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Quantum Information and Computation

The decreasing property of relative entropy and the strong superadditivity of quantum channels

Grigori Gennadievich Amosov, Stefano Mancini

Published Jul 1, 2009• 11 citations
Publisher

Abstract

We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.

DOI
10.5555/2011814.2011817

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The decreasing property of relative entropy and the strong superadditivity of quantum channels

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The decreasing property of relative entropy and the strong superadditivity of quantum channels

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On estimating the output entropy of a tensor product of the quantum phase-damping channel with an arbitrary channel

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Strong superadditivity conjecture holds for the quantum depolarizing channel in any dimension

Grigori Gennadievich Amosov

Given a quantum channel $\ensuremath{\Phi}$ in a Hilbert space $H$, set ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Phi}}(\ensuremath{\rho})={\mathrm{min}}_{{\ensuremath{\rho}}_{av}=\ensuremath{\rho}}{\ensuremath{\Sigma}}_{j=1}^{k}{\ensuremath{\pi}}_{j}S\mathbf{(}\ensuremath{\Phi}({\ensuremath{\rho}}_{j})\mathbf{)}$, where ${\ensuremath{\rho}}_{av}={\ensuremath{\Sigma}}_{j=1}^{k}{\ensuremath{\pi}}_{j}{\ensuremath{\rho}}_{j}$, the minimum is taken over all probability distributions $\ensuremath{\pi}={{\ensuremath{\pi}}_{j}}$ and states ${\ensuremath{\rho}}_{j}$ in $H$, and $S(\ensuremath{\rho})=\ensuremath{-}\mathrm{Tr}\ensuremath{\rho}\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}\ensuremath{\rho}$ is the von Neumann entropy of a state $\ensuremath{\rho}$. The strong superadditivity conjecture states that ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Phi}\ensuremath{\bigotimes}\ensuremath{\Psi}}(\ensuremath{\rho})\ensuremath{\geqslant}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Phi}}\mathbf{(}{\mathrm{Tr}}_{K}(\ensuremath{\rho})\mathbf{)}+{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Psi}}\mathbf{(}{\mathrm{Tr}}_{H}(\ensuremath{\rho})\mathbf{)}$ for two channels $\ensuremath{\Phi}$ and $\ensuremath{\Psi}$ in Hilbert spaces $H$ and $K$, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in any dimensions.

arXiv:0707.1097•Physical Review A•Jun 18, 2007•22 citations
We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.
We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.
We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.
Given a quantum channel ΦΦΦ in a Hilbert space HHH put H^Φ(ρ)=min⁡ρav=ρΣj=1kπjS(Φ(ρj))\hat H_Φ(ρ)=\min \limits_{ρ_{av}=ρ}Σ_{j=1}^{k}π_{j}S(Φ(ρ_{j}))H^Φ​(ρ)=ρav​=ρmin​Σj=1k​πj​S(Φ(ρj​)), where ρav=Σj=1kπjρjρ_{av}=Σ_{j=1}^{k}π_{j}ρ_{j}ρav​=Σj=1k​πj​ρj​, the minimum is taken over all probability distributions π={πj}π=\{π_{j}\}π={πj​} and states ρjρ_{j}ρj​ in HHH, S(ρ)=−Trρlog⁡ρS(ρ)=-Trρ\logρS(ρ)=−Trρlogρ is the von Neumann entropy of a state ρρρ. The strong superadditivity conjecture states that H^Φ⊗Ψ(ρ)≥H^Φ(TrK(ρ))+H^Ψ(TrH(ρ))\hat H_{Φ\otimes Ψ}(ρ)\ge \hat H_Φ(Tr_{K}(ρ))+\hat H_Ψ(Tr_{H}(ρ))H^Φ⊗Ψ​(ρ)≥H^Φ​(TrK​(ρ))+H^Ψ​(TrH​(ρ)) for two channels ΦΦΦ and ΨΨΨ in Hilbert spaces HHH and KKK, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in prime dimensions. The estimation of the quantity H^Φ⊗Ψ(ρ)\hat H_{Φ\otimes Ψ}(ρ)H^Φ⊗Ψ​(ρ) for the special class of Weyl channels ΦΦΦ of the form Φ=Ξ∘ΦdepΦ=Ξ\circ Φ_{dep}Φ=Ξ∘Φdep​, where ΦdepΦ_{dep}Φdep​ is the quantum depolarizing channel and ΞΞΞ is the phase damping is given.
Given a quantum channel $\ensuremath{\Phi}$ in a Hilbert space HHH, set ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Phi}}(\ensuremath{\rho})={\mathrm{min}}_{{\ensuremath{\rho}}_{av}=\ensuremath{\rho}}{\ensuremath{\Sigma}}_{j=1}^{k}{\ensuremath{\pi}}_{j}S\mathbf{(}\ensuremath{\Phi}({\ensuremath{\rho}}_{j})\mathbf{)}$, where ${\ensuremath{\rho}}_{av}={\ensuremath{\Sigma}}_{j=1}^{k}{\ensuremath{\pi}}_{j}{\ensuremath{\rho}}_{j}$, the minimum is taken over all probability distributions $\ensuremath{\pi}={{\ensuremath{\pi}}_{j}}$ and states ${\ensuremath{\rho}}_{j}$ in HHH, and $S(\ensuremath{\rho})=\ensuremath{-}\mathrm{Tr}\ensuremath{\rho}\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}\ensuremath{\rho}$ is the von Neumann entropy of a state $\ensuremath{\rho}$. The strong superadditivity conjecture states that ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Phi}\ensuremath{\bigotimes}\ensuremath{\Psi}}(\ensuremath{\rho})\ensuremath{\geqslant}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Phi}}\mathbf{(}{\mathrm{Tr}}_{K}(\ensuremath{\rho})\mathbf{)}+{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\ensuremath{\Psi}}\mathbf{(}{\mathrm{Tr}}_{H}(\ensuremath{\rho})\mathbf{)}$ for two channels $\ensuremath{\Phi}$ and $\ensuremath{\Psi}$ in Hilbert spaces HHH and KKK, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in any dimensions.