CONCENTRATION OF EIGENVALUE SUMS AND GENERALIZED LIEB’S CONCAVITY THEOREM
5.4 Proof of Concavity Theorems
5.4.3 Revisiting Previous Proofs in the Trace Case
For each j, by the concavity of (5.20) form =1, we have τφ exp(L+rlogA(j)−rlogC(j))
+(1−τ)φ exp(L+rlogB(j)−rlogC(j))
≤ φ exp(L+rlog(τA(j)+(1−τ)B(j))−rlogC(j))
=φ exp(L). Therefore we obtain that
τφ exp(H+
m
X
j=1
pjlogA(j))+ (1−τ)φ exp(H +
m
X
j=1
pjlogB(j))
≤
m
X
j=1
pj
r φ exp(L)
=φ exp(H+
m
X
j=1
pjlogC(j)),
that is, (5.20) is jointly concave on (H++n )×m for allm ≥ 1.
operator concave, and thus its perspective fromH+n ×H+m toH+nm, (A,B)7→ (In⊗B)12 (In⊗ B)−12(A⊗Im)(In⊗ B)−12p
(In⊗B)12 = Ap⊗ B1−p, is jointly operator concave in(A,B). The simplified expression above results from the fact thatA⊗Im commutes withIn⊗ B. For anyK ∈Cn×m, we have the identity (a variant of Ando’s interpretation)
Tr[K∗ApK B1−p]=D
m
X
j=1
(K e(m)j )⊗e(m)j ,Ap⊗ (BT)1−p
m
X
j=1
(K e(m)j )⊗ e(m)j E
Cn⊗Cm, (5.27) whereBT is the transpose of B, ande(m)j = (0, . . . ,j1th, . . . ,0) ∈Cm. Note that since B ∈ H+n, BT is also inH+n. Since B 7→ BT is linear, the joint operator concavity of (A,B)7→ Ap⊗ B1−pthen implies the joint concavity of(A,B) 7→Tr[K∗ApK B1−p].
As an application, Carlen and Lieb [27] applied the Lieb’s concavity theorem to prove the concavity of A 7→ Tr[(K∗ArK)s] for r ∈ [0,1],s ∈ [1, 1r] (they used a slightly different but equivalent expression) based on a variational characterization of this function (the supremum part of [27, Lemma 2.2]). We here provide a simplified proof that captures the main spirit. For any A,B∈H+n, K ∈Cn×n, let
X = (K∗ArK)s, Y = (K∗BrK)s. Then for anyτ ∈[0,1], note that 1s ≤1,r+(1− 1s) ≤1, we have
τTr[X]+(1−τ)Tr[Y]
=τTr
K∗ArK X1−1s +(1−τ)Tr
K∗BrKY1−1s
≤Tr
K∗(τA+(1−τ)B)rK(τX +(1−τ)Y)1−1s
≤Tr
K∗(τA+(1−τ)B)rKs1sTrτX +(1−τ)Y1−1s
=Tr
K∗(τA+(1−τ)B)rKs1s τTr[X]+(1−τ)Tr[Y]1−1s,
(5.28)
where the first inequality is due to Lieb’s concavity theorem with p = r,q = 1− 1s,p+q =r+1− 1s ≤ 1, and the second inequality is Hölder’s. The above then simplifies to
τTr
(K∗ArK)s]+(1−τ)Tr[(K∗BrK)s ≤ Tr
K∗(τA+ (1−τ)B)rKs, which concludes the concavity of A7→Tr[(K∗ArK)s].
The variational characterizations of Tr[(K ArK)s] in [27] can be abstracted to the following two formulas ([26, Lemma 12]): for anyX ∈H+n,
Tr[Xs]= sup(
sTr[XY]−(s−1)Tr[Ys−s1] : Y ∈H+n)
, ifs> 1 or s <0, (5.29) Tr[Xs]= inf(
sTr[XY]+(1−s)Tr[Y−1−ss ] : Y ∈H++n )
, if 0 < s < 1. (5.30) Further, these variational formulas were used to derive the convexity/concavity of the function (A,B) 7→ Tr[(Bq2K∗ApK Bq2)s] for a partial range of p,q,s (partial to the necessary conditions on p,q,s for the corresponding convexity/concavity to hold). For example, formula (5.30) was used by Carlen et al. to prove the concavity for 0 ≤ p,q ≤ 1,0 < s ≤ 1+1q [25, Theorem 4.4]. Recently, the above formulas were modified by Zhang to the following [136, Theorem 3.3]: for any X,Y ∈ Cn×n and anyr0,r1,r2> 0 such that r10 = r11 + r12,
Tr[|XY|r1]= sup (r1
r0Tr[|X Z|r0]− r1
r2Tr[|Y−1Z|r2] : Z ∈Cn×n )
, (5.31)
Tr[|XY|r0]= inf (r0
r1Tr[|X Z|r1]+ r0
r2Tr[|Z−1Y|r2] : Z ∈Cn×ninvertible )
. (5.32) Zhang then used them to provide a unified variational proof of the joint convex- ity/concavity of (A,B) 7→ Tr[(Bq2K∗ApK Bq2)s] for the full range of p,q,s, finally confirming that the sufficient conditions on p,q,scoincide with the necessary con- ditions.
These arguments using matrix tensors and variational forms were also adopted by Tropp [122] to provide an alternative proof of the concavity of A 7→ Tr[exp(H + logA)]. Tropp’s proof is based on his variational formula for trace,
Tr[M]= sup
T∈H++n
Tr
TlogM−TlogT +T, M ∈H++n , (5.33) which relies on the non-negativeness of the matrix relative entropy
D(T;M) =Tr
T(logT −logM)−(T− M), T,M ∈H++n .
The non-negativeness ofD(T;M)is a classical result of Klein’s inequality (see Petz [93, Proposition 3], Carlen [24, Theorem 2.11] or Tropp [122, Proposition 8.3.5]).
Tropp substitutedM =exp(H +logA)in (5.33) to obtain Tr[exp(H+logA)]= sup
T∈H++n
Tr[T H]+Tr[A]−D(T;A).
The concavity of A7→ Tr[exp(H+logA)] then follows from this variational expres- sion, the joint convexity ofD(T;A)in(T,A), and the fact thatg(x)= supy∈Ω f(x,y)
is concave in xif f(x,y)is jointly concave in(x,y)andΩis convex (see, e.g., [27, Lemma 2.3]). The joint convexity of the relative entropy D(T;A) was first due to Lindblad [70]. One can also see Ando [3], Carlen [24] and Tropp [122] for alternative proofs.
A Methodology of another flavor arose from the use of complex analysis. In the same year of Lieb’s original paper on his concavity theorem, Epstein [37] provided a unified way of proving the concavity of A7→Tr[K∗ApK Aq], A7→Tr[(K∗AsK)1s] and A 7→ Tr[exp(H +logA)], using a derivative argument based on the theory of Herglotz functions (functions that are analytic in the open upper half planeC+ and have a positive imaginary part). Epstein’s method relies on integral representations of matrix powers, which has a deep connection to a profound theorem of Loewner’s : a real-valued function f on(0,+∞) is operator monotone if and only if it admits an analytic continuation to a Herglotz function. Loewner’s theorem provides a convenient tool for understanding trace functions by passing the study of a desired property from the integral to the integrand that has a relatively simpler form. One may see the book of Donoghue [35] for a full account of this theory. Epstein’s approach was further developed by Hiai [47, 48] and, for example, adopted in his first proof of the concavity of (A,B) 7→ Tr[(Bq2K∗ApK Bq2)s] for the full range 0 ≤ p,q ≤ 1,0 ≤ s ≤ p+1q [48, Theorem 2.1]. Specifically, by substituting σ = s(p+q) < 1,X = (Bq2K∗ApK Bq2)p1+q in the integral formula,
Xσ = sin(πσ) π
Z ∞
0 t−1+σ(In+t X−1)−1dt, 0 < σ <1, X ∈H++n , (5.34) Hiai passed the joint concavity of (A,B) 7→ Tr[(Bq2K∗ApK Bq2)s] to the joint con- cavity of (A,B) 7→ Trf
1+t(Bq2K∗ApK Bq2)−p1+qg−1
(the case s(p +q) = 1 was handled differently by directly takingt →+∞). He then proved the latter using the derivative argument introduced by Epstein.
The introduction of complex analysis into the field has also led to another branch of methods based on interpolation theories. In his original proof of the concavity of (A,B) 7→ Tr[K∗ApK Bq] for 0 ≤ p,q ≤ 1,p+ q ≤ 1 [67, Theorem 1], Lieb made use of the maximum modulus principle to concentrate the powers of Ap,Bq to the only powerp+qon Aor B, and then proceeded with the operator concavity of X 7→ Xp+q. This technique, relying on the holomorphicity of Xz (X ∈ H+n) as a function of z, already shed some light on the use of complex interpolation theo- ries. Later, Uhlmann [124] applied interpolation theories explicitly to again prove Lieb’s concavity theorem, by interpreting Tr[K∗ApK B1−p] as an interpolation be-
tween Tr[K AK] and Tr[K K B]. Uhlmann’s quadratic interpolation of seminorms extended the relevant works of Lieb to positive linear forms of arbitrary *-algebras.
Kosaki [58] further explored the idea of quadratic interpolation of seminorms and captured Lieb’s concavity theorem in the frame of general interpolation theories.
The above methodologies all have a unique perspective in understanding complicated trace functions. However, some of them are found hardly generalizable tok-trace, as they more or less rely on the linearity of the normal trace. For example, Ando’s identity (5.27) and Tropp’s variational formula (5.33). Our k-trace function φ for k > 1 is at best sub-additive since it is concave and homogeneous of order 1. Hiai’s use of the integral formula also lives on linearity in that the trace operation can be pulled into the integral. Though we can interpret the k-trace as the trace of an anti-symmetric tensor, the power of 1k will have to stay out of the integral, and the anti-symmetric tensor in the integrand will also bring huge difficulties to the derivative argument that comes after.
The variational method introduced by Carlen and Lieb needs the linearity of trace as well in proving the concavity ofA7→Tr[(K∗ArK)s]. One can see this in the last equality of (5.28), which will become an inequality in the undesirable direction if Tr[·] is replaced by Trk[·]1k, since Trk[X]1k is concave in X. In fact, the argument in (5.28) that reduces the concavity of A7→ Tr[(K∗ArK)s] forr ∈ [0,1],s ∈[1, 1r] to the joint concavity of (A,B) 7→ Tr[K∗ApK Bq] for p,q ∈ [0,1],p+q ≤ 1 was performed in a variational manner in [27] based on the following formula (which is equivalent to the expression of the supremum case in [27, Lemma 2.2]):
Tr[(K∗ArK)s]= sup
X∈H++n
sTr[K∗ArK X1−1s]−(s−1)Tr[X]
= sup
X∈H++n
Ψ(A,X), (5.35) which can be derived using Hölder’s inequality for trace. Recall that g(x) = supy∈Ω f(x,y)is concave inxif f(x,y)is jointly concave in(x,y)andΩis convex.
Since Tr[K∗ArK X1−1s] is jointly concave in(A,X) (asr+(1− 1s) ≤ 1) and Tr[X] is linear in X, the functionΨ(A,X) is jointly concave in(A,X). The concavity of A7→Tr[(K∗ArK)s] then follows from (5.35). It is then natural to consider a similar variational formula for thek-trace that can also be shown by Hölder’s inequality:
Trk[(K∗ArK)s]1k = sup
X∈H++n
sTrk[K∗ArK X1−1s]1k −(s−1)Trk[X]1k
= sup
X∈H++n
Ψk(A,X). (5.36)
Note that for s > 1, we have −(s −1) < 0 and thus −(s−1)Trk[X]k1 is convex in X since Trk[·]k1 is concave (this sign of −(s −1) does not give trouble in the trace case due to the linearity of trace). Therefore, even provided that (A,X) 7→
Trk[K∗ArK X1−1s]k1 is jointly concave, the functionΨk(A,X)is not guaranteed to be jointly concave in(A,X), and the variational formula in (5.36) fails to conclude the concavity of A7→Trk[(K∗ArK)s]1k. As a consequence, we have not found a way to adapt this particular argument into a proof of Lemma 5.3.2.
However, these variational approaches, especially Zhang’s variational characteriza- tions, are conveniently applicable to the derivation of the more general cases given Lemma 5.3.2, as we have seen in the proof of Theorem 5.3.3. One can see that the k-trace version of Hölder’s inequality plays an essential role in the process, which has suggested us to employ complex interpolation theories in the first place as interpolation of operators is based essentially on Hölder’s inequality. In particular, we found the operator interpolation technique (Theorem 5.6.7) developed by Stein [116](1956) nicely compatible to our problem. One can derive a variety of interpola- tion inequalities systematically by choosingG(z)properly in inequality (5.62). The choice ofG(z) we make, inspired by Lieb’s original construction, gives the proof of our Lemma 5.3.2. We remark that, Lemma 5.6.8 does not only apply to trace or k-trace, but also to any continuous matrix functionφ:H+n → [0,+∞) that satisfies Hölder’s inequality and is unitary invariant in the sense thatφ(U∗XU) = φ(X) for arbitrary X ∈ H+n and U ∈ Cn×n unitary. In fact, these two properties suffice to imply that log◦φ◦exp is convex onHn, which, along with a canonical majorization argument, will yield the inequality (5.63) (see, e.g., [50]).