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Hacettepe Journal of Mathematics and Statistics

The matrix Heinz mean and related divergence

Trung Hoa Dinh, Anh Vu LE, Công Trình Lê, Ngoc Yen Phan

Published Nov 29, 2021• 4 citations
Publisher

Abstract

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 $0< \alpha <1$.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 $p$-power mean $\mu_p(t, A, B) = ((1-t)A^p + tB^p)^{1/p}$ satisfies the in-betweenness property with respect to the new divergence.

DOI
10.15672/hujms.902879

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In this paper, we introduce a new quantum divergenceΦ(X,Y)=\Tr[(1−αα+α1−α)X+2Y−X1−αYαα−XαY1−α1−α],\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],Φ(X,Y)=\Tr[(α1−α​+1−αα​)X+2Y−αX1−αYα​−1−αXαY1−α​],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.
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.
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.
It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χκ2 divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states ρ and σ does not increase under the action of any quantum channel E . For a fixed channel E and a state σ , the divergence between output states E(ρ) and E(σ) might be strictly smaller than the divergence between input states ρ and σ , which is characterized by the strong data processing inequality (SDPI). Among various input states ρ , the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum χκ1/22 divergence for arbitrary quantum channels and also for a family of χκ2 divergences (with κ≥κ1/2 ) for arbitrary quantum-classical channels.
In information theory, the data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is given by the strong data processing inequality (SDPI) for a fixed channel and a full-rank reference state. In classical information theory, the strong data processing inequality tensorizes for a large family of fff-divergences. However, its quantum counterpart remains largely unexplored. Tensorization of SDPI has previously been shown only for some special cases, and whether it holds more generally for quantum divergences still remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.