepedet, idetically distributed sequece i the state ρ ad A i SH), i =1, 2,..., the ) lim P A1 + + A µ ρ σ a = 1 a e x2 /2 dx 2π Proof. The distributio fuctios of T coverge to the distributio fuctio of the stadard ormal radom variable Z because the momet geeratig fuctios M ρ,t coverge to M Z t) =e t2 /2 9]. 15

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