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arXiv:1306.5358 · Journal of Mathematical Physics

Monotonicity of a relative Rényi entropy

Rupert L. Frank, Élliott H. Lieb

Published Dec 1, 2013• 218 citations
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Abstract

We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al. [“On quantum Rényi entropies: A new definition, some properties,” J. Math. Phys. 54, 122203 (2013); e-print arXiv:1306.3142v1; see also e-print arXiv:1306.3142].

DOI
10.1063/1.4838835

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Carlen-Frank-Lieb conjecture and monotonicity of $\alpha-z$ R\'enyi relative entropy

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In this paper we study the joint convexity/concavity of the trace functions \[ \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any fixed invertible matrix. We will give full range of $(p,q,s)\in\mathbb{R}^3$ for $\Psi_{p,q,s}$ to be jointly convex/concave for all $K$. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of $(\alpha,z)$ for $\alpha$-$z$ Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of $\Psi_{p,0,1/p}$ for $0<p<1$ which was first proved by Epstein using complex analysis. The key is to reduce the problem to the joint convexity/concavity of the trace functions \[ \Psi_{p,1-p,1}(A,B)=\text{Tr} K^*A^{p}KB^{1-p},~~-1\le p\le 1, \] using a variational method.

•Nov 3, 2018•1 citation
We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al. [“On quantum Rényi entropies: A new definition, some properties,” J. Math. Phys. 54, 122203 (2013); e-print arXiv:1306.3142v1; see also e-print arXiv:1306.3142].
Itis well known that in quantum information quantum relative is monotonically decreasing under the completely positive and trace-preserving maps. For a new proposed sandwiched R\'enyi quantum relative, a map that is linear trace-preserving and whose Hilber-Schmidt adjoint map satisfies Schwarz inequality, monotonicity still holds. We give a new proof of monotonicity of sandwiched R\'enyi relative entropy for α∈[12,1)\alpha\in[\frac{1}{2},1)α∈[21​,1). The proof is based on complex interplotation techniques, which already has been used to prove monotonicity under trace-preserving and positive map for α∈(1,∞)\alpha\in(1,\infty)α∈(1,∞).
Abstract The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the Rényi α-relative entropy, which present in [Phys. Rev. A 94 (2016) 052336]. We show that the Rényi α-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement, which presents in [Phys. Rev. A 93 (2016) 032136]. Due to the Rényi α-relative entropy of coherence can act as a coherence monotone quantifier, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of Rényi 2-relative entropy of coherence are derived .
Carlen-Frank-Lieb conjecture and monotonicity of α−z\alpha-zα−z R\'enyi relative entropy
In this paper we study the joint convexity/concavity of the trace functions Ψp,q,s(A,B)=Tr(Bq2K∗ApKBq2)s,  p,q,s∈R, \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, Ψp,q,s​(A,B)=Tr(B2q​K∗ApKB2q​)s,  p,q,s∈R, where AAA and BBB are positive definite matrices and KKK is any fixed invertible matrix. We will give full range of (p,q,s)∈R3(p,q,s)\in\mathbb{R}^3(p,q,s)∈R3 for Ψp,q,s\Psi_{p,q,s}Ψp,q,s​ to be jointly convex/concave for all KKK. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (α,z)(\alpha,z)(α,z) for α\alphaα-zzz Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψp,0,1/p\Psi_{p,0,1/p}Ψp,0,1/p​ for 0<p<10<p<10<p<1 which was first proved by Epstein using complex analysis. The key is to reduce the problem to the joint convexity/concavity of the trace functions Ψp,1−p,1(A,B)=TrK∗ApKB1−p,  −1≤p≤1, \Psi_{p,1-p,1}(A,B)=\text{Tr} K^*A^{p}KB^{1-p},~~-1\le p\le 1, Ψp,1−p,1​(A,B)=TrK∗ApKB1−p,  −1≤p≤1, using a variational method.