Mathematical Programming manuscript No. (will be inserted by the editor) Operator convexity along lines, self-concordance, and sandwiched Rényi entropies Kerry He · James Saunderson · Hamza Fawzi Received: date / Accepted: date Abstract Barrier methods play a central role in the theory and practice of convex optimization. One of the most general and successful analyses of bar- rier methods for convex optimization, due to Nesterov and Nemirovskii, relies on the notion of self-concordance. While an extremely powerful concept, prov- ing self-concordance of barrier functions can be very difficult. In this paper we give a simple way to verify that the natural logarithmic barrier of a con- vex nonlinear constraint is self-concordant via the theory of operator convex functions. Namely, we show that if a convex function is operator convex along any one-dimensional restriction, then the natural logarithmic barrier of its epi- graph is self-concordant. We apply this technique to construct self-concordant barriers for the epigraphs of functions arising in quantum information the- ory. Notably, we apply this to the sandwiched Rényi entropy function, for which no self-concordant barrier was known before. Additionally, we utilize our sufficient condition to provide simplified proofs for previously established self-concordance results for the noncommutative perspective of operator con- vex functions. An implementation of the convex cones considered in this paper is now available in our open source interior-point solver QICS. Keywords Operator convexity · Self-concordant barrier · Interior-point methods · Sandwiched Rényi entropy Kerry He E-mail: kerry.he1@monash.edu Department of Electrical and Computer Systems Engineering, Monash University, Clayton VIC 3800, Australia James Saunderson E-mail: james.saunderson@monash.edu Department of Electrical and Computer Systems Engineering, Monash University, Clayton VIC 3800, Australia Hamza Fawzi E-mail: h.fawzi@damtp.cam.ac.uk Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom. https://github.com/kerry-he/qics 2 Kerry He et al. Mathematics Subject Classification (2020) 90C25 · 90C51 · 90C90 · 81P17 1 Introduction Let f : Rn → R be a convex function. Convex optimization problems involving this function can often be expressed in terms of its epigraph, i.e., epi f := {(t, x) ∈ R× dom f : t ≥ f(x)}. If we have a self-concordant barrier for this set, then this allows us to incor- porate the function f into the broader Nesterov-Nemirovskii framework for interior-point methods [35]. However, constructing an efficiently computable self-concordant barrier for a set is not always straightforward. If a function G is a self-concordant barrier for the domain of f , then it would be natural to hope that a self-concordant barrier for the epigraph of f is (t, x) 7→ − log(t− f(x)) +G(x). (1) However, this is not true in general, e.g., when f(x) = ex (see [35, Proposition 5.3.3] for an actual self-concordant barrier for the epigraph of the exponen- tial). Moreover, even in cases where (1) is self-concordant, proving this can be difficult. For example, functions arising in quantum information theory, such as the sandwiched Rényi entropy of the title, are often spectral functions of multiple Hermitian matrices. Although it is possible to obtain explicit ex- pressions for the derivatives of these functions (see, e.g., Section 5.1), they typically depend, in a complicated way, on the eigendecompositions of the matrices involved. This makes it challenging to prove self-concordance of (1), which requires uniformly bounding the third derivative in terms of the second derivative. In this paper, instead of proving self-concordance by directly working with the derivatives of the barrier function, we instead relate self-concordance to operator convexity, i.e., univariate functions that are convex with respect to the Loewner order when extended to spectral functions of Hermitian matrices (see Section 2.1 for a precise definition). In particular, we show that if a function, when restricted to any line within its domain, is operator convex, then the natural logarithmic barrier (1) is self-concordant. This allows us to prove self- concordance by instead using tools from the rich literature of operator convex functions. Below, we summarize these main ideas. We first formalize the definition of being operator convex along lines, then state a theorem relating operator convexity along lines with self-concordant barriers. Note that this theorem is a simplification of the main technical result of our paper which we present later in Theorem 4. Definition 1 Let V be a finite-dimensional real inner product space, and f : dom f → R be a function with open convex domain dom f ⊂ V. We say Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 3 that f is operator convex along lines if for all x ∈ dom f and h ∈ V such that x± h ∈ cl dom f , the function t 7→ f(x+ th), is operator convex on (−1, 1). We say that f is operator concave along lines if −f is operator convex along lines. Remark 1 Given a function f : dom f → R with open convex domain dom f ⊂ V, an alternative way to define operator convexity along lines would be to require that for all x ∈ dom f and all h ∈ V, the univariate function t 7→ f(x+ th) is operator convex on the open interval {t ∈ R : x+ th ∈ dom f}. It turns out that this alternative definition is equivalent to Definition 1. We provide a proof for this in Appendix A.1. Theorem 1 Let V be a finite-dimensional real inner product space, and f : dom f → R be a C3 function with open domain dom f ⊂ V. Let G be a ν-self- concordant barrier for cl dom f . If f is operator convex along lines then (t, x) 7→ − log(t− f(x)) +G(x), defined on R× dom f is a (1 + ν)-self-concordant barrier for cl epi f . Proof See Section 3. ⊓⊔ We prove this theorem by showing that if a function is operator convex along lines, then it satisfies another previously known sufficient condition for (1) to be a self-concordant barrier, i.e., it is compatible with respect to its domain, in the sense of Nesterov and Nemirovskii. We provide a more de- tailed background on this concept in Section 2.2. In previous works [15,5,13], it was shown that operator convex functions and their noncommutative per- spectives were compatible with their domains. Our key assumption of being operator convex along lines is a weaker condition which not only generalizes the results from these works, but also allows us to prove compatibility of more complicated expressions. Specifically, we focus on constructing self-concordant barriers for epigraphs (and hypographs) of functions called sandwiched Rényi (quasi-relative) en- tropies. These functions have remarkable convexity and concavity properties, and play a prominent role in quantum information theory. However, until now they have not been amenable to optimization via off-the-shelf interior point methods. We discuss these functions in more detail in the following section. 1.1 Sandwiched Rényi entropies Consider the following trace function, sometimes referred to as the sandwiched α-quasi-relative entropy, Ψα(X,Y ) := tr [( Y 1−α 2α XY 1−α 2α )α] , (2) 4 Kerry He et al. Table 1 Summary of the convexity properties of the sandwiched quasi-relative entropy Ψα (see (2)) and sandwiched Rényi entropy Dα (see (3)) for various ranges of α, and whether an explicit self-concordant barrier function is known for each function. α Function Convexity Operator convexity along lines Explicit self-concordant barrier known [ 1 2 , 1) Ψα concave concave yes Dα convex convex yes α → 1 Ψα linear linear yes Dα convex convex yes [13] (1, 2] Ψα convex convex yes Dα neither neither n/a (2,∞) Ψα convex neither no Dα neither neither n/a defined on Hn ++×Hn ++, i.e., pairs of positive definite n×n Hermitian matrices. The sandwiched α-quasi-relative entropy is jointly concave for α ∈ [1/2, 1] and jointly convex for α ∈ [1,∞) (see, e.g., [16]). Note that Ψα is neither convex nor concave for α ∈ (0, 1 2 ) (see Appendix B). The sandwiched quasi-relative entropy is used to define the sandwiched α-Rényi entropy [32,40] Dα(X ∥Y ) := 1 α− 1 log(Ψα(X,Y )), (3) where α ∈ [ 12 , 1)∪ (1,∞). These sandwiched Rényi entropies are used to quan- tify how dissimilar two quantum states are, and arise in applications such as quantum hypothesis testing [31] and quantum cryptography [7]. The sand- wiched Rényi entropy Dα is jointly concave for α ∈ [1/2, 1) and is (jointly) quasi-convex (but neither convex nor concave) for α ∈ (1,∞). If we wish to minimize the sandwiched Rényi entropy, due to monotonicity of the logarithm, it suffices to maximize the sandwiched quasi-relative entropy for α ∈ [1/2, 1), or minimize the sandwiched quasi-relative entropy for α ∈ (1,∞). Therefore it suffices to develop efficient optimization techniques to minimize or maximize Ψα for appropriate corresponding ranges of α. Optimizing Rényi entropies Currently, there is a lack of efficient optimization techniques available to optimize the sandwiched Rényi entropy. A first-order method known as entropic mirror descent was proposed in [41] to minimize these functions. However the algorithm is not guaranteed to converge to the optimal solution, only to a neighborhood around it. For some choices of α, there are relatively well-known techniques to op- timize the sandwiched Rényi entropy. When α = 1 2 , the sandwiched quasi- relative entropy Ψ1/2 is equal to the square root of the fidelity function F (X,Y ) = ∥ √ X √ Y ∥21 (where ∥·∥1 denotes the trace norm), which has a well-known Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 5 semidefinite programming representation [39] given by ∥ √ X √ Y ∥1 = min Z∈Cn×n 1 2 tr[Z + Z∗] subj. to [ X Z Z∗ Y ] ⪰ 0. When α → 1, the (normalized) sandwiched Rényi entropy converges to the (normalized) quantum relative entropy, i.e., lim α→1 Dα(tr[X];X ∥Y ) = D1(X ∥Y ), where Dα is the perspective of the sandwiched Rényi entropy (see (4) for a precise definition), and D1(X ∥Y ) = tr[X log(X)]− tr[X log(Y )], is the (Umegaki) quantum relative entropy, which is known to be jointly con- vex [9]. A self-concordant barrier was given for this function in [13], allowing us to solve optimization problems involving the quantum relative entropy us- ing interior-point methods, see, e.g., [19,20,29,6]. Alternatively, it was shown in [14] and [12] that the quantum relative entropy could be approximated using linear matrix inequalities, and could therefore be optimized using semidefinite programming software. Main results Using Theorem 1, we show that the natural logarithmic barrier functions for the hypographs of Ψα for α ∈ [ 12 , 1] and epigraphs of Ψα for α ∈ [1, 2] are self-concordant with optimal barrier parameter, as summarized below (see Section 2.2 for terminology related to self-concordant barriers). Theorem 2 Let n be any positive integer. (i) If α ∈ [ 12 , 1], then Ψα is operator concave along lines and the function (t,X, Y ) ∈ R×Hn ++×Hn ++ 7→ − log(Ψα(X,Y )−t)−log det(X)−log det(Y ), is a (1 + 2n)-logarithmically homogeneous self-concordant barrier for cl hypoΨα = cl{(t,X, Y ) ∈ R×Hn ++ ×Hn ++ : t ≤ Ψα(X,Y )}. (ii) If α ∈ [1, 2], then Ψα is operator convex along lines and the function (t,X, Y ) ∈ R×Hn ++×Hn ++ 7→ − log(t−Ψα(X,Y ))−log det(X)−log det(Y ), is a (1 + 2n)-logarithmically homogeneous self-concordant barrier for cl epiΨα = cl{(t,X, Y ) ∈ R×Hn ++ ×Hn ++ : t ≥ Ψα(X,Y )}. Moreover, these barriers are optimal in the sense that any self-concordant bar- rier for cl hypoΨα when α ∈ [ 12 , 1] and cl epiΨα when α ∈ [1, 2] has parameter at least 1 + 2n. Proof See Section 4. ⊓⊔ 6 Kerry He et al. For α ∈ (2,∞) the function Ψα is not operator convex along lines since x 7→ xα is not operator convex for α ∈ (2,∞) [23]. Therefore, for this range of α the methods developed in this paper do not give a self-concordant barrier for cl epiΨα. We briefly discuss this range of α further in Section 6. If α ∈ [ 12 , 1), the sandwiched Rényi entropyDα is convex. In this setting, we can directly give a self-concordant barrier for the (conic hull of the) epigraph of Dα, i.e., the epigraph of the perspective of the sandwiched Rényi entropy Dα(u;X ∥Y ) := uDα(u −1X ∥u−1Y ), (4) which is defined on R++ ×Hn ++ ×Hn ++. Theorem 3 For any positive integer n and α ∈ [ 12 , 1), Dα is operator convex along lines and the function (t, u,X, Y ) ∈ R× R++ ×Hn ++ ×Hn ++ 7→ − log(t−Dα(u;X ∥Y ))− log(u)− log det(X)− log det(Y ), is a (2 + 2n)-self-concordant barrier for cl epiDα = cl{(t, u,X, Y ) ∈ R× R++ ×Hn ++ ×Hn ++ : t ≥ Dα(u;X ∥Y )}. Moreover, this barrier is optimal in the sense that any self-concordant barrier for cl epiDα when α ∈ [ 12 , 1) has parameter at least 2 + 2n. Proof See Section 4. ⊓⊔ 1.2 Related work Previous results have established self-concordance of barriers for functions related to the sandwiched Rényi entropy, such as the Petz Rényi entropy [36], which is defined as D̂α(X ∥Y ) = log(Ψ̂α(X,Y ))/(α− 1) where Ψ̂α(X,Y ) = tr[XαY 1−α]. The function Ψ̂α is sometimes referred to as the α-quasi-relative entropy. Note that Ψα and Ψ̂α are not equivalent in general, but agree when their matrix arguments commute [40]. The quasi-relative entropy Ψ̂α is jointly concave for α ∈ [0, 1] and jointly convex for α ∈ [−1, 0]∪ [1, 2], results which directly follow from theorems of Lieb [30] and Ando [1]. Like the quantum relative entropy, it was shown in [13] that natural barriers for the epigraphs or hypographs of the quasi-relative entropy Ψ̂α are self-concordant, and therefore optimiza- tion problems minimizing these functions could also be efficiently solved using interior-point methods. Section 4 involves establishing operator concavity along lines for Ψα and (α − 1)Dα for α ∈ [1/2, 1) and operator convexity along lines for Ψα for α ∈ [1, 2]. Our approach builds on techniques that have been developed to prove convexity and concavity results (in the usual sense) for Ψα and related Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 7 trace functions. We comment on three notable existing techniques. First, in the works which originally introduced the sandwiched Rényi entropy [32,40], convexity of Ψα for α ∈ [1, 2] was shown by expressing the function as an appropriate composition between noncommutative perspectives of operator convex functions and positive linear maps. Second, a complex analysis tech- nique based on Epstein’s method [10] was used by Hiai [22,24,25] to prove concavity of a more general class of trace functions. These results specialize to give the concavity of Ψα for α ∈ [ 12 , 1]. Third, a variational technique was used in [16] (see, also, [42]), to prove concavity or convexity of Ψα for the full range α ∈ [ 12 ,∞). 2 Preliminaries We denote the set of n×n Hermitian matrices as Hn with trace inner product ⟨X,Y ⟩ = tr[X∗Y ], where X∗ denotes the conjugate transpose of a complex matrixX. Similarly, we denote the positive semidefinite cone as Hn +, and its in- terior as Hn ++. In the remainder of this section, we provide some background on operator monotone and operator convex functions, as well as self-concordant barriers. 2.1 Operator monotonicity and convexity Consider a finite-dimensional real inner product space V and a proper convex cone K ⊂ V. We define the partial ordering x ⪰K y to mean x − y ∈ K for x, y ∈ V. When we omit the subscript, we refer to the Loewner ordering, i.e., we use X ⪰ Y to mean X − Y ∈ Hn + for X,Y ∈ Hn. We also define the dual cone K∗ of K as K∗ := {z ∈ V : ⟨z, x⟩ ≥ 0, ∀x ∈ K}. Now consider a second finite-dimensional real inner product space V′, a proper convex cone K′ ⊂ V′, and a function f : V′ → V. We say that f is (K′,K)- monotone if for all x, y ∈ dom f we have x ⪰K′ y =⇒ f(x) ⪰K f(y). We say that the function f is K-convex if for all x, y ∈ dom f we have λf(x) + (1− λ)f(y) ⪰K f(λx+ (1− λ)y), ∀λ ∈ [0, 1]. Similarly, we say that f is K-concave if −f is K-convex. Now consider a real-valued function g defined on the interval (a, b) where −∞ ≤ a < b ≤ ∞. We can extend this function to be defined on Hermitian matrices X ∈ Hn whose eigenvalues are in (a, b) as follows. If X has the spectral decomposition X = ∑n i=1 λiviv ∗ i where λi ∈ (a, b) for all i = 1, . . . , n, then we define g(X) = ∑n i=1 g(λi)viv ∗ i . Given this, we say that the function 8 Kerry He et al. g is operator monotone if for all positive integers n and matrices X,Y ∈ Hn with eigenvalues in (a, b), we have X ⪰ Y =⇒ g(X) ⪰ g(Y ), i.e., g is (Hn +,Hn +)-monotone for all positive integers n. We say that g is oper- ator convex if for all positive integers n and matrices X,Y ∈ Hn with eigen- values in (a, b), we have λg(X) + (1− λ)g(Y ) ⪰ g(λX + (1− λ)Y ), ∀λ ∈ [0, 1], i.e., g is Hn +-convex for all positive integers n. Similarly, we say g is operator concave if −g is operator convex. Important examples of these functions include x 7→ log(x), x 7→ xp for p ∈ [0, 1], and x 7→ −xp for p ∈ [−1, 0], which are all operator monotone and operator concave on (0,∞). Similarly, the function x 7→ xp for p ∈ (1, 2] is operator convex on (0,∞), but is not operator monotone. See, e.g., [4, Theorem 2.6], for a proof of these results. There is an elegant theory behind operator monotone and operator convex functions (see, e.g., [23,38]). One important result is Loewner’s theorem, which relates operator monotone functions to Pick functions, and which provides us with an integral representation for the class of operator monotone functions. Lemma 1 (Loewner’s Theorem) For a real-valued function g defined on the interval (a, b), where −∞ ≤ a < b ≤ ∞, the following statements are equivalent: (i) g is operator monotone on (a, b). (ii) g has an analytic continuation from (a, b) to the upper half-plane C+ := {z ∈ C : Im z > 0} that maps C+ into C+ (i.e., g is a Pick function). (iii) g has the following integral representation g(x) = α+ βx+ ∫ R\(a,b) 1 s− x − s s2 + 1 dµ(s), ∀x ∈ (a, b), where α ∈ R, β ≥ 0, and µ is a positive finite Borel measure on R \ (a, b). It is also well known that operator convex functions have a similar integral representation. Lemma 2 ([23, Theorem 2.7.6]) Let g be an operator convex function de- fined on the interval (−1, 1). Then there exists a unique positive finite Borel measure µ on [−1, 1] such that g(x) = g(0) + g′(0)x+ 1 2 g′′(0) ∫ 1 −1 x2 1− sx dµ(s), ∀x ∈ (−1, 1). Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 9 We also recall that the noncommutative perspective of a function g : (0,∞) → R is defined as Pg(X,Y ) := X 1 2 g ( X− 1 2Y X− 1 2 ) X 1 2 , on the domain Hn ++ × Hn ++. A remarkable fact about the non-commutative perspective is that if g : (0,∞) → R is operator concave then Pg is (jointly) Hn +-concave for all positive integers n [8, Theorem 2.2]. We also recall that the noncommutative perspective satisfies the identity Pg(X,Y ) = Pĝ(Y,X) for all X,Y ∈ Hn ++, where ĝ(x) = xg(1/x) is the transpose of g [26, Lemma 2.1]. 2.2 Self-concordant barriers For a finite-dimensional real inner product space V, consider a C3, closed, strictly convex function F with open domain domF := {x ∈ V : F (x) < ∞}. We say that F is self-concordant if |D3F (x)[h, h, h]| ≤ 2(D2F (x)[h, h])3/2, for all x ∈ domF and h ∈ V, where DkF (x)[h1, . . . , hk] = ∂k ∂t1 · · · ∂tk ∣∣∣∣ t1=...=tk=0 F (x+ t1h1 + · · ·+ tkhk), denotes the k-th directional derivative at x along the directions h1, . . . , hk. Since domF is open and the epigraph of F is closed, F is a barrier function for cl domF , as any sequence xk ∈ domF converging to the boundary of domF satisfies F (xk) → ∞ [34, Theorem 5.13]. Additionally, F is a ν-self-concordant barrier if 2DF (x)[h]− D2F (x)[h, h] ≤ ν, for all x ∈ domF and h ∈ V. When domF is a convex cone, we say that F is ν-logarithmically homogeneous if F (tx) = F (x)− ν log(t), for all x ∈ domF and t > 0. If F is a ν-logarithmically homogeneous self- concordant barrier, then it is also a ν-self-concordant barrier [34, Lemma 5.4.3]. Similar to [13], the main technique we will use to construct logarithmically homogeneous self-concordant barriers is to use compatibility of functions with respect to their domains, which we define below. Note that, by convention, compatibility is defined in terms of concave functions rather than convex func- tions. Because of this, in what follows we will typically formulate our general results in terms of concave functions and their hypographs. 10 Kerry He et al. Definition 2 ([35, Definition 5.1.1]) Let V and V′ be finite-dimensional real inner product spaces, let K ⊂ V′ be a closed, convex cone, and let f : dom f → V′ be a K-concave C3 function with open domain dom f ⊂ V. Then f is (K, β)-compatible with the domain cl dom f if there exists β ≥ 0 such that D3f(x)[h, h, h] ⪯K −3βD2f(x)[h, h], for all x ∈ dom f and h ∈ V such that x± h ∈ cl dom f . Once we have established compatibility of a function, we can use the following result to construct a self-concordant barrier for the hypograph of the function. Lemma 3 ([34, Theorem 5.4.4]) Let V and V′ be finite-dimensional real inner product spaces, and let K ⊂ V′ be a closed, convex cone. Let f : dom f → V′ be a K-concave C3 function with open domain dom f ⊂ V, and which is (K, β)-compatible with cl dom f . Let G be a ν-self-concordant barrier for cl dom f , and H be an η-self-concordant barrier for K. Then (t, x) 7→ H(f(x)− t) + β3G(x), defined on the domain V′ × dom f is an (η + β3ν)-self-concordant barrier for the set cl hypo f = cl{(t, x) ∈ V′ × dom f : t ⪯K f(x)}. Proof Using the notation from [34, Theorem 5.4.4], let ξ = f , E1 = V, E2 = E3 = V′, Q = dom f , Q2 = {(y, z) ∈ V′ × V′ : y ⪰K z}, Φ(y, z) = H(y − z), and F = G. Clearly, any element of K × {0} is a recession direction for Q2, and additionally cl hypo f = cl{(z, x) ∈ V′ × dom f : ∃ y ∈ V′, f(x) ⪰K y, (y, z) ∈ Q2}, which is enough to recognize that [34, Theorem 5.4.4] implies our desired result. ⊓⊔ 3 Operator concavity and self-concordance In this section, we present the main technical result of the paper which estab- lishes a relationship between operator concavity along lines and compatibility. This is summarized in the following theorem. Note that this strengthens the statement in Theorem 1 as, by considering functions which are not necessarily scalar-valued, it applies in greater generality and leads to a stronger conclu- sion. We begin by introducing a generalization of operator convexity along lines to vector-valued functions before stating our theorem. Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 11 Definition 3 Let V and V′ be finite-dimensional real inner product spaces, let K ⊂ V′ be a proper, convex cone, and let the dual cone of K be denoted as K∗ ⊂ V′. Let f : dom f → V′ be a function with open convex domain dom f ⊂ V. We say that f is K-operator convex along lines if for all z ∈ K∗, x ∈ dom f and h ∈ V such that x± h ∈ cl dom f , the function t 7→ ⟨z, f(x+ th)⟩, is operator convex on (−1, 1). We say that f is K-operator concave along lines if −f is K-operator convex along lines. Theorem 4 Let V and V′ be finite-dimensional real inner product spaces, and let K ⊂ V′ be a proper, convex cone. Let f : dom f → V′ be a C3 function with open domain dom f ⊂ V. If f is K-operator concave along lines, then f is (K, 1)-compatible with respect to cl dom f . Proof Let F : (−1, 1) → R be defined as F (t) = ⟨z, f(x+ th)⟩, for any z ∈ K∗, x ∈ dom f and h ∈ V such that x ± h ∈ cl dom f . It follows from K-operator concavity along lines of f that F is operator concave. We first show that f is K-concave. Since F is operator concave, and there- fore concave in the standard sense, it follows that ⟨z,D2f(x)[h, h]⟩ = F ′′(0) ≤ 0. As this holds for all z ∈ K∗, then D2f(x)[h, h] ⪯K 0 for all x ∈ dom f and h ∈ V such that x± h ∈ cl dom f . By scaling h by an arbitrary constant, and using the fact that K is full-dimensional, we extend this result to hold for all h ∈ V, which shows that f is K-concave, see, e.g., [35, Lemma 5.1.2]. We now prove the desired compatibility result. Given that F is operator concave on (−1, 1), we can use Lemma 2 to show that F has the following integral representation F (t) = F (0) + F ′(0)t+ 1 2 F ′′(0) ∫ 1 −1 t2 1− st dµ(s), ∀ t ∈ (−1, 1), for a unique Borel probability measure µ on [−1, 1]. Let us define the integrand ξs : (−1, 1) → R by ξs(t) = t2 1− st , for s ∈ [−1, 1]. A straightforward computation shows that ξ′s(t) = 2t− st2 (1− st)2 , ξ′′s (t) = 2 (1− st)3 , ξ′′′s (t) = 6s (1− st)4 , (5) and therefore ξ′′′s (0) = 6s ≥ −6 = −3ξ′′s (0), (6) 12 Kerry He et al. for all s ∈ [−1, 1]. Next, we recognize that ⟨z,Dkf(x)[h, . . . , h]⟩ = F (k)(0) = 1 2 F ′′(0) dk dtk ∣∣∣∣ t=0 ∫ 1 −1 ξs(t) dµ(s) = 1 2 F ′′(0) ∫ 1 −1 ξ(k)s (0) dµ(s), (7) for k = 2 and k = 3. We can exchange the order of differentiation and in- tegration using a similar argument to [13, Theorem A.3], as follows. Using the dominated convergence theorem (see, e.g., [2, Corollary 5.9]), it suffices to show for k = 1, 2 and 3 that there are constants ε > 0 and Ck > 0 such that |ξ(k)s (t)| ≤ Ck for all (t, s) ∈ [−ε, ε]× [−1, 1]. Using the expressions (5), we can concretely set C1 = 2ε+ ε2 (1− ε)2 , C2 = 2 (1− ε)3 , C3 = 6 (1− ε)4 , for any 0 < ε < 1. This satisfies the assumptions required for us to apply the dominated convergence theorem, and thus we can repeatedly swap the order of differentiation and integration to obtain (7). Finally, noting that F ′′(0) ≤ 0 as F is concave, we combine (6) and (7) to show that ⟨z,D3f(x)[h, h, h]⟩ ≤ −3⟨z,D2f(x)[h, h]⟩. As this holds for all z ∈ K∗, it follows that D3f(x)[h, h, h] ⪯K −3D2f(x)[h, h]. This is true for all x ∈ dom f and h ∈ V such that x±h ∈ cl dom f , which allows us to obtain the desired result by appealing to the definition of compatibility. ⊓⊔ Note that Theorem 1, which we introduced earlier in Section 1, is a simple corollary of this result. Proof of Theorem 1 This follows directly from Theorem 4 and Lemma 3. ⊓⊔ We also introduce some composition rules for operator convexity along lines and linear maps, which follow as relatively simple consequences of the definition. Proposition 1 Let U, V, V′ and V′′ be finite-dimensional real inner product spaces, and let K′ ⊂ V′ and K′′ ⊂ V′′ be two proper convex cones. Let f : dom f ⊂ V → V′ have open domain dom f ⊂ V, and be K′-operator convex along lines. (i) Let Φ : V′ → V′′ be a positive linear map (i.e., Φ(K′) ⊆ K′′). Then Φ ◦ f is K′′-operator convex along lines. Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 13 (ii) Let Ξ : U → V be a linear map satisfying imΞ ∩ dom f ̸= ∅. Then f ◦ Ξ with domain Ξ−1(dom f) := {x ∈ U : Ξ(x) ∈ dom f} is K′-operator convex along lines. Proof To show (i), we need to show that for any z ∈ K′′ ∗ , x ∈ dom f and h ∈ V such that x± h ∈ cl dom f , the function t 7→ ⟨z, Φ(f(x+ th))⟩ = ⟨Φ∗(z), f(x+ th)⟩, is operator convex on (−1, 1), where Φ∗ denotes the adjoint of the linear map Φ. The desired result directly follows from K′-operator convexity along lines of f , combined with the fact that the adjoint of a positive linear map is also positive, i.e., Φ∗(K′′ ∗) ⊆ K′ ∗. Similarly, to show (ii), we need to show that for any w ∈ K′ ∗, y ∈ Ξ−1(dom f) and u ∈ U such that y ± u ∈ clΞ−1(dom f), the function t 7→ ⟨w, f(Ξ(y + tu))⟩ = ⟨w, f(Ξ(y) + tΞ(u)))⟩, is operator convex on (−1, 1). This also follows directly from K′-operator convexity along lines of f , combined with the fact that Ξ(y) ∈ dom f and Ξ(y) ± Ξ(u) ∈ cl dom f (as clΞ−1(dom f) ⊆ Ξ−1(cl dom f) [33, Theorem 18.1]). ⊓⊔ As an important example of how we can apply these ideas, we can recover all of the results from [13], which studies noncommutative perspective func- tions, by showing that the noncommutative perspective belongs to the class of functions which are operator concave along lines. Proposition 2 Let g : (0,∞) → R be an operator concave function and let Pg : Hn ++ × Hn ++ → Hn be its non-commutative perspective. Then Pg is Hn +- operator concave along lines for all positive integers n. Proof See Section 4.2. ⊓⊔ Applying Theorem 4 then allows us to recover all of the compatibility results from [13]. 4 Proof of Theorems 2 and 3 To prove these results, we will make use of the following operator convexity along lines results. Item (i) of Lemma 4 is established in Section 4.1. Item (ii) of Lemma 4 is established in Section 4.2. Item (iii) of Lemma 4 is established in Section 4.3. Lemma 4 For any positive integer n, the following statements hold: (i) For α ∈ [ 12 , 1], the function Ψα is operator concave along lines. (ii) For α ∈ [1, 2], the function Ψα is operator convex along lines. (iii) For α ∈ [ 12 , 1), the function Dα is operator convex along lines. 14 Kerry He et al. Self-concordance of the barrier functions in Theorems 2 and 3 are then a direct consequence of Theorem 1. Optimality of the barrier parameters in Theorem 2 follow directly from [13, Corollary 3.13], while optimality of the barrier parameter in Theorem 3 is proven in Appendix A.2. We remark that although the statements of item (i) and (ii) are very sim- ilar, the proofs we provide differ quite significantly. This is reflective of the fact that some of the proofs for the convexity or concavity of Ψα can also be very different for different ranges of α, as discussed in Section 1.2. In fact, our proofs for operator convexity and concavity along lines follow very similar ideas to these convexity proofs, which we discuss in more detail in the following sections. 4.1 Proof of Lemma 4(i) To prove that Ψα is operator concave along lines for α ∈ [ 12 , 1], we follow a similar complex analysis approach as [22,24,25] which was used to prove concavity of Ψα for the same range of α. The key idea we use is that if F (t) is the restriction of Ψα to a line, then its transpose tF (1/t) turns out to be a Pick function. This allows us to show that Ψα is operator concave along lines by using Loewner’s theorem. Concretely, we are required to show that F (t) := Ψα(X + tH, Y + tV ) = tr [( (Y + tV ) 1−α 2α (X + tH)(Y + tV ) 1−α 2α )α] , (8) is operator concave on the interval (−1, 1) for all X,Y ∈ Hn ++ and H,V ∈ Hn satisfying X ± H ⪰ 0 and Y ± V ⪰ 0. To do this, we first show that the transpose of F , i.e., F̂ (t) := tF (1/t), is operator monotone by using the following lemma. Lemma 5 Let X,Y ∈ Hn ++ and H,V ∈ Hn be any Hermitian matrices satis- fying X ±H ⪰ 0 and Y ± V ⪰ 0, and let α ∈ [ 12 , 1]. Consider the function F̂ (t) := tr [( (tY + V ) 1−α 2α (tX +H)(tY + V ) 1−α 2α )α] , defined on the interval (1,∞). Then F̂ has an analytic continuation to the upper half-plane C+ := {z ∈ C : Im z > 0} that maps C+ into C+. Proof This is an intermediate result in the proof of [24, Theorem 2.1], where, in the notation of [24], Φ and Ψ are the identity operators, p = (1 − α)/α, q = 1, and s = α. ⊓⊔ Given this result, Loewner’s theorem implies that F̂ is also operator mono- tone on (1,∞). We then use the following lemma to show that F is operator concave on (0, 1). Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 15 Lemma 6 Let f be an operator monotone function on (γ,∞) for some γ > 0, and let f̂(x) = xf(1/x) be the transpose of f . Then f̂ is operator concave on (0, 1/γ). Proof See Appendix A.3. ⊓⊔ By swapping out H,V for −H,−V , an identical argument shows that F is also operator concave on (−1, 0). Finally, we use the following lemma to extend operator concavity of F to t = 0. Lemma 7 Let a, b, c be real numbers satisfying a < b < c, and let f : (a, c) → R be a C2n function. If f is Hn +-convex on (a, b) and Hn +-convex on (b, c), then f is also Hn +-convex (a, c). Proof See Appendix A.4. ⊓⊔ As F is C∞, we can apply the above lemma for all positive integers n, and conclude that F is operator concave on the entire domain (−1, 1). This concludes the proof. 4.2 Proof of Lemma 4(ii) To prove that Ψα is operator convex along lines for α ∈ [1, 2], we use similar ideas as those used in [32,40] to prove joint convexity of Ψα for this range of α. In particular, we use the fact that Ψα can be expressed as a suitable compo- sition of noncommutative perspectives and linear maps. We then show, using a combination of composition rules for operator convexity, that this composi- tion of noncommutative perspectives and linear maps is operator convex along lines. To do this, we first focus on proving a more general result about operator convexity along lines. We will use ⊕ to denote the direct sum of two matrices, and ⊗ to denote the Kronecker product of two matrices, i.e., for any two matrices X ∈ Cn×m and Y ∈ Cp×q, we have X ⊕ Y = [ X 0 0 Y ] , and X ⊗ Y = x11Y . . . x1mY ... . . . ... xn1Y . . . xnmY  , where xij denotes the (i, j)-th entry of X. An important property of the Kro- necker product is that if X,Y ∈ Hn are any two Hermitian matrices with diagonalizations X = U1Λ1U ∗ 1 and Y = U2Λ2U ∗ 2 (where U1, U2 are unitary matrices and Λ1, Λ2 are real diagonal matrices) then X ⊗ Y is also Hermitian with the diagonalization X ⊗ Y = (U1 ⊗ U2)(Λ1 ⊗ Λ2)(U1 ⊗ U2) ∗. 16 Kerry He et al. Using this, it is also easy to verify for any p ∈ R that (X ⊗ Y )p = Xp ⊗ Y p. We are now ready to present the following general theorem showing that certain families of multivariate matrix concave functions are operator concave along lines. Theorem 5 Let fn : (Hn ++) d → Hn be a family of functions indexed by a positive integer n. Let fn satisfy the following two properties: – Respects direct sums, i.e., fn+m(X1 ⊕ Y1, . . . , Xd ⊕ Yd) = fn(X1, . . . , Xd)⊕ fm(Y1, . . . , Yd), for all d-tuples of n × n Hermitian matrices Xi and m × m Hermitian matrices Yi, and positive integers n and m. – Respects simultaneous unitaries, i.e., fn(UX1U ∗, . . . , UXdU ∗) = Ufn(X1, . . . , Xd)U ∗, for all n × n unitaries U , d-tuples of n × n Hermitian matrices Xi, and positive integers n. If fn is Hn +-concave for all positive integers n, then fn is Hn +-operator concave along lines for all positive integers n. Remark 2 Functions satisfying assumptions like those in Theorem 5 arise nat- urally in non-commutative or free probability, operator theory, and systems theory. In the language, for instance, of [21], the disjoint union of all posi- tive definite cones (of all sizes) is an example of a non-commutative domain. Moreover, a family of functions defined on a non-commutative domain, that is compatible with direct sums and similarities, is a free mapping. Proof Our task is to show that F (t) := tr[A · fn(X1 + tH1, . . . , Xd + tHd)], is operator concave on (−1, 1) for all A ∈ Hn +, Xi ∈ Hn ++ and Hi ∈ Hn satisfying Xi ± Hi ⪰ 0 for i = 1, . . . , d, and positive integers n. To prove this, we will show that the extension of F to matrix arguments is Hm + -concave on the set of m × m Hermitian matrices with eigenvalues in (−1, 1), for all positive integers m. We do this by showing that, for any matrix T ∈ Hm with eigenvalues in (−1, 1), the matrix F (T ) can be expressed in the form F (T ) = Ξ(fnm(X1(T ), . . . ,Xd(T ))), (9) where Xi(T ) are affine maps for i = 1, . . . , d, and Ξ is a positive linear map. The desired result then follows from Hnm + -concavity of fnm. We now establish that F has a representation of the form (9). Consider a matrix T ∈ Hm with eigenvalues in (−1, 1) and diagonalization T = UΛU∗ (where U is unitary and Λ is diagonal and real). Let us denote Xi(T ) = Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 17 I ⊗Xi + T ⊗Hi and Xi(λ) = Xi + λHi for i = 1, . . . , d. Using this notation, note that Xi(T ) = (U ⊗ I)Xi(Λ)(U ∗ ⊗ I), and additionally that Xi(Λ) = Xi(λ1) . . . Xi(λm)  . Now using the two key properties of fn outlined in the statement of the theo- rem, we can show that fnm(X1(T ), . . . ,Xd(T )) = (U ⊗ I)fnm(X1(Λ), . . . ,Xd(Λ))(U ∗ ⊗ I) = (U ⊗ I) fn(X1(λ1), . . . , Xd(λ1)) . . . fn(X1(λm), . . . , Xd(λm))  (U∗ ⊗ I), where the first equality uses compatibility with simultaneous unitaries, and the second equality uses compatibility with direct sums. Finally, consider the linear map Ξ : Hnm → Hm defined by Ξ(M) = n∑ i=1 µi(I⊗ v∗i )M(I⊗ vi), where A has the eigendecomposition A = ∑n i=1 µiviv ∗ i . This is the linear map which applies the operation Mij 7→ tr[AMij ] to each n× n block of an m×m block matrix. Note that this is a positive linear map as A ∈ Hn +, which implies that µi ≥ 0 for all i = 1, . . . , n. It is also relatively straightforward to confirm that Ξ((U ⊗ I)M(U∗ ⊗ I)) = UΞ(M)U∗, for any unitary matrix U . Using these properties of Ξ, we can see that Ξ(fnm(X1(T ), . . . ,Xd(T ))) = U F (λ1) . . . F (λm) U∗ = UF (Λ)U∗ = F (T ). Thus, we have shown that T 7→ F (T ) is the composition between an Hnm + - concave function and a positive linear map from Hnm to Hm, and therefore that T 7→ F (T ) is Hm + -concave on m×m Hermitian matrices with eigenvalues in (−1, 1). As this is true for any positive integer m, we conclude that F is operator concave on (−1, 1), as desired. ⊓⊔ We next introduce the following composition rule involving noncomutative perspectives. 18 Kerry He et al. Lemma 8 Consider the functions g : (0,∞) → R and h : domh → Hn ++ where domh is a convex set. If g is operator concave and operator monotone, and h is Hn +-concave, then the function (X, y) 7→ Pg(X,h(y)), defined on Hn ++ × domh is jointly Hn +-concave. Proof See Appendix A.5. ⊓⊔ We are now ready to state a corollary of Theorem 5 which deals with noncommutative perspectives, and which we will use to prove Lemma 4(ii). Corollary 1 For the functions g : (0,∞) → R and h : (0,∞) → (0,∞), consider the composed noncommutative perspective function Pg,h(X,Y, Z) := Pg(X,Ph(Y,Z)), defined on the domain Hn ++ ×Hn ++ ×Hn ++. If either – g is operator concave, and h is affine, or – g is operator concave and operator monotone, and h is operator concave, then Pg,h is Hn +-operator concave along lines for all positive integers n. Proof First, it is relatively straightforward to confirm that the noncommu- tative perspective, and therefore the composed noncommutative perspective Pg,h, respects direct sums and simultaneous unitaries as required in Theo- rem 5. Therefore, by appealing to this theorem, it suffices to show that Pg,h is jointly Hn +-concave for all positive integers n under the given assumptions. If h is affine, i.e., of the form h(x) = ax+b, then Pg,h(X,Y, Z) = Pg(X, aY +bZ) and joint Hn +-concavity of Pg,h follows immediately from joint Hn +-concavity of Pg when g is operator concave. When h is operator concave and g is both operator concave and operator monotone, Ph is jointly Hn +-concave, and there- fore joint concavity of Pg,h follows directly from Lemma 8. This concludes the proof. ⊓⊔ Proof of Lemma 4(ii) Using Corollary 1 with g(x) = −x1−α and h(x) = x 1 α , we conclude that Pg,h is operator concave along lines. By considering the positive linear map Φ : Hn2 → R which satisfies Φ(X ⊗ Ȳ ) = tr[XY ] for all X,Y ∈ Hn, where Ȳ denotes the elementwise conjugate of the matrix Y , a Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 19 straightforward computation shows that Φ(Pg,h(X ⊗ I, Y ⊗ I, I⊗ Ȳ )) = Φ(Pĝ(Ph(Y ⊗ I, I⊗ Ȳ ), X ⊗ I)) = Φ ( Pĝ ( (Y 1 2 ⊗ I)(Y −1 ⊗ Ȳ ) 1 α (Y 1 2 ⊗ I), X ⊗ I )) = Φ ( Pĝ ( Y α−1 α ⊗ Ȳ 1 α , X ⊗ I )) = −Φ (( Y α−1 2α ⊗ Ȳ 1 2α )( Y 1−α 2α XY 1−α 2α ⊗ Ȳ − 1 α )α ( Y α−1 2α ⊗ Ȳ 1 2α )) = −Φ ( Y α−1 2α ( Y 1−α 2α XY 1−α 2α )α Y α−1 2α ⊗ Ȳ 1−α α ) = −Ψα(X,Y ), where we have used the identity Pg(X,Y ) = Pĝ(Y,X) and ĝ(x) = xg(1/x) = −xα in the first line. Therefore, using Proposition 1 gives the desired operator convexity along lines result. ⊓⊔ We can also use Theorem 5 to prove Proposition 2 and recover the existing results from [13]. Proof of Proposition 2 This follows directly from Theorem 5 by recognizing that the noncommutative perspective respects direct sums and simultaneous unitaries, and g being operator concave implies Pg is Hn +-concave for all posi- tive integers n. ⊓⊔ 4.3 Proof of Lemma 4(iii) The proof of operator convexity along lines of Dα for α ∈ [ 12 , 1) is a simple extension of the proof for Lemma 4(i). We need to show that G(t) := Dα(u+ ts;X + tH ∥Y + tV ) = u+ ts α− 1 log ( Ψα ( X + tH u+ ts , Y + tV u+ ts )) , is operator convex on (−1, 1) for all u ∈ R++, X,Y ∈ Hn ++, s ∈ R, and H,V ∈ Hn which satisfy u± s ≥ 0, X ±H ⪰ 0, and Y ±V ⪰ 0. To prove this, we first use the fact that Ψα is homogeneous of degree one to show that G(t) = u+ ts α− 1 log ( F (t) u+ ts ) = 1 α− 1 Plog(u+ ts, F (t)), where F is defined in (8), and which we proved was operator concave on (−1, 1) in Section 4.1. By applying Lemma 8 where g(x) = log(x) and h(x) = F (x), it follows that the extension of G to matrix arguments, i.e., G(T ) = 1 α− 1 Plog(uI+ sT, F (T )) where T ∈ Hm has eigenvalues in (−1, 1), is Hm + -convex for any positive integer m. Therefore, we conclude that G is operator convex on (−1, 1). 20 Kerry He et al. Remark 3 By setting u = 1 and s = 0, a straightforward corollary of this result is that the (non-homogenized) sandwiched Rényi entropy Dα is also operator convex along lines. This can be used to construct a (1 + 2n)-self- concordant barrier for cl epiDα by appealing to Theorem 1. However, as Dα is not homogeneous of degree one, this epigraph is not a cone, and optimality of the barrier parameter cannot be easily established using [13, Corollary 3.13]. Remark 4 By combining the ideas discussed in this section, together with the ideas presented in the proofs of Corollary 1, Proposition 2, and Corollary [13, Corollary 1.8], we can also show that the (perspective) of the Rényi entropy D̂α is operator convex along lines for α ∈ [0, 1). Therefore, suitable barrier functions for its epigraph can also be constructed by appealing to Theorem 1. 5 Implementation We implement the barrier function for the epigraphs and hypographs proposed in Theorems 2 and 3 in the primal-dual interior-point solver QICS [20]. These cones are accessible using the optimization modeling software PICOS [37]. In Section 5.1, we give some details about the derivative oracles required for these barriers. In Section 5.2, we present some numerical experiments to evaluate the performance of using the proposed barrier to solve optimization problems involving the sandwiched Rényi entropy. 5.1 Derivatives We provide brief derivations of expressions for the first and second derivatives of the trace function Ψg,h(X,Y ) := tr [ g ( h(Y ) 1 2Xh(Y ) 1 2 )] , (10) defined on Hn ++ × Hn ++, for twice continuously differentiable functions g : (0,∞) → R and h : (0,∞) → R. Note that Ψα(X,Y ) = Ψg,h(X,Y ) when g(x) = xα and h(x) = x 1−α α . The derivatives of the barrier functions proposed in Theorem 2 are then a straightforward consequence of these results. We refer the reader to [20, Section 4.1.1] for a discussion of how to obtain derivatives for the barriers from the expressions presented in this section. First, by using the fact thatAB andBA share the same nonzero eigenvalues for any A ∈ Cn×m and B ∈ Cm×n [28, Theorem 1.32], we can show that Ψg,h(X,Y ) = tr [ g ( X 1 2h(Y )X 1 2 )] , for all X,Y ∈ Hn ++. Using this expression together with the original expres- sion (10), the first derivatives of Ψg,h are relatively easy to derive using the Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 21 chain rule [28, Theorem 3.4] combined with the derivative for trace functions, D(tr ◦ g)(X)[H] = ⟨H, g′(X)⟩, see, e.g., [27, Theorem 3.23]. Doing this gives DXΨg,h(X,Y )[H] = 〈 h(Y ) 1 2Hh(Y ) 1 2 , g′ ( h(Y ) 1 2Xh(Y ) 1 2 )〉 DY Ψg,h(X,Y )[V ] = 〈 X 1 2Dh(Y )[V ]X 1 2 , g′ ( X 1 2h(Y )X 1 2 )〉 , where X,Y ∈ Hn ++ and H,V ∈ Hn. From these expressions, the second deriva- tives can be found by using the chain rule together with the product rule [28, Theorem 3.3] to obtain D2 XXΨg,h(X,Y )[H1, H2] = 〈 h(Y ) 1 2H1h(Y ) 1 2 , D(g′) ( h(Y ) 1 2Xh(Y ) 1 2 )[ h(Y ) 1 2H2h(Y ) 1 2 ]〉 D2 Y Y Ψg,h(X,Y )[V1, V2] = 〈 X 1 2Dh(Y )[V1]X 1 2 , D(g′) ( X 1 2h(Y )X 1 2 )[ X 1 2Dh(Y )[V2]X 1 2 ]〉 + 〈 X 1 2D2h(Y )[V1, V2]X 1 2 , g′ ( X 1 2h(Y )X 1 2 )〉 . whereX,Y ∈ Hn ++ andH1, H2, V1, V2 ∈ Hn. Although the derivative D2 XY Ψg,h(X,Y )[H,V ] is not as straightforward to derive, we show how to do this using a similar method to the proof of [20, Lemma 4.2]. Lemma 9 Consider the trace function Ψg,h defined in (10), where g : (0,∞) → R is twice continuously differentiable, and h : (0,∞) → R is continuously dif- ferentiable. If g̃(x) = xg′(x), X,Y ∈ Hn ++, and H,V ∈ Hn, then D2 XY Ψg,h(X,Y )[H,V ] = 〈 Dh(Y )[V ], h(Y )− 1 2Dg̃ ( h(Y ) 1 2Xh(Y ) 1 2 )[ h(Y ) 1 2Hh(Y ) 1 2 ] h(Y )− 1 2 〉 . Proof First, assume that g′ is a function of the form g′(x) = xp for any positive integer p. Then DY Ψg,h(X,Y )[V ] = 〈 Dh(Y )[V ], X 1 2 ( X 1 2h(Y )X 1 2 )p X 1 2 〉 = 〈 Dh(Y )[V ], h(Y )− 1 2 ( h(Y ) 1 2Xh(Y ) 1 2 )p+1 h(Y )− 1 2 〉 = 〈 Dh(Y )[V ], h(Y )− 1 2 g̃ ( h(Y ) 1 2Xh(Y ) 1 2 ) h(Y )− 1 2 〉 . Taking the derivative of this expression in the variable X and the direction H gives the desired result when g′(x) = xp for a positive integer p. By linearity, this is also true when g is any polynomial function. This result is then extended to all twice continuously differentiable functions g by using a similar continuity argument as [3, Theorem V.3.3]. ⊓⊔ The third derivatives of Ψg,h can similarly be derived by using the chain and product rules on the second derivative expressions, which we omit the details of for berevity. Concrete expressions for the derivatives of the spectral functions Dk(g′)(X)[H] and Dkh(Y )[V ] can be found in, e.g., [27, Theorem 3.33]. 22 Kerry He et al. 5.2 Numerical experiments Here, we provide numerical experiments using our implementation of our bar- rier functions in QICS to solve optimization problems involving the sand- wiched Rényi entropy function. Note that in all experiments, we reformulate minimization of the sandwiched Rényi entropy as optimizing the sandwiched quasi-relative entropy Ψα (see the discussion in Section 1.1). In the following, we define the partial traces tr1 : Hn2 → Hn and tr2 : Hn2 → Hn to be the unique linear maps which satisfy tr1(X ⊗ Y ) = tr[X]Y, tr2(X ⊗ Y ) = tr[Y ]X. for all X,Y ∈ Hn. Note that the partial traces are the adjoint operators of the linear maps X 7→ I⊗X and X 7→ X ⊗ I, respectively, where we recall that ⊗ denotes the Kronecker product. Sandwiched Rényi mutual information First, we consider computing the sand- wiched Rényi mutual information, which is defined as the optimal value of min X∈Hn Dα(A∥ tr2(A)⊗X) subj. to tr[X] = 1, X ⪰ 0, (11) for some matrix A ∈ Hn2 + . When α → 1, it is known that the minimum is attained at X∗ = tr1(A). However, in general a closed-form expression for the sandwiched Rényi mutual information is not known. Instead, [17, Lemma 5] shows that the minimum is attained at an X∗ ∈ Hn + which is the unique fixed-point of the following map X∗ = tr1(Z(X∗))/ tr[Z(X∗)], (12) where Z(X∗) = ( (tr2(A)⊗X∗) 1−α 2α A(tr2(A)⊗X∗) 1−α 2α )α . We confirm this by solving (11) using our implementation of the barrier func- tions in Theorem 2 in QICS. The results from running these experiments are summarized in Table 2. Quantum rate-distortion We next present some experiments to approximate the solution of quantum relative entropy programs by using the fact that the (normalized) sandwiched Rényi entropy converges to the quantum relative en- tropy as α → 1. In particular, we estimate the optimal rate-distortion tradeoff for the maximally entangled state, which for a given constant 0 ≤ δ ≤ 1 is given as the solution to min X∈Hn2 Dα(X ∥I⊗ tr1(X)) subj. to tr2(X) = 1 n I, δ ≥ ⟨X,∆⟩, X ⪰ 0, (13) Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 23 Table 2 Results of computing the Sandwiched Rényi mutual information (11) for randomly generated unit trace n2 × n2 Hermitian matrices A. The reported residual is the Frobenius norm of the residual matrix of (12). Also shown is the code snippet used to solve (11) using QICS and PICOS. Note that the constraint X ⪰ 0 is implied by the domain of picos.sandquasientr. import numpy , picos # Generate problem data n, alpha = 4, 0.75 A = numpy.random.rand(n*n, 2*n*n).view(complex) A = A @ A.conj().T # Define problem P = picos.Problem () X = picos.HermitianVariable("X", n) tr2_A = picos.partial_trace(A, 1, (n, n)) obj = picos.sandquasientr(A, tr2_A @ X, alpha) P.set_objective("max", obj) P.add_constraint(picos.trace(X) == 1) # Solve problem P.solve(solver="qics") n α Time (s) Residual 4 0.75 0.09 4.3× 10−8 4 1.50 0.12 3.5× 10−9 8 0.75 3.40 1.7× 10−7 8 1.50 5.68 8.3× 10−10 16 0.75 693.50 1.8× 10−8 16 1.50 1557.06 2.3× 10−10 as α → 1, where ∆ = I− 1 n n∑ i=1 n∑ j=1 eie ⊤ j ⊗ eie ⊤ j , and {ei ∈ Rn}ni=1 represent the standard basis. In [18, Theorem 4.16], it was shown that the optimal value of this problem when α → 1 is log(n) + (1− δ) log(1− δ) + δ log ( δ n2 − 1 ) , (14) whenever 0 ≤ δ ≤ 1 − 1/n2, and is zero otherwise. We verify this by solv- ing (13) for values of α which converge to 1. These problems are solved using our implementation of the barrier functions in Theorem 2 in QICS, and we summarize these results in Table 3. Note that the sandwiched Rényi entropy is monotone in α for fixed matrix arguments [32, Theorem 7], which is reflected in these results. 6 Conclusion In this paper, we have established a close relationship between compatibility and operator concavity of a function, and used this result to prove the self- concordance of natural logarithmic barrier functions of the hypograph of Ψα for α ∈ [ 12 , 1] and the epigraph of Ψα for α ∈ [1, 2]. This allows us to solve optimization problems involving the sandwiched Rényi entropy using interior- point methods. 24 Kerry He et al. Table 3 Results of estimating the optimal rate-distortion tradeoff of the maximally mixed state for n = 4 and δ = 0.25 by solving (13) for values of α converging to 1. The optimal value reported for α = 1, shown in bold, is obtained from the closed form expression (14). Also shown is the code snippet used to solve (13) using QICS and PICOS. Note that the constraint X ⪰ 0 is implied by the domain of picos.sandquasientr. import numpy , picos # Generate problem data n, alpha , delta = 4, 0.99, 0.25 Delta = numpy.eye(n*n) Delta [::n+1, ::n+1] -= 1/n # Define problem P = picos.Problem () X = picos.SymmetricVariable("X", n*n) tr1_X = picos.partial_trace(X, 0, (n, n)) tr2_X = picos.partial_trace(X, 1, (n, n)) obj = picos.sandquasientr(X, picos.I(n) @ tr1_X , alpha) P.set_objective("max", obj) P.add_constraint(tr2_X == picos.I(n) / n) P.add_constraint (( X | Delta ) <= delta) # Solve problem P.solve(solver="qics") α Optimal value 0.9 0.0027555 0.99 0.1332757 0.999 0.1455750 0.9999 0.1467922 1 0.1469467 1.0001 0.1470813 1.001 0.1481874 1.01 0.1604453 1.1 0.2740472 Whether there exists a self-concordant barrier for the epigraph of Ψα for α ∈ (2,∞) with optimal barrier parameter 1 + 2n remains an open question. Since xα for α ∈ (2,∞) is not operator convex [23] it follows that Ψα is not operator convex along lines for α ∈ (2,∞). Therefore we cannot appeal to Theorem 1 to construct a self-concordant barrier for the epigraph of Ψα for α ∈ (2,∞). An alternative approach could be to directly try to establish compatibility properties of Ψα for α ∈ (2,∞). For the scalar case, we can prove the following compatibility result. Proposition 3 For α ∈ [2,∞), the function (x, y) 7→ −xαy1−α is (R+, (2α− 1)/3)-compatible with the domain R+ × R+. Proof See Appendix A.6. ⊓⊔ This compatibility parameter is tight in the sense that it cannot be improved, and so serves as a lower bound for the compatibility parameter for Ψα for α ∈ [2,∞). Based on numerical experiments, we believe this compatibility result extends to the situation when Ψα is defined on matrices of any dimension. Conjecture 1 For α ∈ [2,∞), the function −Ψα is (Hn +, (2α−1)/3)-compatible with the domain Hn + ×Hn + for any positive integer n. If this were true, then we could still use Lemma 3 to construct self-concordant barriers for the epigraph of Ψα for α ∈ [2,∞), albeit with a barrier parameter which increases with increasing α. As an alternative, it may be possible to find Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 25 a lifted representation (see, e.g., [11]) for the epigraph of Ψα which admits a barrier function with barrier parameter that does not grow with α. Another possible extension of our results is to consider the general class of trace functions Ψp,q,s(X,Y ) = tr [( Y q 2XpY q 2 )s] , for p ≥ q and s > 0. This generalizes the sandwiched Rényi entropy when p = 1, q = 1−α α , and s = α, and the Rényi entropy when p = α, q = 1 − α, and s = 1. The convexity properties of Ψp,q,s for the full range of p, q and s are summarized in [42], which we repeat below for convenience. (i) Ψp,q,s is jointly concave for 0 ≤ q ≤ p ≤ 1 and 0 < s ≤ 1 p+q . (ii) Ψp,q,s is jointly convex for −1 ≤ q ≤ p ≤ 0 and s > 0. (iii) Ψp,q,s is jointly convex for −1 ≤ q ≤ 0, 1 ≤ p < 2, (p, q) ̸= (1,−1), and s ≥ 1 p+q . An interesting question is, to which range of parameters p, q and s do our compatibility results and techniques extend, thus allowing us to give self- concordant barriers for this larger class of functions. For scenario (i) and the subset of scenario (ii) where −1 ≤ q ≤ p ≤ 0 and 0 < s ≤ − 1 p+q , we can prove that Ψp,q,s is (R+, 1)-compatible with respect to the domain Hn +×Hn + by using virtually the same proof as for Lemma 4(i) presented in Section 4.1. The only modification required is to use the insights from the proof of [25, Theorem 2.1] (see, also, [4, Section 3]) in place of Lemma 5. It remains an open question how to construct self-concordant barriers for the remaining scenarios, i.e., the subset of (ii) where −1 ≤ q ≤ p ≤ 0 and s > − 1 p+q , and scenario (iii). We note that scenario (iii) contains the α ∈ [1,∞) range as a special case, and therefore a subset of this range may be amenable to a generalization of our approach in Section 4.2. Funding This research was supported in part by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (project DP250104201). K. He was supported by an Australian Government Research Training Program (RTP) Scholarship. H. Fawzi was partially funded by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee EP/X032051/1. Declarations Conflicts of interest The authors have no relevant financial or non-financial interests to disclose. 26 Kerry He et al. A Auxiliary proofs A.1 Proof of Remark 1 For some x′ ∈ dom f and h′ ∈ V, let us define the function F : (a, b) → R, F (t) = f(x′ + th′), where a ∈ R ∪ {−∞}, b ∈ R ∪ {∞}, and (a, b) = {t ∈ R : x′ + th′ ∈ dom f}. If F is operator convex on (a, b) for every x′ ∈ dom f and h′ ∈ V, then it is clear that the condition from Definition 1 is also satisfied (as it only requires F be operator convex for a subset of possible x′ and h′, and functions which are operator convex on an interval are clearly also operator convex on any sub-interval). To show the opposite direction, let us first assume that a, b ∈ R, i.e., F is defined on a finite interval. If f satisfies Definition 1, then we can show that F is operator convex on (a, b) by choosing x = x′ + (a + b)h′/2 and h = (b − a)h′/2, applying the affine transform t 7→ (a+ b− 2t)/(a− b), and using the fact that operator convexity is preserved under affine transforms. Now consider the case where domF = (a,∞), and consider a sequence {bi}∞i=0 satisfying bi = a+ i for every i ∈ N. The domain domF can be represented as the union of overlapping open intervals domF = ⋃∞ i=0(bi, bi+2). If f satisfies Definition 1, then from the argument in the previous paragraph, the function F is operator convex on (bi, bi+2) for every i ∈ N. Using [38, Theorem 9.3], we conclude that F is operator convex on the union of these intervals, i.e., on the entire domain domF . The cases where domF = (−∞, b) and domF = R follow from nearly identical argu- ments, which concludes the proof. A.2 Proof of optimality of barrier parameter in Theorem 3 Here, we prove that any barrier for cl epiDα must have parameter at least 2 + 2n by using the same technique as the proof of [13, Corollary 3.13]. To do this, we first introduce the following result. Lemma 10 ([13, Proposition 3.11]) Let n be a positive integer, and let h : Rn ++ → R be convex and positively homogeneous of degree one. Then any self-concordant barrier for cl epih has barrier parameter at least 1 + n. Now consider the function h : R++ × Rn ++ × Rn ++ defined by h(u, x, y) = Dα(u; diag(x)∥ diag(y)), where diag(x) is the diagonal matrix with diagonal elements given by x. As Dα is convex and positively homogeneous of degree one, so is h, and therefore Lemma 10 tells us that any self-concordant barrier for cl epih has barrier parameter at least 2 + 2n. This implies that any self-concordant barrier (t, u,X, Y ) 7→ F (t, u,X, Y ) for cl epih must also have parameter at least 2+2n, as otherwise (t, u, x, y) 7→ F (t, u, diag(x),diag(y)) would be a self-concordant barrier for cl epih with parameter less than 2 + 2n, which completes the proof. A.3 Proof of Lemma 6 As f is operator monotone on (γ,∞), Loewner’s theorem (see Lemma 1) tells us that f has the integral representation f(x) = α+ βx+ ∫ γ −∞ 1 s− x − s s2 + 1 dµ(s), ∀x ∈ (γ,∞), Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 27 for some constants α ∈ R and β ≥ 0 and a positive Borel measure µ on (−∞, γ]. The transpose of f is therefore equal to f̂(x) = xf(1/x) = αx+ β + ∫ γ −∞ x2 sx− 1 − sx s2 + 1 dµ(s), ∀x ∈ (0, 1/γ). It suffices to show that the integrand is operator concave on the domain (0, 1/γ), for each s ∈ (−∞, γ]. When s = 0, the integrand is a negative quadratic, which is operator concave. When s ̸= 0, we have x2 sx− 1 = x s + 1 s2 + 1 s2 1 sx− 1 . Since x 7→ x−1 is operator convex on (0,∞), it follows that h(x) := (sx− 1)−1 is operator concave on (−∞, 1/s) if s ≥ 0, and is operator concave on (1/s,∞) if s ≤ 0. Therefore, h is operator concave on (0, 1/γ) for each s ∈ (−∞, 0)∪ (0, γ], and therefore the entire integrand is operator concave on (0, 1/γ) for each s ∈ (−∞, γ], from which the desired result follows. A.4 Proof of Lemma 7 We will use the following lemma to prove the desired result. Lemma 11 ([38, Theorem 9.2]) Let f be a C2n function on the interval (a, b), and let n be a fixed integer. Then f is Hn +-convex if and only if the Hansen-Tomiyama matrix Hn(x; f) with entries Hn(x; f)ij = f (i+j)(x) (i+ j)! , 1 ≤ i, j ≤ n, is positive semidefinite for all x ∈ (a, b). Using Lemma 11, we know the Hansen-Tomiyama matrix Hn(x; f) is positive semidef- inite for all x ∈ (a, b) ∪ (b, c). As f is C2n, all entries of the Hansen-Tomiyama matrix are continuous on (a, c), and therefore Hn(b; f) must also be positive semidefinite. Appealing to Lemma 11 again in the other direction gives the desired Hn +-convexity result. A.5 Proof of Corollary 8 Let X1, X2 ∈ Hn ++ and y1, y2 ∈ domh. For some λ ∈ [0, 1], let X = λX1 + (1 − λ)X2 and y = λy1 + (1− λ)y2. As h is Hn +-concave, we have by definition that h(y) ⪰ λh(y1) + (1− λ)h(y2). Using operator monotonicity of g, we can show that Pg(X,h(y)) = X 1 2 g ( X− 1 2 h(y)X− 1 2 ) X 1 2 ⪰ X 1 2 g ( X− 1 2 (λh(y1) + (1− λ)h(y2))X − 1 2 ) X 1 2 = Pg(X,λh(y1) + (1− λ)h(y2)). Finally, we use joint concavity of Pg to show Pg(X,h(y)) ⪰ λPg(X1, h(y1)) + (1− λ)Pg(X2, h(y2)), which shows that the desired function is jointly Hn +-concave, as required. 28 Kerry He et al. A.6 Proof of Proposition 3 First, it is straightforward to compute the second derivatives of f as ∂2f ∂x2 = −α(α− 1)xα−2y1−α, ∂2f ∂x∂y = α(α− 1)xα−1y−α, ∂2f ∂y2 = −α(α− 1)xαy−α−1, and the third derivatives of f as ∂3f ∂x3 = −α(α− 1)(α− 2)xα−3y1−α, ∂3f ∂x2∂y = α(α− 1)2xα−2y−α, ∂3f ∂x∂y2 = −α2(α− 1)xα−1y−α−1, ∂3f ∂y3 = α(α− 1)(α+ 1)xαy−α−2. Now consider x, y > 0 and h, v ∈ R which satisfy x ± h ≥ 0 and y ± v ≥ 0, and let us denote x̂ = h/x and ŷ = v/y, which satisfy −1 ≤ x̂, ŷ ≤ 1. The second and third directional derivatives can be expressed as D2f(x, y)[(h, v), (h, v)] = −α(α− 1)xαy1−α(x̂− ŷ)2 D3f(x, y)[(h, v), (h, v), (h, v)] = −α(α− 1)xαy1−α(x̂− ŷ)2((α− 2)x̂− (α+ 1)ŷ). Then using the fact that α > 2, we can show that (α− 2)x̂− (α+ 1)ŷ ≤ 2α− 1. Appealing to the definition of compatibility then gives the desired result. B Non-convexity of sandwiched quasi-relative entropy for α ∈ (0, 1 2 ) To show that Ψα is neither convex nor concave for α ∈ (0, 1 2 ), we first consider the matrices X = 1 2 [ 1 1 1 1 ] and Yy = [ y 0 0 1 ] , for y > 0. Noting that X2 = X, a direct calculation shows that Ψα(X,Yy) = tr [( X 1 2 Y 1−α α y X 1 2 )α] = 1 2α ( y 1−α α + 1 )α . Now consider the function f(y) = (ya + 1)b on the domain (0,∞), where a = 1−α α and b = α. Another direct calculation shows that the second derivative of this function can be expressed as f ′′(y) = abya−2(ya + 1)b−2((ab− 1)ya + (a− 1)). When α ∈ (0, 1 2 ), we can see that ab− 1 = −α < 0 and a− 1 = 1 α − 2 > 0. Therefore, there exists some positive y at which f ′′(y) changes sign, and so f is neither convex nor concave. Since f is continuous, it follows that there exist y1, y2, y3, y4 ∈ (0,∞) such that f ( y1 + y2 2 ) > 1 2 (f(y1) + f(y2)) and yet f ( y3 + y4 2 ) < 1 2 (f(y3) + f(y4)) . We now consider the family of positive definite matrices Xϵ = X+ϵI for ϵ > 0. By continuity, we have that for sufficiently small ϵ > 0, Ψα ( Xϵ, 1 2 (Yy1 + Yy2 ) ) > 1 2 (Ψα(Xϵ, Yy1 ) + Ψα(Xϵ, Yy2 )) and yet Ψα ( Xϵ, 1 2 (Yy3 + Yy4 ) ) < 1 2 (Ψα(Xϵ, Yy3 ) + Ψα(Xϵ, Yy4 )) . This shows that Ψα is neither convex nor concave on Hn ++×Hn ++ for α ∈ (0, 1 2 ), as desired. Operator convexity along lines, self-concordance, and sandwiched Rényi entropies 29 References 1. 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In: 2022 IEEE International Symposium on Information Theory (ISIT), pp. 252–257. IEEE (2022) 42. Zhang, H.: From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture. Advances in Mathematics 365, 107053 (2020) Introduction Preliminaries Operator concavity and self-concordance Proof of Theorems 2 and 3 Implementation Conclusion Auxiliary proofs Non-convexity of sandwiched quasi-relative entropy for (0, 12)