arXiv:2011.03045v1 [math.OA] 5 Nov 2020 ilsums tial itiuin hysoe that showed they distribution, ntesbetada xesv ito eeecs eoest Before references. of list framework. extensive W necessary an the fields. and recall other subject several 19 the to the on connections in with [13]) (see area Voiculescu established by initiated was which theory M and nta t ibr pc opeinL completion space Hilbert its instead iheult if equality with uia)sub-algebras (unital) epciesub- respective ofr on confers scnee” i.e., centered”, is of notion the recall we ( of f(iait)dpnec.The dependency. (bivariate) of i τ fihuns) lmnsof Elements (faithfulness). 1 ( AIA ORLTO N OOOIIYO REENTROPY FREE OF MONOTONICITY AND CORRELATION MAXIMAL xy n20,Dmo aa,adSep[]ivsiae hsmaxim this investigated [5] Shepp and Kagan, Dembo, 2001, In e ( Let the in statement analogous the prove to is note this of aim The ero’ orlto coefficient correlation Pearson’s h ne product inner The 6= x qipdwt a with equipped X = ) 1 i x , . . . , 2 2 i , ,τ M, p uaiepoaiiysubspaces probability mutative Abstract. ierno aibe.Ti stefe-rbblt analog free-probability the is This variables. random tive ftemntnct ftefe nrp n reFse infor Fisher free and entropy free the est this of theorem. use monotonicity we the application, an of As probability. classical in ewe h u-lersgnrtdby generated sub-algebras the between ste aual enda h urmmof supremum the as defined naturally then is τ 2 m/n ( S yx 6= M n eannomttv,fihu,tailpoaiiyspace probability tracial faithful, noncommutative, a be ) n taeproperty), (trace ) ) i : for = 3 aua L natural a ∈ ∗ i , . . . , agba hysa r free. are span they -algebras σ m M X ( eitouetemxmlcreaincoefficient correlation maximal the introduce We τ X 1 ≤ n ( 1 x + n ) r stefml of family the is M ∗ 1 − hr ( where , < lna form -linear · · · freeness · · · 1 1 M , ∞ 6= x EJMNDDU N IREYOUSSEF PIERRE AND DADOUN BENJAMIN 2 + M srcuewt norm with -structure . r 2 i whenever 0 = ) 1. X r . . . , x nti en ocmuaierno aibe r rei t if free are variables random noncommutative vein, this In . r called are n hc stecutrato lsia needne esay we independence: classical of counterpart the is which , aia correlation maximal i h nrdcinadmi result main and Introduction ) ,y x, τ i fiid admvariables random i.i.d. of ∈ ( ⊆ R N x ρ M ( i ∗ ( sasqec ffe n dnial itiue noncommut distributed identically and free of sequence a is M S τ X 1 x : moments n = ) and 1 : S , are ≥ X , 2 τ ocmuaierno variables random noncommutative ( M m ( M nnngtvt) and (non-negativity), 0 s M x 2 free n ) ) ∗ 2 .Mr eeal,L generally, More ). ≤ y → : : = cov( = n hwta h aia orlto coefficient correlation maximal the that show and τ ) , r ( ( f“n lentn rdc fcnee elements centered of product alternating “any if 1 x x τ C 1 j ( m k n + o l 1 all for 0 = ) x x uhthat such i R m , 1 X . . . k ( ,y x, 2 ( · · · 1 ρ X + X , : ( eo eutb Dembo–Kagan–Shepp by result a of ue = mt opoieaohrsml proof simple another provide to imate 1 f x ) x X , ( ≤ i 2 n ∈ p r X ) ): X 0sadhssnefluihdit an into flourished since has and 80’s /σ R and 2 n, M 1 ee o[,8 o nintroduction an for 8] [7, to refer e h aini h recnrllimit central free the in mation ewe w admvariables random two between ) ) ( i ,x x, tn u anrsl,ltu first us let result, main our ating τ r M aey eadeso the of regardless Namely, . g , ( 2 1 n o all for and 1 = (1) X ≥ , s 1 ( m M , 1 i X 2 τ 1 ow a ....consider w.l.o.g. may we so , ) ≤ ( ( otx ffe rbblt,a probability, free of context := , 2 M σ 2 x 1 )oe L over )) ewe w noncom- two between ) ( lcreainfrtepar- the for correlation al ∗ j hti unital a is that , X ′ x x ≤ ildnt h closure the denote will ) ≤ 1 fadol if only and if 0 = ) 2 + i sasadr measure standard a is ) 1 r i , . . . , . . . n the and , and 2 + -functions x r m x ≤ j equals ∈ distribution n ,y x, .Finally, ). M ∗ a- -algebra i ,g f, j ∈ x with that X 0 = . M X he i ’s 1 , 2 BENJAMINDADOUNANDPIERREYOUSSEF

2 ′ ′ in L (M) of the sub-algebra M M. If M = C x1,...,xn is the sub- -algebra of noncommu- tative polynomials in x ,...,x ⊆ M and x∗,...,xh ∗ M, wei refer to L2∗(M ′) asL2(x ,...,x ). 1 n ∈ 1 n ∈ 1 n Let cov(x,y) := x τ(x) 1,y τ(y) 1 and σ(x)2 := cov(x,x). Just like in classical probability, we can defineh − the Pearson· − correlation· i coefficient between x and y by cov(x,y) ρ(x,y) := . σ(x) σ(y)

Note that we have ρ(x,y) 1 by the Cauchy–Schwarz inequality. Given M1, M2 M two subspaces, we call | | ≤ ⊆

R(M1, M2) := sup ρ M1×M2 2 2 the maximal correlation coefficient between M1 and M2. For M1 = L (x) and M2 = L (y) we simply write R(x,y), and we call it the maximum correlation between the noncommutative random variables x and y. We are now ready to state our main result.

Theorem 1.1. Let (xi)i∈N be a sequence of free, non-zero, identically distributed, noncommu- tative random variables, and let s := x + + x . Then for any m n, we have n 1 · · · n ≤ m R(s ,s )= . n m r n

As in the classical setting, the interesting feature of the above statement is its universality as it holds regardless of the distribution of the noncommutative random variables. The proof of Theorem 1.1, which is carried in Section 2, adapts the approach of [5] to a noncommutative setting. A celebrated result of Artstein et al [1] provided a solution to Shannon’s problem regarding the monotonicity of entropy in the classical central limit theorem. In the context of free probability, the concept of free entropy and information was developped by Voiculescu in a series of papers (see for example [15]). Two approaches were given for the definition of free entropy, referred to as microstates and non-microstates and denoted by χ and χ∗ respectively (see [8, Chapters 7 and 8]). As these coincide in the one-dimensional setting considered in this paper, the free entropy of a compactly supported probability measure µ is given by 3 1 χ(µ)= χ∗(µ) := log a b µ(da)µ(db)+ + log(2π). ZZR2 | − | 4 2 Given a noncommutative probability space (M,τ) and a self-adjoint element z M, we de- ∈ fine χ(z) as χ(µz) where µz denotes the distribution of z, i.e., the probability measure charac- terized by p dµ = τ p(z) for all polynomials p C[X]. z ∈ In [11],R Shlyakhtenko proved
the monotonicity of the free entropy in the free central limit theorem providing an analogue of the result of [1] in the noncommutative setting. As an ap- plication of our maximal correlation estimate, we recover the monotonicity property which we state in the next corollary.

Corollary 1.2. Given (xi)i∈N a sequence of free, identically distributed, self-adjoint random variables in (M,τ), one has s s χ m χ n , (1) √m ≤ √n for every integers m n. ≤ MAXIMAL CORRELATION AND MONOTONICITY OF FREE ENTROPY 3

As in the classical setting, the monotonicity of the entropy follows from that of the Fisher information. In Section 3, we prove the latter as a consequence of Theorem 1.1. The idea of using the maximal correlation inequality in this context goes back to Courtade [4] who used the result of Dembo, Kagan, and Shepp [5] to provide an alternative proof of the monotonicity of entropy in the classical setting. Formulated alternatively, the above corollary states that given a compactly supported prob- ability measure µ and positive integers m n, one has ≤ ⊞ ⊞ χ(m−1/2 µ m) χ(n−1/2 µ n), (2) ∗ ≤ ∗ where ⊞ denotes the free convolution operation (so µ⊞n is the distribution of the sum of n free copies of an element x with distribution µ), and α∗ is the pushforward operation by the dilation t αt. As a matter of fact, it is possible to make sense of µ⊞t for all real t 1 (see [9]). Very recently,7→ Shlyakhtenko and Tao [12] extended (2) to real-valued exponents while≥ providing two different proofs. It would be interesting to see if the argument in this paper based on maximal correlation could be extended to cover non-integer exponents.

Aknowledgement: The authors are grateful to Roland Speicher for helpful comments and for bringing to their attention the recent preprint [12]. The second named author is thankful to Marwa Banna for helpful discussions.

2. Proof of Theorem 1.1

Let us fix x ,x ,... M. For every finite set I N, we denote 1 2 ∈ ⊂ 2 2 LI :=L (xi : i I) and projI (z) := proj 2 (z) ∈ LI 2 the orthogonal projection of z M onto LI , which is nothing else but the conditional expecta- 2 ∈ tion of z given LI: y = proj (z) y L2 and x L2, τ(xy)= τ(xz). I ⇐⇒ ∈ I ∀ ∈ I In particular proj (z)= z if z L2 and by the trace property I ∈ I proj (xzy)= x proj (z)y for all x,y L2. I I ∈ I Note that proj∅(z)= τ(z) 1 and projJ projI = projJ for every J I (tower property). When freeness is further involved,· we can say◦ a bit more. The following⊆ lemma appears in different forms in the literature ([3], [8, Section 2.5]), we include its proof for completeness. Lemma 2.1. Let I, J N be finite sets and suppose (x : k I J) is free. Then: ⊂ k ∈ ∪ (i) if z is a (noncommutative) polynomial in variables in C x : j J , then proj (z) is a h j ∈ i I polynomial in only those variables that are actually in C xk : k I J ; (ii) the projections commute: proj proj = proj . h ∈ ∩ i I ◦ J I∩J

Proof. (i) By linearity of projI , we may suppose without loss of generality that z = a1 ar with a C x and consecutively distinct indices i ,...,i J. From the moment-· · · j ∈ h ij i 1 r ∈ cumulant formula [8, Definition 9.2.7] w.r.t. the conditional expectation projI , we can write

I projI (z) := κπ(a1, . . . , ar), π∈XNC(r) 4 BENJAMINDADOUNANDPIERREYOUSSEF

I where the summation ranges over all non-crossing partitions π NC(r) of 1, . . . , r and the κπ ∈ { }I n 2 are nestings (consistently with the blocks of π) of the free (conditional) cumulants κn : M LI (which can be defined inductively). For instance, if → I = 1, 4 , J = 1, 2, 3, 4 , z = x4 x3 x x∗ x∗ x2 x x∗ x3 x5 x (r = 10), { } { } 1 · 2 · 4 · 1 · 2 · 3 · 1 1 · 4 · 3 · 4

and π = , 1 2 3 4 5 6 7 8 9 10 then

κI (a , . . . , a )= κI x4 κI x3 κI x ,x∗ ,x∗ κI x2 κI x x∗,x3 ,x5 ,x . π 1 r 2 1 · 3 2 · 2 4 1 2 · 1 3 · 2 1 1 4 3 4 


 We now invoke [10, Theorem 3.6] with F := τ 2 and the sub-algebra N := C xj : j J I free |LI h ∈ \ i 2 C from B :=LI over D := 1 to see that all conditional cumulants involving any variable in N I· I ∗ 2 I ∗ 3 3 ∗ 5 reduce to constants; e.g., κ3(. . .)= τ κ2(x4,x1) τ x3 τ κ2(x1x1,x4) κ3(x2,x2,x3) in the above display, where κ is the (unconditionned) cumulant
M
3 C. This shows
that proj (z) is in fact 3 → I a polynomial only in the subset of the variables a1, . . . , ar that belong to C xk : k I J (in the I I ∗ 2 I ∗ 3 3 ∗ 5 h I 4 ∈ ∩ i example, κπ(a1, . . . , ar)= τ κ2(x4,x1) τ x3 τ κ2(x1x1,x4) κ3(x2,x2,x3) κ2(x1,x4) is indeed a polynomial in the C x ,x -variables a
= x
4, a = a = x
, a = x∗, a = x x∗, a = x3). h 1 4i 1 1 3 10 4 4 1 7 1 1 8 4 (ii) Clearly τ x proj proj (z) = τ proj (proj (xz)) = τ(xz) for all x L2 , so we only I ◦ J I J ∈ I∩J need to check that proj proj
(z) L2 . By a closure
argument, we may suppose that I J ∈ I∩J proj (z) belongs to C x : j J .
In this case proj proj (z) C x : k I J L2 J h j ∈ i I J ∈ h k ∈ ∩ i⊆ I∩J by the previous point.


We now set, for every z M, ∈ z := ( 1)|I|−|J| proj (z) L2, (3) I − J ∈ I JX⊆I where is the cardinal notation. The following decomposition will play a crucial role in the proof of| · Theorem | 1.1. Lemma 2.2 (Efron–Stein decomposition). For every finite set I N, ⊂ projI (z)= zJ . JX⊆I

Proof. We repeat in a compact way the argument of Efron and Stein [6]: z = ( 1)|J|−|K| proj (z) J − K JX⊆I JX⊆I KX⊆J

=  ( 1)|J|−|K| proj (z) − K KX⊆I KX⊆J⊆I   = (1 − 1)|I\K| | {z }  = projI (z).

The elements zJ will be orthogonal thanks to this direct consequence of Lemma 2.1:

Lemma 2.3. Suppose that x1,x2,... are free, and let I, J N be finite sets such that I J = . Then proj (z ) = 0 for every z M. In particular, z is orthogonal⊂ to z L2 . \ 6 ∅ J I ∈ I J ∈ J MAXIMAL CORRELATION AND MONOTONICITY OF FREE ENTROPY 5

Proof. Apply Lemma 2.1 and gather the subsets K I that have same intersection L := J K with J: ⊆ ∩ proj (z )= ( 1)|I|−|K| proj proj (z) J I − J ◦ K KX⊆I

= ( 1)|I|−|L|  ( 1)|K| proj (z) − − L L⊆XI∩J KX⊆I\J   =(1−1)|I\J| = 0. | {z } 

To prove Theorem 1.1 we shall finally exploit the fact that the partial sum s := x + + x n 1 · · · n is symmetric in (x1,...,xn). Our next proposition is tailored for this purpose.

Proposition 2.4. Suppose that x1,...,xn are free and identically distributed.

(1) For every symmetric polynomial z = p(x1,...,xn) in x1,...,xn and every I 1,...,n , ∗ ∗ ′ ⊆ { } the pair (zI , zI ) has the same distribution as (zI′ , zI′ ) where I := 1,..., I . Consequently, if τ(z) = 0, then { | |} I proj (z) | | z . I 2 ≤ r n k k2

(2) For every m n and every polynomial p, ≤ 2 2 projL (x1,...,xm) p(sn) = projL (sm) p(sn) .

Proof. (1) Let m = I and let ς be a permutation of 1,...,n mapping I′ to I. Since p is | | { } symmetric, we can write z = p(xς(1),...,xς(n)). By Lemma 2.1, we see from its definition ∗ in (3) that zI is a polynomial q(xς(1),...,xς(m)) in the xi, i I. Now the pairs (zI , zI )= ∗ ∈ ∗ (q(xς(1),...,xς(m)),q(xς(1),...,xς(m)) ) and (zI′ , zI′ ) = (q(x1,...,xm),q(x1,...,xm) ) have the same moments because x1,...,xn are free and identically distributed. In par- ticular zI 2 = zI′ 2, and the stated inequality becomes clear by combining this with Lemmask 2.2k andk 2.3:k proj (z) 2 = z 2 I 2 k J k2 JX⊆I m m = z 2  k k {1,...,k}k2 Xk=1 n m n z 2 ≤ n kk {1,...,k}k2 Xk=1 m = proj (z) 2 n {1,...,n} 2

m = z 2, n k k2 m m n 2 using that z∅ = 0, that k n k for all 1 k m n, and that z L (x1,...,xn). ≤ ≤ ≤ 2≤ ∈ (2) We only need to check that
proj2
(p(s )) L (s ) since L (x1,...,xm) n ∈ m 2 y L (s ), τ y proj 2 (p(s )) = τ proj 2 (yp(s )) = τ yp(s ) ∀ ∈ m L (x1,...,xm) n L (x1,...,xm) n n (as L2(s ) L2(x ,...,x )). But p(s
) is a polynomial in s ,x
,...,x and freeness
m ⊂ 1 m n m m+1 n of s ,x ,...,x again implies by Lemma 2.1 that proj 2 (p(s )) C s .  m m+1 n L (x1,...,xm) n ∈ h mi 6 BENJAMINDADOUNANDPIERREYOUSSEF

Proof of Theorem 1.1. The lower bound R(sn,sm) m/n is straightforward since, by free- 2 2 2 ≥ 2 ness, σ(sn) = nσ(x1) and cov(sn,sm) = σ(sm) = mσp (x1) . For the upper bound, we must ′ 2 ′ 2 show that ρ(z, z ) m/n for all z L (sn) and z L (sm). W.l.o.g., we may suppose that ′ ≤ ∈ ∈ τ(z) = τ(z ) = 0 and,p by another closure argument, that z is a polynomial in sn (and thus a symmetric polynomial in x1,...,xn). Then by the Cauchy–Schwarz inequality and Proposi- tion 2.4,

cov(z, z′)= z, z′ h i 2 ′ = projL (sm)(z), z ′ proj 2 (z) z ≤ L (sm) 2 k k2 m z z′ ≤ r n k k2 k k2 m = σ(z) σ(z′), r n and the proof is complete. 

3. Monotonicity of the free entropy and free Fisher information

The goal of this section is to prove Corollary 1.2. Let us start by noting that the free entropy and free Fisher information of a self-adjoint element z M are related through the integral formula (see [8, Chapter 8]) ∈

1 ∞ 1 1 χ(z)= Φ z + √tx dt + log(2π e), (4) 2 Z0 1+ t −   2 where x is a standard semi-circular variable free from z, and Φ denotes the free Fisher informa- tion. After [8, Chapter 8], the free Fisher information of a noncommutative, self-adjoint random 2 variable z M is defined as Φ(z) := ξ 2 where the so called conjugate variable ξ := ξ(z) is any element of∈ L2(z) such that, for everyk integerk r 0, ≥ r−1 τ ξzr = τ zk τ zr−1−k . (5)
Xk=0

(If such a ξ does not exist, we set Φ(z) := .) We note from (5) that τ(ξ(z)) = 0 and the homogeneity property Φ(αz)= α−2 Φ(z),α>∞0. In the next Corollary, we show how the monotonicity of the free Fisher information follows easily from Theorem 1.1.

Corollary 3.1. Let (xi)i∈N be a sequence of free, identically distributed, self-adjoint random variables in (M,τ) and denote sk := x1 + . . . + xk for every positive integer k. Then for all positive integers m n, we have ≤ s s Φ n Φ m . √n ≤ √m

Proof. Assume the existence of ξ(s ), as otherwise Φ(s ) = and there is nothing to prove. 1 1 ∞ According to [8, p. 206], the free sum s = s + (s s ) admits ξ(s ) = proj 2 (ξ(s )) as n m n − m n L (sn) m MAXIMAL CORRELATION AND MONOTONICITY OF FREE ENTROPY 7 conjugate variable. Therefore, by Theorem 1.1, 2 2 2 Φ(sn)= projL (sn) ξ(sm) 2 = cov projL (sn) ξ(sm) ,ξ(sm)

 m proj 2 ξ(s ) ξ(s ) , ≤ r n L (sn) m 2 m 2
i.e., Φ(s ) m Φ(s ). We conclude by the homogeneity property.  n ≤ n m In view of (4) and the divisibility of the semicircular distribution w.r.t. the free convolution, the above corollary readily implies (1), thus proving Corollary 1.2.

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Benjamin Dadoun, Mathematics, Division of Science, New York University Abu Dhabi, UAE E-mail: [email protected]

Pierre Youssef, Mathematics, Division of Science, New York University Abu Dhabi, UAE E-mail: [email protected]