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Linear and Multilinear Algebra

On new quantum divergences

Trung Hoa Dinh, Hong Du, A. N. D. Nguyen, Trung-Dung Vuong

Published May 8, 2023• 7 citations
Publisher

Abstract

In this paper, we introduce new quantum divergences of the form Φ(A,B)=Tr(AσB−AτB),A,B>0,where σ and τ are Kubo–Ando operator means such that σ≥τ. More precisely, we show that Φ(A,B) is a quantum divergence when σ is the weighted Kubo–Ando matrix power mean and τ is the weighted geometric mean. In addition, we construct a new quantum Hellinger-type divergence using the linear approximation of the function x. We also study the least-squares problem, the data processing inequality, and in-betweenness property of the matrix means with respect to the quantum Hellinger-type divergence.

DOI
10.1080/03081087.2023.2209272

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In this paper, we introduce new quantum divergences of the form Φ(A,B)=Tr(AσB−AτB),A,B>0,where σ and τ are Kubo–Ando operator means such that σ≥τ. More precisely, we show that Φ(A,B) is a quantum divergence when σ is the weighted Kubo–Ando matrix power mean and τ is the weighted geometric mean. In addition, we construct a new quantum Hellinger-type divergence using the linear approximation of the function x. We also study the least-squares problem, the data processing inequality, and in-betweenness property of the matrix means with respect to the quantum Hellinger-type divergence.
In this paper, we study a matrix equation involving the matrix geometric mean A♮tB=A1/2(A−1/2BA−1/2)tA1/2,A\natural_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2},A♮t​B=A1/2(A−1/2BA−1/2)tA1/2, t∈(1,2)t\in(1,2)t∈(1,2) . We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean A♮tBA\natural_{t}BA♮t​B to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality.
In this paper, we introduce a new quantum divergence$$\Phi (X,Y) = \Tr \left[\left(\dfrac{1-\alpha}{\alpha}+ \dfrac{\alpha}{1-\alpha}\right)X+2Y - \dfrac{X^{1 -\alpha}Y^{\alpha}}{\alpha}- \dfrac{X^{\alpha}Y^{1-\alpha}}{1-\alpha} \right],$$where 0<α<10< \alpha <10<α<1.We study the least square problem with respect to this divergence. We also show that the new quantum divergence satisfies the Data Processing Inequality in quantum information theory. In addition, we show that the matrix ppp-power mean μp(t,A,B)=((1−t)Ap+tBp)1/p\mu_p(t, A, B) = ((1-t)A^p + tB^p)^{1/p}μp​(t,A,B)=((1−t)Ap+tBp)1/p satisfies the in-betweenness property with respect to the new divergence.
Quantum Algorithms for Computing Maximal Quantum fff-divergence and Kubo-Ando means
The development of quantum computation has resulted in many quantum algorithms for a wide array of tasks. Recently, there is a growing interest in using quantum computing techniques to estimate or compute quantum information-theoretic quantities such as Renyi entropy, Von Neumann entropy, matrix means, etc. Motivated by these results, we present quantum algorithms for computing the maximal quantum fff-divergences and the operator-theoretic matrix Kubo--Ando means. Both of them involve Renyi entropies, matrix means as special cases, thus implying the universality of our framework.
Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality ∥A+B+2rA♯1−αB∥p≤∥A+B+2rQα,z(A,B)∥p, holds for p = 1, 2, α=1,z, and r≥0, among other related inequalities.
Let Pn\mathbb{P}_nPn​ be the cone of n×nn\times nn×n complex positive definite matrices. For 0<θ<10<θ<10<θ<1 and r≥0r\ge0r≥0, let Mr(θ)M_r^{(θ)}Mr(θ)​ denote the weighted matrix power mean, with the logarithmic endpoint at r=0r=0r=0, defined by Mr(θ)(A,B)={((1−θ)Ar+θBr)1/r,r>0,exp⁡((1−θ)log⁡A+θlog⁡B),r=0. M_r^{(θ)}(A,B) = \begin{cases} \big((1-θ)A^r+θB^r\big)^{1/r}, & r>0,\\[2mm] \exp\big((1-θ)\log A+θ\log B\big), & r=0. \end{cases} Mr(θ)​(A,B)=⎩⎨⎧​((1−θ)Ar+θBr)1/r,exp((1−θ)logA+θlogB),​r>0,r=0.​ For p>q≥0p>q\ge0p>q≥0, we introduce Φp,q(θ)(A,B)=Tr⁡[Mp(θ)(A,B)−Mq(θ)(A,B)]. Φ_{p,q}^{(θ)}(A,B) = \operatorname{Tr}\big[M_p^{(θ)}(A,B)-M_q^{(θ)}(A,B)\big]. Φp,q(θ)​(A,B)=Tr[Mp(θ)​(A,B)−Mq(θ)​(A,B)]. We prove that Φp,q(θ)Φ_{p,q}^{(θ)}Φp,q(θ)​ is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for 0≤q≤1≤p≤2,p>q, 0\le q\le1\le p\le2,\qquad p>q, 0≤q≤1≤p≤2,p>q, it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.
Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies DqxD^{q_x}Dqx​. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all DqxD^{q_x}Dqx​ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the DqxD^{q_x}Dqx​ are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.