Large Deviations in Quantum Information Theory

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1 Large Deviatios i Quatum Iformatio Theory R. Ahlswede ad V. Bliovsky Abstract We obtai asymptotic estimates o the probabilities of evets of special types which are usefull i quatum iformatio theory, especially i the theory of idetificatio for oisy chaels. 1 Itroductio Let T be the topological Hausdorff space with Borel σ algebra B. Also let (P ) =1 be a sequece of distributios o T. We say that the large deviatios priciple (LDP) for (P ) =1 is valid if there exists a fuctioal I : T R, I 0, such that for arbitrary B B the followig relatios are valid l P (B) if I(ξ) if ξ Bo l P (B) if I(ξ), (1) ξ B where B (B o ) is the closure (ope kerel) of the set B. I the case where T is a metric space we say that the local LDP for (P ) =1 is valid if there exists a fuctioal I : T R, I 0, such that for arbitrary z T l P (B z,δ ) δ 0 = I(z), where B z,δ = {y T : d(y,z) δ} is the ball of radius δ cetered i z. Let H d be the space of Hermitia d d matrices ad let P d be a distributio o it. Space H d is aturally isomorphic to the liear space R d2 ad the dimesio of it equals d 2. The large deviatios theory i R d2 is properly developed. We recall it. Let t R d2 ad let The the value Λ(t) = Ee (t,z) <. (2) Λ (ξ) = sup t R d2 ((t,ξ) l Λ(t)) is the rate fuctio for the sequece of probabilities (P ) =1, where ( ) 1 P (B) = Z i B 1 i=1

2 Z i H d,b B. If we cosider i additio the expoetial tightess of the sequece of probabilities (P ) =1 the the LDP for (P ) =1 is still valid (with rate fuctio Λ ), whe d =. Actually oe of the iterestig problems for Quatum Iformatio Theory is the estimatio of the probability of the evets (see [2]) B(C) = {Z C}, (3) B (C) = {Z C}, (4) where Z = 1 Z = 1 ad (Z i ) =1 is the sequece of i.i.d. radom Hermitia matrices of fiite dimesio d d ad C is some Hermitia matrix. Note that B (C) is ope ad whe d < the B (C) is closed. The we also cosider the case whe Z i is radom Hilbert-Schmidt operator o a ifiite dimesioal separable Hilbert space H. Let s cosider at first that the dimesio d of the groud space H is fiite. Note that i this case sequece (P ) =1 is expoetially compact. 1 It is easy to see that the sets B (C),B (C) are Borel sets. Next we prove that if EZ i B (C), (5) the if Λ l P (B(C)) (C Iδ,x) if x C d δ 0 l P (B(C)) if x C d Λ (C,x), (6) ad where l P (B (C)) = if x C d Λ (C,x), (7) Λ (C,x) = sup(t(x,cx) l Λ(t,x)), Λ(x) = Ee t(x,z i x) t R ad I is the uit matrix. It is eough to optimize the expressios i the last relatios oly over uit vectors x C d. Note that if EZ i B(C), the l P (B(C)) = l P (B (C)) = 0. We will produce the proof of (6), (7) i such a way that it will be valid for d = after mior chages. First of all we prove that the it i the LHS of (6) exists. Note that the 1 This meas that for every oegative real umber δ there exists a compact set K δ with P (K δ ) < e δ for all large. 2

3 relatios mea that for some x R d or corresspodigly. Z = Z = Z i C, i=1 Z i C i=1 (x,z x) > (x,cx) (x,z x) (x,cx) The closure of the set S = {Z C} is cotaied i the set x R d {(x,zx) > ((x,cx) ǫ x )},t > 0, where ǫ x are choose i such a way that (x,ez i x) < (x,cx) ǫ x. Sice the sequece (P ) =1 is expoetially compact, for every L < oe ca choose the compact set K R d2 such that for large eough, l(p (R d2 \ K L )) < L. (8) The the set S = S K L is also compact ad oe ca choose fiitely may x 1,x 2,...,x m R d such that m S {(x j,zx j ) > (x j,cx j ) ǫ xj }. j=1 For every set {(x j,zx j ) > (x j,cx j )} the LDP for (P ) =1 is valid with rate fuctio Λ (C,x j ). Ideed i this case we deal with the oe dimesioal radom variable (x j,zx j ) ad by the previous coditio Ee ((x j,z i,x j )t) <. Therefore ad Λ (C Iδ,x j ) if tag9 l P (t(x j,z x j ) > t(x j,cx j )) l P (t(x j,z x j ) > t(x j,cx j )) ( l P t(xj,z x j ) > t((x j,cx j ) ǫ xj ) ) Λ (C Iǫ,x j ) 3

4 Λ l P (t(x j,z x j ) t(x j,cx j )) (C,x j ) if l P (t(x j,z x j ) t(x j,cx j )) ( l P t(xj,z x j ) t((x j,cx j ) ǫ xj ) ) Λ (C Iǫ xj,x j ). (10) The RHS equalities i the relatios (9) (10) are a cosequece of the Chebyshev iequality. The LHS equalities i the relatios (9) (10) are a cosequece Cramer s theorem for semiifiite itervals (see [1]: for a arbitrary a R ad a sequece of i.i.d. radom variables (χ i ) i=1 where for every i Now we choose ad the we have l P ( 1 i=1 χ i [a, ) ) Γ(x) = sup(xt l Ee tχ i ). t R if Λ l P (B(C)) (C Iδ,x) if x C d δ 0 l ( P (B(C) K L ) + P (R d \ K L ) ) = if x a Γ(x), L > if x C d Λ (C,x) (11) l P (B(C)) (12) if x j (Λ (C Iǫ xj ),x j )) + o(1), (13) l ( P (B (C) K L ) + P (R d \ K L ) ) l P (B (C)) if x C Λ (C,x) (14) if(λ (C Iǫ xj,x j )) + o(1), (15) x j where o(t) is the rest term which follows from the relatio (11) ad the fact that we take if xj. Because we ca chage the rage over which the ifimum at the RHS of the last chai of relatios is take to x R d, we obtai that l P (B (C)) = if x C d Λ (C,x), if Λ (C Iδ,x) if x C d δ 0 l P (B(C)) l P (B(C)) if x j (Λ (C Iǫ xj,x j )). Sice for every x j, (x j,ez i x j ) < (x j,cx j ) ǫ xj, the fuctio Λ (C Iǫ xj,x j ) mootoically icrease as ǫ xj decreases we ca omit ǫ i the RHS of the relatios (13), (15). Now we 4

5 cosider the case whe d = ad we cosider the matrices, which corresspod to selfadjoit Hilbert-Schmidt operators o Hilbert space H. We suppose that EZ i is also H-S matrix. I this case all previous cosideratios are still valid uder the assumptio that (P ) =1 is expoetially tight. Hece we should fid out i which cases this assumptio is valid. Recall that for the self-adjoit Hilbert-Schmidt matrix Z = (z ij ) the followig relatios are valid Z Z 2 Z 1, where is uiform orm, Z 2 2 = tr(z 2 ) = i,j a ij 2 is the square of the Hilbert-Schmidt orm ad Z 1 = tr Z is the trace-orm (if it exists). Hece to every Hilbert-Schmidt (H-S) matrix corresspods i the atural order the sequece of reals which is l 2 sequece ad vice versa. If oe cosiders the set of all l 2 sequeces A = {a} such that a j b j for some give l 2 sequece b of positive reals, the the set A is compact (i l 2 orm). Hece the expoetial tightess of the sequece (P ) =1 is a cosequece of the followig coditio. Let b = {b 1,b 2,...,b,...,} be some l 2 sequece of positive reals ad let Z = (z i ) be the R - represetatio of self-adjoit H-S matrix Z. If for some atural the sum ( ) π = e Ω+ j (bj) + e Ω j (b j), (16) where Ω ± i j=1 (b) = sup((t,ez j ± b j ) l Ee (t,zj) ), (17) t R coverges, the the sequece (P ) =1 is expoetially compact. Note that the RHS of iequality (16) is othig else but the additive upper boud for the probability that Ez i z i > b i for some i. It is easy to see that if (17) is valid, the for ay give L > 0 oe ca choose i such that ( ) e Ω+ j (bj) + e Ω j (b j) < 1 i + 1 e L. j>i The we use the relatio, which follows from the fiiteess of Λ(t), Ω ± m(b) b b to choose the values b 1, b 2,..., b i > 0 such that = e Ω± m( b m) < 1 i + 1 e L,m = 1, 2,...,i. 5

6 The we have π < e L (18) ad b 1, b 2,..., b i,b i+1,... is l 2 sequece. From (18) it follows that P is expoetially compact. Note that eve if oe cosiders the diagoal matrices A ad distributio P o them (all elemets o the diagoals of matrices from A are l 2 sequeces) ad we have o iformatio about the joit distributios of differet diagoal elemets the oly possible upper boud for the probability P( Z EZ C), where C is some self-adjoit bouded operator, is the additive boud (16), which ca be obtaied by usig the Chebyshev iequality for the estimatio of deviatios P( z j Ez j > c j ). The last thig we should like to ote is a coveiet upper asymptotic boud o the probability of the evet Z C, which follows from our cosideratios. As i estimatios (13) we use the Chebyshev estimatio (x is uit vector) P (B (C)) e (if x R d(t(x,cx) l Eet(x,Z j,x) +ǫ)) = = e ǫ sup x R d ( Ee t(x,z j,x) e t(x,cx)) e ǫ sup x R d ( (x,ee t(z j C),x) ) e ǫ Ee tz j tc. Here the secod iequality follows from the iequality e (x,zx) (x,e Z x), which i tur is a cosequece of the covexity of e y. Sice ǫ > 0 is arbitrary we have sup l P (B(C)) if t 0 l EtZ j tc. This formula uder the assumptio of expoetial compactess is also valid for d =. Note also that the uitary trasformatio U = {u p,q } preserve the compactess ad if for some H-S matrices A = {a i,j },B = {b i,j }, a i,j b i,j, the for the matrices A = UAU = {a i,j} ad B = UBU = {b i,j} we have a i,j = p,q u i,p a p,q u q,j p,q u i,p a p,q u q,j p,q u i,p b p,q u q,j ad a i,j 2 p,q i,j b p,q 2 <. Hece, we ca view { a i,j } as a sequece (i ay order). It has l 2 majorat { b i,j } i some basis the cojugate sequece a i,j has the property that a i,j has the majorat UBU ij. Hece oe could check the validess of the relatios (16), (17) i some most coveiet basis for the correspodig estimatios. 6

7 At the ed we should like to state a ope problem. Let {Z j } =1 be a sequece of i.i.d. Hermitia d d matrices. The the followig chai of relatios is valid ( ( Tr E e P )) ( ( = E Tr e P )) ( ( E Tr e P ) ( m Tr e P ) ) ( ( i=m+1 Z i ) = E Tr e P )) ( ( m E Tr e P )) i=m+1 Z i = ( ( = Tr E e P )) ( ( m Tr E e P )) i=m+1 Z i. Here the first iequality is a cosequece of the Golde - Thompso iequality (see [4], [3]) Hece the sequece (a ) =1 with is subadditive ad there exists the it Tr ( e A+B) Tr ( e A e B). a = ltr ( ( E e P )) a a m + a m a = a. The problem ow is to fid a explicit expressio for this it i terms of a sigle letter characterizatio i.e. i terms of the margial distributio of Z j. If Z j are commutative, the l ( Ee ) Z i a l ( d E Z i ). (19) Because E = Ee Z i is A self-adjoit operator E 2 = E 2 ad hece from (19) ad the covergece of (a /) =1 it follows that a = a 2 k k 2 = l k EeZ i. Also i ay case ad hece From the iequalities a l d Ee Z i a l EeZ i. Ee P TrEe P d Ee P 7

8 it follows that l Ee P = a. Refereces [1 ] J.D. Deuschel ad D.W. Strook, Large Deviatios, Academic press, Bosto, [2 ] R. Ahlswede ad A. Witer, Strog coverse for idetificatio via quatum chaels, IEEE Tras. o If. Theory, Vol. 48, No. 3, , [3 ] M. Reed ad B. Simo, Fuctioal Aalysis, Academic Press. Vol. 1,2, New York ad Lodo, [4 ] C.J. Thompso, Iequality with applicatios i statistical mechaics, J. Math. Phys. Vol. 6, No. 11, ,

Math Solutions to homework 6

Math Solutions to homework 6 Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there