From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture (New title)
aa r X i v : . [ m a t h . F A ] F e b FROM WIGNER-YANASE-DYSON CONJECTURE TOCARLEN-FRANK-LIEB CONJECTURE
HAONAN ZHANG
Abstract.
In this paper we study the joint convexity/concavity of the tracefunctions Ψ p,q,s ( A, B ) = Tr( B q K ∗ A p KB q ) s , p, q, s ∈ R , where A and B are positive definite matrices and K is any fixed invertiblematrix. We will give full range of ( p, q, s ) ∈ R for Ψ p,q,s to be jointly con-vex/concave for all K . As a consequence, we confirm a conjecture of Carlen,Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaertand Datta and obtain the full range of ( α, z ) for α - z R´enyi relative entropiesto be monotone under completely positive trace preserving maps. We also givesimpler proofs of many known results, including the concavity of Ψ p, , /p for0 < p < p, − p, ( A, B ) = Tr K ∗ A p KB − p , − ≤ p ≤ , using a variational method. Introduction
The joint convexity/concavity of the trace functions(1.1) Ψ p,q,s ( A, B ) = Tr( B q K ∗ A p KB q ) s , p, q, s ∈ R , has played an important role in mathematical physics and quantum information.Its study can be traced back to the celebrated Lieb’s Concavity Theorem [Lie73],which states that Ψ p,q, is jointly concave for all 0 ≤ p, q ≤ , p + q ≤ K . Using this, Lieb confirmed the Wigner-Yanase-Dyson conjecture [WY63]: for < p < and any self-adjoint K , the function (1.2) S p ( ρ, K ) := 12 Tr[ ρ p , K ][ ρ − p , K ] = − Tr ρK + Tr ρ p Kρ − p K, is concave in ρ , where [ A, B ] = AB − BA . We refer to [WY63, Lie73] for moredetails about the skew information − S p ( ρ, K ).Since then, a lot of work around the joint convexity/concavity of Ψ p,q,s hasemerged [And79, Bek04, CFL16, CL08, CL99, Eps73, FL13, Hia13, Hia16], follow-ing [Lie73]. Through this line of research many methods have been developed.Two main methods are the “analytic method” and the “variational method”. Werefer to a very nice survey paper [CFL18] for more historical information and theexplanation of these two methods.Another motivation to study the joint convexity/concavity of Ψ p,q,s comes fromquantum information theory. Indeed, the joint convexity/concavity of Ψ p,q, / ( p + q ) is closely related to the monotonicity (or Data Processing Inequality ) of the α - z R´enyi relative entropies, which has become a frontier topic in recent years. We
Mathematics Subject Classification.
Primary 15A15, 81P45; Secondary 47A56, 94A17.
Key words and phrases.
Joint convexity/concavity, quantum relative entropy, Data ProcessingInequality. shall recall this in Section 2. Starting from this Audenaert and Datta conjecturedthat:
Conjecture 1. [AD15, Conjecture 1] If 1 ≤ p ≤ , − ≤ q < p, q ) = (1 , − K , the functionΨ p,q, / ( p + q ) ( A, B ) = Tr( B q K ∗ A p KB q ) p + q , is jointly convex in ( A, B ), where A and B are positive definite matrices.We cheat a little bit here, since the original form of their conjecture concernsthe convexity of A Tr( A q K ∗ A p KA q ) p + q for all K . However, by doublingdimension, a standard argument shows that they are equivalent. See the discussionsafter [CFL18, Conjecture 1] for example.In this paper we confirm a stronger conjecture of Carlen, Frank and Lieb: Conjecture 2. [CFL18, Conjecture 4] If 1 ≤ p ≤ , − ≤ q < , ( p, q ) = (1 , − s ≥ p + q , then for any matrix K , the functionΨ p,q,s ( A, B ) = Tr( B q K ∗ A p KB q ) s , is jointly convex in ( A, B ), where A and B are positive definite matrices. pq o concave for 0 ≤ s ≤ p + q convex for s ≥ p + q convex for s ≥ convex for s ≥ p + q -1 -1 1 212 q=p Figure 1.1.
Joint convexity/concavity (for all K ) of Ψ p,q,s Consequently, we give the full range of ( p, q, s ) for Ψ p,q,s to be jointly convex orjointly concave for any invertible K . See Figure 1.1 (note that (1 , −
1) and ( − , ROM WIGNER-YANASE-DYSON CONJECTURE TO CARLEN-FRANK-LIEB CONJECTURE3
Theorem 1.1.
Fix any invertible matrix K . Suppose that p ≥ q and s > . Then Ψ p,q,s defined in (1.1) is(1) jointly concave if ≤ q ≤ p ≤ and < s ≤ p + q ;(2) jointly convex if − ≤ q ≤ p ≤ and s > ;(3) jointly convex if − ≤ q ≤ , ≤ p ≤ , ( p, q ) = (1 , − and s ≥ p + q . We remark here that the symmetric property of Ψ p,q,s allows us to assume p ≥ q and s >
0. See the discussions before Proposition 2.2. Moreover, the above resultis sharp, in view of Proposition 2.3.As a corollary of Theorem 1.1, Proposition 2.1 and Proposition 2.3, we obtainall ( α, z ) such that D α,z is monotone under completely positive trace preservingmaps (or satisfies Data Processing Inequality, see (2.6) for the precise definition). Theorem 1.2.
The α - z relative R´enyi entropy D α,z is monotone under completelypositive trace preserving maps if and only if one of the following holds(1) < α < and z ≥ max { α, − α } ;(2) < α ≤ and α ≤ z ≤ α ;(3) ≤ α < ∞ and α − ≤ z ≤ α . As we mentioned earlier, in the history two main methods have been developedto study the convexity/concavity of the trace functions Ψ p,q,s : the analytic methodand the variational method. The analytic method, which is the methodology em-ploying the theory of Herglotz functions, was first introduced by Epstein [Eps73].The variational method was first used by Carlen and Lieb in [CL08]. Both of themhave their own advantages, as the authors wrote in [CFL18, Page 8]: “It appearsthat the analyticity method is especially useful for proving concavity and the vari-ational method is more useful for proving convexity, but this is not meant to bean absolute distinction.” In this paper we confirm Conjecture 2 by developing onlythe variational method.The main value of this paper is twofold. Firstly, we develop the variationalmethod in a very simple way such that it is useful to prove both convexity andconcavity, and it reduces the convexity/concavity of Ψ p,q,s to three very particularcases, which were already known (see Theorem 3.7). In this way we obtain the fullrange of ( p, q, s ) such that Ψ p,q,s is jointly convex/concave and confirm Conjecture1 and Conjecture 2. Secondly, using our variational method in a slightly differentway, we can furthermore reduce these three very particular cases to Lieb’s concavityresult [Lie73] of Ψ p, − p, for 0 < p ≤ p, − p, for − ≤ p <
0. In other words, from Lieb’s and Ando’s classical jointconvexity/concavity results (which admit many simple proofs) onΨ p, − p, ( A, B ) = Tr K ∗ A p KB − p , the subsequent results on joint convexity/concavity of Ψ p,q,s can be derived easilyvia our variational method. In this way we recover many classical results immedi-ately. Moreover, we emphasize here that the analytic method can be avoided.In the past half a century we have developed a lot of tools to tackle the con-vexity/concavity of trace functions, and have witnessed a number of applicationsof the convexity/concavity of trace functions to many areas, like mathematicalphysics and quantum information. Now our variational method helps us to reducethe Carlen-Frank-Lieb conjecture (in fact the joint convexity/concavity of the wholefamily Ψ p,q,s ) to the convexity/concavity of the trace function (1.2) (in which theessential part is Ψ p, − p, ) in the Wigner-Yanase-Dyson conjecture. This brings usback to the origin of the whole story. HAONAN ZHANG
This paper is organized as follows. In Section 2 we recall the background ofConjecture 1 and Conjecture 2. In Section 3 we give the proof of our main resultTheorem 1.1.We fix some notations in this paper. We use H to denote a finite-dimensionalHilbert space. We use B ( H ) to denote the family of bounded linear operatorson H , P ( H ) to denote the family of positive linear operators on H (or n -by- n positive semi-definite matrices with dim H = n ), and D ( H ) to denote the familyof density operators, i.e., positive linear operators on H with unit trace (or n -by- n positive semi-definite matrices having unit trace with dim H = n ). Moreover, weuse B ( H ) × (reps. P ( H ) × and D ( H ) × ) to denote the family of invertible operatorsin B ( H ) (resp. P ( H ) and D ( H )). We use Tr to denote the usual trace on matrixalgebra and we use I to denote the identity matrix. For any matrix A we use | A | to denote its modulus ( A ∗ A ) .We close this section with a remark. In this paper we are mainly dealing withthe invertible matrices, to avoid some technical problems and make the paper morereadable. In this case for A ∈ P ( H ) × and α ∈ R , A α is always well-defined.Some results in this paper are still valid in the non-invertible case, by using anapproximation argument. For example, in Conjectures 1 and 2, K is not assumedto be invertible, since X s is always well-defined for positive semi-definite X and s >
0. When K is not invertible, one can approximate K with invertible K ǫ = K + ǫI , where ǫ > B q K ∗ A p KB q ) s ,which is the limit of Tr( B q K ∗ ǫ A p K ǫ B q ) s as ǫ tends to 0, follows from that ofTr( B q K ∗ ǫ A p K ǫ B q ) s , since the convexity is stable under taking limits.2. Background
In this section we collect necessary background information for this paper. Mostof them are borrowed from the survey paper [CFL18]. One can refer to [CFL18]and the references therein for further details. Experts may skip this section withoutany difficulty.Given two probability density functions P and Q on R , the relative entropy , or Kullback-Leibler divergence of P with respect to Q is given by(2.1) S ( P || Q ) := ˆ R P ( x )(log P ( x ) − log Q ( x )) dx. For α ∈ (0 , ∪ (1 , ∞ ), the α -R´enyi relative entropy of P with respect to Q isdefined as [R´en61](2.2) S α ( P || Q ) := 1 α − ˆ R P ( x ) α Q ( x ) − α dx. Both classical relative entropies (2.1) and (2.2) have been generalized to quantumsetting, where the density functions are replaced by the density operators, and theintegral is replaced by the trace, respectively. However, their quantum analoguesmight take various forms.Fix ρ, σ ∈ D ( H ) × with H being any finite-dimensional Hilbert space. A naturalquantum analogue of (2.1), is the so-called Umegaki relative entropy [Ume62](2.3) D ( ρ || σ ) := Tr ρ (log ρ − log σ ) . It is monotone under completely positive trace preserving (CPTP) maps [Lin75].That is,(2.4) D ( E ( ρ ) ||E ( σ )) ≤ D ( ρ || σ ) , for all CPTP maps E : B ( H ) → B ( H ) and all density operators ρ, σ ∈ D ( H ) × . ROM WIGNER-YANASE-DYSON CONJECTURE TO CARLEN-FRANK-LIEB CONJECTURE5
The inequality (2.4) is known as the
Data Processing Inequality (DPI). As oneof the most fundamental inequalities in quantum information, DPI has strong linkswith the Strong Subadditivity (SSA) of the von Neumann entropy [LR73], the un-certainty principle [TR11], the quantum hypothesis testing [MO15] and the Holevobound for the accessible information [Hol73]. Not every quantum analogue of (2.1)satisfies DPI. For example, it is known that [CL18] D ′ ( ρ || σ ) := Tr ρ log( σ − ρσ − ) , as a generalization of (2.1), does not satisfy DPI.A natural generalization of (2.2) is the family of quantum α -R´enyi relative en-tropies D α ( ρ || σ ) := 1 α − ρ α σ − α ) , α ∈ (0 , ∪ (1 , ∞ ) . Another important generalization of (2.2), introduced by M¨uller-Lennert, Dupuis,Szehr, Fehr, Tomamichel [MLDS +
13] and Wilde, Winter, Yang [WWY14], are the sandwiched α -R´enyi entropies : e D α ( ρ || σ ) := 1 α − σ − α α ρσ − α α ) α , α ∈ (0 , ∪ (1 , ∞ ) . Audenaert and Datta [AD15] introduced a new family of quantum R´enyi relativeentropies by using two parameters, called the α - z R´enyi relative entropies :(2.5) D α,z ( ρ || σ ) := 1 α − σ − α z ρ αz σ − α z ) z , α ∈ ( −∞ , ∪ (1 , ∞ ) , z > . It unifies D α and e D α by taking z = 1 and z = α , respectively. We comment herethat the α - z R´enyi relative entropies have appeared earlier in a paper by Jaksic,Ogata, Pautrat and Pillet [JOPP12].A natural question is, for which ( α, z ) does the α - z R´enyi relative entropy D α,z satisfy DPI, that is,(2.6) D α,z ( E ( ρ ) ||E ( σ )) ≤ D α,z ( ρ || σ ) , for any CPTP map E on B ( H ) and all density operators ρ, σ ∈ D ( H ) × ? Thisremained open for some range of ( α, z ) before the present paper. It is well-knownthat DPI is essentially equivalent to the joint convexity/concavity of the tracefunctions inside the definition of D α,z . Proposition 2.1. [CFL18, Proposition 7]
Let α ∈ ( −∞ , ∪ (1 , ∞ ) and z > . Set p = αz and q = − αz . Then (2.6) holds for any CPTP map E : B ( H ) → B ( H ) , alldensity operators ρ, σ ∈ D ( H ) × and any finite-dimensional Hilbert space H if andonly if one of the following holds(1) α < and Ψ p,q, / ( p + q ) with K = I is jointly concave;(2) α > and Ψ p,q, / ( p + q ) with K = I is jointly convex. For the reader’s convenience, we present its proof in the end of this section. Fromsome known results on the joint convexity/concavity of Ψ p,q, / ( p + q ) with K = I ,Audenaert and Datta obtained DPI for D α,z for some—but not full—range of ( α, z )[AD15, Theorem 1]. By saying full we mean necessary and sufficient conditions on( α, z ). It is then natural to ask whether DPI holds for the remaining range of ( α, z ).This motivated Audenaert and Datta to raise Conjecture 1.More generally, consider the joint convexity/concavity of trace functionsΨ p,q,s ( A, B ) = Tr( B q K ∗ A p KB q ) s , where A, B ∈ P ( H ) × , K ∈ B ( H ) × and p, q, s ∈ R . Note that Ψ q,p,s ( B, A ) =Ψ p,q,s ( A, B ) with K replaced by K ∗ , and Ψ − p, − q, − s ( A, B ) = Ψ p,q,s ( A, B ) with K replaced by ( K − ) ∗ . So in the sequel we assume that p ≥ q and s > HAONAN ZHANG
The knowledge of the joint convexity/concavity of Ψ p,q,s before the survey paper[CFL18] is summarized in the following proposition in [CFL18] or the figure therein.
Proposition 2.2. [CFL18, Theorem 2]
Fix K ∈ B ( H ) × . Then Ψ p,q,s is(1) jointly concave if ≤ q ≤ p ≤ and < s ≤ p + q ;(2) jointly convex if − ≤ q ≤ p ≤ and s > ;(3) jointly convex if − ≤ q ≤ , ≤ p < , ( p, q ) = (1 , − and s ≥ min { p − , q +1 } or p = 2 , − ≤ q ≤ and s ≥ q +2 . For more historical details of these results, see the discussions after [CFL18,Theorem 2]. We only comment here that the case s = 1, which was first studiedin the history, is due to Lieb [Lie73] for 0 ≤ q ≤ p ≤ p + q ≤
1, as wellas for − ≤ q ≤ p ≤
0, and due to Ando [And79] for − ≤ q ≤ , ≤ p < p + q ≥
1. Their work played an important role in the development of matrixanalysis.The following proposition, due to Hiai [Hia13], gives the necessary conditions forΨ p,q,s to be jointly convex or jointly concave.
Proposition 2.3. [Hia13, Propositions 5.1(2) and 5.4(2)][CFL18, Proposition 3]
Let p ≥ q and s > . Suppose that ( p, q ) = (0 , and K = I .(1) If Ψ p,q,s is jointly concave for H = C , then ≤ q ≤ p ≤ and < s ≤ p + q .(2) If Ψ p,q,s is jointly convex for H = C , then either − ≤ q ≤ p ≤ and s > or − ≤ q ≤ , ≤ p ≤ , ( p, q ) = (1 , − and s ≥ p + q . From the above two propositions, Carlen, Frank and Lieb raised Conjecture2. Some partial results were known before the present paper, as pointed out inProposition 2.2 (3).We close this section with the proof of Proposition 2.1. It comes from [CFL18,Proposition 7], following a well-known argument due to Lindblad [Lin75] and Uhlmann[Uhl73].
Proof of Proposition 2.1.
We use Ψ to denote Ψ p,q, / ( p + q ) with K = I . We onlyprove the case α >
1, since the proof for α < E ( ρ ) , E ( σ )) ≤ Ψ( ρ, σ ) , for any CPTP map E on B ( H ), for all ρ, σ ∈ D ( H ) × and for all H if and only if Ψis jointly convex.To show the “if” part, take any CPTP map E : B ( H ) → B ( H ). Then we canwrite E as E ( γ ) = Tr U ( γ ⊗ δ ) U ∗ , where δ ∈ D ( H ′ ), U is unitary on H ⊗ H ′ , and H ′ is a Hilbert space such that N ′ := dim H ′ ≤ (dim H ) . Here Tr denotes the usual partial trace over H ′ . For aproof, see for example [Lin75, Lemma 5]. It origins in the celebrated Stinespring’sTheorem [Sti55]. Let du denote the normalized Haar measure on the group of allunitaries on H ′ , then(2.7) E ( γ ) ⊗ I H ′ N ′ = ˆ ( I H ⊗ u ) U ( γ ⊗ δ ) U ∗ ( I H ⊗ u ∗ ) du, where I H and I H ′ are the identity maps over H and H ′ , respectively. By the tensorproperty of Ψ, we haveΨ( E ( ρ ) , E ( σ )) = Ψ (cid:18) E ( ρ ) ⊗ I H ′ N ′ , E ( σ ) ⊗ I H ′ N ′ (cid:19) . ROM WIGNER-YANASE-DYSON CONJECTURE TO CARLEN-FRANK-LIEB CONJECTURE7
From the joint convexity of Ψ and (2.7) it follows thatΨ( E ( ρ ) , E ( σ )) ≤ ˆ Ψ(( I H ⊗ u ) U ( ρ ⊗ δ ) U ∗ ( I H ⊗ u ∗ ) , ( I H ⊗ u ) U ( σ ⊗ δ ) U ∗ ( I H ⊗ u ∗ )) du. By the unitary invariance and the tensor property of Ψ we obtain thatΨ( E ( ρ ) , E ( σ )) ≤ Ψ( ρ, σ ) , as desired.To show the “only if” part, for any ρ , ρ , σ , σ ∈ D ( H ) × and any 0 < λ < ρ = (cid:18) λρ
00 (1 − λ ) ρ (cid:19) and σ = (cid:18) λσ
00 (1 − λ ) σ (cid:19) , in D ( H ⊕ H ) × . Since the map(2.8) E (cid:18) a bc d (cid:19) = 12 (cid:18) a + d a + d (cid:19) , is a CPTP map, we obtain from the monotonicity of Ψ thatΨ( E ( ρ ) , E ( σ )) ≤ Ψ( ρ, σ ) , which is nothing butΨ( λρ + (1 − λ ) ρ , λσ + (1 − λ ) σ ) ≤ λ Ψ( ρ , σ ) + (1 − λ )Ψ( ρ , σ ) . This finishes the proof of the joint convexity of Ψ. (cid:3) The proofs
This section is devoted to the proof of Theorem 1.1. The following classicalresults will serve as the building blocks to achieve the joint convexity/concavity ofΨ p,q,s . The concavity result is due to Lieb [Lie73] and the convexity result is due toAndo [And79]. They have now many simple proofs, see for example [NEE13]. Weonly comment here that they are based on the operator convexity of A A p when − ≤ p < ≤ p ≤
2, and the operator concavity of A A p when 0 < p ≤ Lemma 3.1. [Lie73, And79]
For any K ∈ B ( H ) × , the function Ψ p, − p, ( A, B ) = Tr K ∗ A p KB − p , A, B ∈ P ( H ) × , is (1) jointly concave if < p ≤ ;(2) jointly convex if − ≤ p < . Theorem 1.1 will be reduced to Lemma 3.1 in three steps, using a variationalmethod. The idea of the variational method is based on the following lemma[CFL18, Lemma 13]. We give the proof here for the reader’s convenience.
Lemma 3.2.
Let
X, Y be two convex subsets of vector spaces and f : X × Y → R a function.(1) If f ( · , y ) is convex (resp. concave) for any y ∈ Y , then x sup y ∈ Y f ( x, y ) (resp. x inf y ∈ Y f ( x, y ) ) is convex (resp. concave).(2) If f is jointly convex (resp. concave) on X × Y , then x inf y ∈ Y f ( x, y ) (resp. x sup y ∈ Y f ( x, y ) ) is convex (resp. concave).Proof. (1) This follows immediately from the definition. HAONAN ZHANG (2) We only prove the convexity here. The proof of the concavity is similar.For any x , x ∈ X and any 0 < λ <
1, set x := λx + (1 − λ ) x . Thenfor any ǫ > i = 1 ,
2, there exists y i ∈ Y such that f ( x i , y i ) ≤ inf y ∈ Y f ( x i , y ) + ǫ . By the joint convexity of f , we haveinf y ∈ Y f ( x, y ) ≤ f ( x, λy + (1 − λ ) y ) ≤ λf ( x , y ) + (1 − λ ) f ( x , y ) ≤ λ inf y ∈ Y f ( x , y ) + (1 − λ ) inf y ∈ Y f ( x , y ) + ǫ. Then the proof finishes by letting ǫ → + . (cid:3) The following variational method is the key of the proof. It originates in [CL08]and the special cases (either r = 1 or r = 1) have been widely used [CFL18]. Theorem 3.3.
For r i > , i = 0 , , such that r = r + r , we have for any X, Y ∈ B ( H ) × that (3.1) Tr | XY | r = min Z ∈B ( H ) × (cid:26) r r Tr | XZ | r + r r Tr | Z − Y | r (cid:27) , and (3.2) Tr | XY | r = max Z ∈B ( H ) × (cid:26) r r Tr | XZ | r − r r Tr | Y − Z | r (cid:27) . Proof.
For any p > k · k p as k A k pp := Tr | A | p . For any Z ∈ B ( H ) × , we haveby H¨older’s inequality thatTr | XY | r ≤ k XZ k r r k Z − Y k r r = [Tr | XZ | r ] r r [Tr | Z − Y | r ] r r . For a proof of H¨older’s inequality, see [Bha97, Exercise IV.2.7]. Actually it is aspecial case of [Bha97, Exercise IV.2.7] by choosing the unitarily invariant norm |||·||| to be k · k . And [Bha97, Exercise IV.2.7] can be proved by almost the sameargument as the proof of [Bha97, Corollary IV.2.6], since [Bha97, Theorem IV.2.5]is valid for all r > x α y β ≤ αx + βy for positive x, y and positive α, β such that α + β = 1, it follows that(3.3) Tr | XY | r ≤ [Tr | XZ | r ] r r [Tr | Z − Y | r ] r r ≤ r r Tr | XZ | r + r r Tr | Z − Y | r . By exchanging Y and Z , we have(3.4) Tr | XY | r ≥ r r Tr | XZ | r − r r Tr | Y − Z | r . In view of (3.3), to prove (3.1) it suffices to find a minimizer. For this let Y ∗ X ∗ = U | Y ∗ X ∗ | be the polar decomposition of Y ∗ X ∗ , then XY U = | Y ∗ X ∗ | . Set Z := Y U | Y ∗ X ∗ | − r r r , then we have XZ = XY U | Y ∗ X ∗ | − r r r = | Y ∗ X ∗ | r r r , Z − Y = | Y ∗ X ∗ | r r r U ∗ . Using the facts that k · k p is unitarily invariant and k A k p = k A ∗ k p for all A , wehave Tr | XZ | r = Tr | Y ∗ X ∗ | r r r r = Tr | XY | r r r r = Tr | XY | r , and Tr | Z − Y | r = Tr | Y ∗ X ∗ | r r r r = Tr | XY | r r r r = Tr | XY | r . Hence Tr | XY | r = r r Tr | XZ | r + r r Tr | Z − Y | r , which proves (3.1). ROM WIGNER-YANASE-DYSON CONJECTURE TO CARLEN-FRANK-LIEB CONJECTURE9
In view of (3.4), to prove (3.2) it suffices to find a maximizer. For this let U beas above and choose Z to be Y U | Y ∗ X ∗ | r r , then XZ = XY U | Y ∗ X ∗ | r r = | Y ∗ X ∗ | r r r , Y − Z = U | Y ∗ X ∗ | r r . It follows thatTr | XZ | r = Tr | Y ∗ X ∗ | ( r r r r = Tr | Y ∗ X ∗ | r = Tr | XY | r , and Tr | Y − Z | r = Tr | Y ∗ X ∗ | r = Tr | XY | r . Hence Tr | XY | r = r r Tr | XZ | r − r r Tr | Y − Z | r and the proof of (3.2) is finished. (cid:3) Remark . It is possible to generalize this variational method to the infinitedimensional case or to more general norm functions, which is beyond the aim ofthis paper. It is also possible to apply this variational method to trace functionswith n ≥ r j > , j = 0 , , . . . , n such that r = P nj =1 1 r j . Then wehave for X , . . . , X n ∈ B ( H ) × thatTr | X · · · X n | r = min r r Tr | X Z | r + n − X j =2 r r j Tr | Z − j − X j Z j | r j + r r n Tr | Z − n − X n | r n , (3.5)and Tr | X · · · X n | r = max r r Tr | X Z | r − n − X j =2 r r j Tr | Z − j X − j Z j − | r j − r r n Tr | X − n Z n − | r n , (3.6)where min and max run over all Z , . . . , Z n − ∈ B ( H ) × . The proof is similar to thetwo variables case. We only explain here that min is indeed achieved for (3.5). Let X ∗ n · · · X ∗ = U | X ∗ n · · · X ∗ | be the polar decomposition of X ∗ n · · · X ∗ . Then set Z j := X j +1 · · · X n U | X ∗ n · · · X ∗ | α j , α j = j X k =1 r r k − ≤ j ≤ n −
1. One can check thatTr | X · · · X n | r = r r Tr | X Z | r + n − X j =2 r r j Tr | Z − j − X j Z j | r j + r r n Tr | Z − n − X n | r n . Now we are ready to proceed with the three steps of reductions. Note that
Step1 is enough to finish the proof of Theorem 1.1 and confirm Conjectures 1 and 2.
Step 1:
In the first step we reduce the joint convexity/concavity of Ψ p,q,s to theconvexity/concavity ofΥ p,s ( A ) := Tr( K ∗ A p K ) s , A ∈ P ( H ) × , for all K ∈ B ( H ) × , which has already been thoroughly studied. Theorem 3.5. [CFL18, Proposition 5]
For any K ∈ B ( H ) × , Υ p,s is(1) concave if < p ≤ and < s ≤ p ;(2) convex if − ≤ p ≤ and s > ;(3) convex if ≤ p ≤ and s ≥ p . See the discussions after Proposition 5 in [CFL18] for more historical information.We only comment here that the proof of concavity for 0 < p ≤ s = p is dueto Epstein [Eps73]. His analytic method is nowadays developed as an importanttool in matrix analysis, in particular to deal with concavity (rather than convexity)of trace functions. We will give a simpler proof of this theorem later, without usingEpstein’s analytic approach. Proof of Theorem 1.1 given Theorem 3.5.
Before proceeding with the proof notefirst that Ψ p,q,s ( A, B ) = Tr( B q K ∗ A p KB q ) s = Tr | A p KB q | s . (1) If q = 0, then the claim reduces to Theorem 3.5 (1). To show the case0 < q ≤ p ≤ < s ≤ p + q , set λ := s ( p + q ) ∈ (0 ,
1] and we apply (3.1) to( r , r , r ) = (2 s, λp , λq ) and ( X, Y ) = ( A p K, B q ):(3.7) Ψ p,q,s ( A, B ) = min Z ∈B ( H ) × (cid:26) pp + q Tr | A p KZ | λp + qp + q Tr | Z − B q | λq (cid:27) . Since 0 < λp ≤ p and 0 < λq ≤ q , from Theorem 3.5 (1) it follows that the maps A pp + q Tr | A p KZ | λp = pp + q Tr( Z ∗ K ∗ A p KZ ) λp and B qp + q Tr | Z − B q | λq = qp + q Tr( Z − B q ( Z − ) ∗ ) λq are both concave. Hence they are both jointly concave as functions in ( A, B ) andso is Ψ p,q,s by Lemma 3.2 (1) and (3.7).(2) If p = 0, then the claim reduces to Theorem 3.5 (2). Suppose − ≤ q ≤ p < s >
0, then we apply (3.2) to ( r , r , r ) = (2 t, s, − q ) with t = s − q and( X, Y ) = ( A p K, B q ):(3.8) Ψ p,q,s ( A, B ) = max Z ∈B ( H ) × n st Tr | A p KZ | t + sq Tr | B − q Z | − q o . Note that t > sq < < − q ≤
1. By Theorem 3.5 (1) and (2), the maps A st Tr | A p KZ | t = st Tr( Z ∗ K ∗ A p KZ ) t and B sq Tr | B − q Z | − q = sq Tr( Z ∗ B − q Z ) − q are both convex. Hence they are both jointly convex as functions in ( A, B ) and sois Ψ p,q,s by Lemma 3.2 (1) and (3.8).(3) If q = 0, then the claim reduces to Theorem 3.5 (3). Suppose − ≤ q < , ≤ p ≤ , ( p, q ) = (1 , −
1) and s ≥ p + q , then we apply (3.2) to ( r , r , r ) = (2 t, s, − q )with t = s − q and ( X, Y ) = ( A p K, B q ):(3.9) Ψ p,q,s ( A, B ) = max Z ∈B ( H ) × n st Tr | A p KZ | t + sq Tr | B − q Z | − q o . Since sq <
0, 0 < − q ≤ t = s − − q ≥ p , we have by Theorem 3.5 (1) and (3)that the maps A st Tr | A p KZ | t = st Tr( Z ∗ K ∗ A p KZ ) t and B sq Tr | B − q Z | − q = sq Tr( Z ∗ B − q Z ) − q are both convex. Hence they are both jointly convex as functions in ( A, B ) and sois Ψ p,q,s by Lemma 3.2 (1) and (3.9). (cid:3)
ROM WIGNER-YANASE-DYSON CONJECTURE TO CARLEN-FRANK-LIEB CONJECTURE11
Remark . One can understand this step of reduction in the following heuristicway. In Figure 1.1, the green region [0 , × [0 ,
1] is generated by two intervals ofthe p -axis and the q -axis: [0 , × { } and { } × [0 , p,q,s (Theorem 1.1 (1)) from the concavity of Υ p,s (Theorem3.5 (1)) in the above proof. The proof of the yellow region of the Figure 1.1 can beunderstood in a similar way. Step 2:
In our second step we reduce Theorem 3.5 to three particular cases.
Theorem 3.7. [Eps73, Hia13, CL08]
Fix K ∈ B ( H ) × , then(1) Υ p, /p is concave when < p ≤ (Epstein);(2) Υ p,s is convex when − ≤ p < and < s ≤ (Hiai);(3) Υ p, /p is convex when ≤ p ≤ (Carlen-Lieb).Proof of Theorem 3.5 given 3.7. Indeed, when 0 < p ≤
1, 0 < s < p and s = p + t ,by applying (3.1) to ( r , r , r ) = (2 s, p , t ) and ( X, Y ) = ( A p , K ) we obtain thatTr( K ∗ A p K ) s = min Z ∈B ( H ) × n sp Tr( Z ∗ A p Z ) p + st Tr( K ∗ ( Z − ) ∗ Z − K ) t o . Then by Lemma 3.2 (1), the concavity of Υ p, /p implies the concavity of Υ p,s .When − ≤ p < s >
1, by applying (3.2) to ( r , r , r ) = (2 , s, ss − ) and( X, Y ) = ( A p , K ) we obtain thatTr( K ∗ A p K ) s = max Z ∈B ( H ) × (cid:8) s Tr Z ∗ A p Z − ( s − Z ∗ ( K − ) ∗ K − Z ) ss − (cid:9) . Then by Lemma 3.2 (1), the convexity of Υ p, implies the convexity of Υ p,s .When 1 ≤ p ≤ s > p and p = s + t , by applying (3.2) to ( r , r , r ) =( p , s, t ) and ( X, Y ) = ( A p , K ) we obtain thatTr( K ∗ A p K ) s = max Z ∈B ( H ) × n sp Tr( Z ∗ A p Z ) p − st Tr( Z ∗ ( K − ) ∗ K − Z ) t o . Then by Lemma 3.2 (1), the convexity of Υ p, /p implies the convexity of Υ p,s . (cid:3) Step 3:
In the last step we reduce Theorem 3.7 to Lemma 3.1.
Proof of Theorem 3.7 given Lemma 3.1.
The proof is inspired by the proof of (2)in [CFL18]. Let us recall it first. If s = 1, the convexity of Υ p, follows from theoperator convexity of A A p for − ≤ p <
0. If 0 < s <
1, by applying (3.1) to( r , r , r ) = (2 s, , s − s ) and ( X, Y ) = ( A p K, I ), we haveTr( K ∗ A p K ) s = min Z ∈B ( H ) × n s Tr | A p KZ | + (1 − s )Tr | Z − | s − s o = min Z ∈P ( H ) × (cid:8) s Tr K ∗ A p KZ + (1 − s )Tr Z ss − (cid:9) = min Z ∈P ( H ) × n s Tr K ∗ A p KZ − p + (1 − s )Tr Z s (1 − p ) s − o . Since s (1 − p ) s − <
0, the function t t s (1 − p ) s − is convex. Thus Z Tr Z s (1 − p ) s − is convex(see for example [Car10, Theorem 2.10]). This, together with Ando’s convexityresult (Lemma 3.1 (2)) and Lemma 3.2 (2), yields the convexity of Υ p,s . Now we prove (1). There is nothing to prove when p = 1. For 0 < p <
1, byapplying (3.2) to ( r , r , r ) = (2 , p , − p ) and ( X, Y ) = ( A p K, I ), we haveTr( K ∗ A p K ) p = max Z ∈B ( H ) × (cid:26) p Tr | A p KZ | − − pp Tr | Z | − p (cid:27) = max Z ∈P ( H ) × (cid:26) p Tr K ∗ A p KZ − − pp Tr Z − p (cid:27) = max Z ∈P ( H ) × (cid:26) p Tr K ∗ A p KZ − p − − pp Tr Z (cid:27) . Then by Lieb’s concavity result (Lemma 3.1 (1)) and Lemma 3.2 (2), Υ p, /p isconcave.(3) can be shown similarly. Indeed, the case p = 1 is trivial. For 1 < p ≤
2, byapplying (3.1) to ( r , r , r ) = ( p , , p − ) and ( X, Y ) = ( A p K, I ), we haveTr( K ∗ A p K ) p = min Z ∈B ( H ) × (cid:26) p Tr | A p KZ | + p − p Tr | Z − | p − (cid:27) = min Z ∈P ( H ) × (cid:26) p Tr K ∗ A p KZ + p − p Tr Z − p (cid:27) = min Z ∈P ( H ) × (cid:26) p Tr K ∗ A p KZ − p + p − p Tr Z (cid:27) . Then by Ando’s convexity result (Lemma 3.1 (2)) and Lemma 3.2 (2), Υ p, /p isconvex. (cid:3) Remark . Although the variational methods (3.1) and (3.2) admit analogues(3.5) and (3.6) of n ( ≥
3) variables, the joint convexity/concavity of P ( H ) × × · · · × P ( H ) × ∋ ( A , . . . , A n ) Tr( A pn n K ∗ n − · · · K ∗ A p K · · · K n − A pn n ) s can not be derived directly from Theorem 3.5 because of the appearance of theterm Tr | Z − j − X j Z j | r j . For example, we haveTr | X X X | r = min Z ,Z ∈B ( H ) × (cid:26) r r Tr | X Z | r + r r Tr | Z − X Z | r + r r Tr | Z − X | r (cid:27) . (3.10)To obtain the joint concavity of P ( H ) × × P ( H ) × P ( H ) × ∋ ( A , A , A ) Tr( A p K ∗ A p K ∗ A p K A p K A p ) s , via the variational method (3.10), the concavity of the function of the form P ( H ) × ∋ A Tr | Y A p Y | r = Tr( Y ∗ A p Y ∗ Y A p Y ) r is required. Unfortunately, little is known for general Y ∗ Y = I . Indeed, Carlen,Frank and Lieb proved that [CFL16, Corollary 3.3] for p, q, r ∈ R \ { } , the function( A, B, C ) Tr C r B q A p B q C r is never concave, and it is convex if and only if q = 2 , p, r < − ≤ p + r < ROM WIGNER-YANASE-DYSON CONJECTURE TO CARLEN-FRANK-LIEB CONJECTURE13
Acknowledgement.
The author would like to thank Quanhua Xu, Adam Skalski,Ke Li and Zhi Yin for their valuable comments. He also would like to thank theanonymous referees for pointing out some errors in an earlier version of this pa-per and for helpful comments and suggestions that make this paper better. Theresearch was partially supported by the NCN (National Centre of Science) grant2014/14/E/ST1/00525, the French project ISITE-BFC (contract ANR-15-IDEX-03), NSFC No. 11826012, and the European Union’s Horizon 2020 research andinnovation programme under the Marie Sk lodowska-Curie grant agreement No.754411.
References [AD15] K. M. R. Audenaert and N. Datta. α - z -R´enyi relative entropies. J. Math. Phys. ,56(2):022202, 16, 2015.[And79] T. Ando. Concavity of certain maps on positive definite matrices and applications toHadamard products.
Linear Algebra Appl. , 26:203–241, 1979.[Bek04] T. N. Bekjan. On joint convexity of trace functions.
Linear algebra and its applica-tions , 390:321–327, 2004.[Bha97] R. Bhatia.
Matrix analysis , volume 169 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1997.[Car10] E. A. Carlen. Trace inequalities and quantum entropy: an introductory course.
En-tropy and the quantum , 529:73–140, 2010.[CFL16] E. A. Carlen, R. L. Frank, and E. H. Lieb. Some operator and trace function convexitytheorems.
Linear Algebra Appl. , 490:174–185, 2016.[CFL18] E. A. Carlen, R. L. Frank, and E. H. Lieb. Inequalities for quantum divergences andthe Audenaert–Datta conjecture.
Journal of Physics A: Mathematical and Theoret-ical , 51(48):483001, 2018.[CL99] E. A. Carlen and E. H. Lieb. A Minkowski type trace inequality and strong subaddi-tivity of quantum entropy. In
Differential operators and spectral theory , volume 189of
Amer. Math. Soc. Transl. Ser. 2 , pages 59–68. Amer. Math. Soc., Providence, RI,1999.[CL08] E. A. Carlen and E. H. Lieb. A Minkowski type trace inequality and strong subaddi-tivity of quantum entropy. II. Convexity and concavity.
Lett. Math. Phys. , 83(2):107–126, 2008.[CL18] E. A. Carlen and E. H. Lieb. Some trace inequalities for exponential and logarithmicfunctions.
Bulletin of Mathematical Sciences , pages 1–40, 2018.[Eps73] H. Epstein. Remarks on two theorems of E. Lieb.
Comm. Math. Phys. , 31:317–325,1973.[FL13] R. L. Frank and E. H. Lieb. Monotonicity of a relative R´enyi entropy.
J. Math. Phys. ,54(12):122201, 5, 2013.[Hia13] F. Hiai. Concavity of certain matrix trace and norm functions.
Linear Algebra Appl. ,439(5):1568–1589, 2013.[Hia16] F. Hiai. Concavity of certain matrix trace and norm functions. II.
Linear AlgebraAppl. , 496:193–220, 2016.[Hol73] A. S. Holevo. Bounds for the quantity of information transmitted by a quantumcommunication channel.
Problemy Peredachi Informatsii , 9:3–11, 1973.[JOPP12] V. Jaksic, Y. Ogata, Y. Pautrat, and C. Pillet. Entropic fluctuations in quantum sta-tistical mechanics. an introduction. In
Quantum Theory from Small to Large Scales:Lecture Notes of the Les Houches Summer School: Volume 95, August 2010 . OxfordUniversity Press, 2012.[Lie73] E. H. Lieb. Convex trace functions and the Wigner-Yanase-Dyson conjecture.
Ad-vances in Math. , 11:267–288, 1973.[Lin75] G. Lindblad. Completely positive maps and entropy inequalities.
Comm. Math.Phys. , 40:147–151, 1975.[LR73] E. H. Lieb and M. B. Ruskai. Proof of the strong subadditivity of quantum-mechanicalentropy.
J. Math. Phys. , 14:1938–1941, 1973.[MLDS +
13] M. M¨uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel. On quan-tum R´enyi entropies: a new generalization and some properties.
J. Math. Phys. ,54(12):122203, 20, 2013. [MO15] M. Mosonyi and T. Ogawa. Quantum hypothesis testing and the operational inter-pretation of the quantum R´enyi relative entropies.
Comm. Math. Phys. , 334(3):1617–1648, 2015.[NEE13] I. Nikoufar, A. Ebadian, and G. M. Eshaghi. The simplest proof of Lieb concavitytheorem.
Adv. Math. , 248:531–533, 2013.[R´en61] A. R´enyi. On measures of entropy and information. In
Proc. 4th Berkeley Sympos.Math. Statist. and Prob., Vol. I , pages 547–561. Univ. California Press, Berkeley,Calif., 1961.[Sti55] W. F. Stinespring. Positive functions on C ∗ -algebras. Proc. Amer. Math. Soc. , 6:211–216, 1955.[TR11] M. Tomamichel and R. Renner. Uncertainty relation for smooth entropies.
Phys. Rev.Lett. , 106(11):110506, 2011.[Uhl73] A. Uhlmann. Endlich-dimensionale Dichtematrizen. II.
Wiss. Z. Karl-Marx-Univ.Leipzig Math.-Natur. Reihe , 22:139–177, 1973.[Ume62] H. Umegaki. Conditional expectation in an operator algebra. IV. Entropy and infor-mation.
Kodai Math. Sem. Rep. , 14:59–85, 1962.[WWY14] M. M. Wilde, A. Winter, and D. Yang. Strong converse for the classical capacityof entanglement-breaking and Hadamard channels via a sandwiched R´enyi relativeentropy.
Comm. Math. Phys. , 331(2):593–622, 2014.[WY63] E. P. Wigner and M. M. Yanase. Information contents of distributions.
Proc. Nat.Acad. Sci. U.S.A. , 49:910–918, 1963.
Laboratoire de Math´ematiques, Universit´e Bourgogne Franche-Comt´e, 25030 Besanc¸on,France and Institute of Mathematics, Polish Academy of Sciences, ul. ´Sniadeckich 8,00-656 Warszawa, PolandCurrent address: Institute of Science and Technology Austria (IST Austria), AmCampus 1, 3400 Klosterneuburg, Austria
E-mail address ::