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The additivity conjecture for the Weyl channels in the dimension d=3

Grigori Gennadievich Amosov

Published Jun 6, 2006• 0 citations
OpenAlex

Abstract

Following the method introduced in quant-ph/0408004 and quant-ph/0605177 we prove that the additivity conjecture for the Holevo-Schumacher-Westmoreland bound of the output of a quantum channel holds for the channels of the form $\\Phi (x)=\\sum \\limits _{m=0}^{2}(r_{m}W_{m,0}xW_{m,0}^{*}+p_{1}W_{m,1}xW_{m,1}^{*}+p_{2}W_{m,2}x W_{m,2}^{*})$. Here $x$ runs the set of states $\\sigma (H)$ in the Hilbert space $H, dimH=3,$ $W_{m,n}$ are the Weyl operators satisfying the relation $W_{m,n}W_{m',n'}=e^{\\frac {2\\pi i(m'n-mn')}{3}}W_{m+m',n+n'}, 0\\le m,n\\le 2,$ and $0\\le r_{m},p_{1},p_{2}\\le 1, \\sum \\limits _{m=0}^{2}r_{m}+3(p_{1}+p_{2})=1$. This class includes the quantum depolarizing channel as well as a number of the other channels being covariant with respect to the maximum commutative group of unitaries.

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Following the method introduced in quant-ph/0408004 and quant-ph/0605177 we prove that the additivity conjecture for the Holevo-Schumacher-Westmoreland bound of the output of a quantum channel holds for the channels of the form Phi(x)=sumlimitsm=02(rmWm,0xWm,0∗+p1Wm,1xWm,1∗+p2Wm,2xWm,2∗)\\Phi (x)=\\sum \\limits _{m=0}^{2}(r_{m}W_{m,0}xW_{m,0}^{*}+p_{1}W_{m,1}xW_{m,1}^{*}+p_{2}W_{m,2}x W_{m,2}^{*})Phi(x)=sumlimitsm=02​(rm​Wm,0​xWm,0∗​+p1​Wm,1​xWm,1∗​+p2​Wm,2​xWm,2∗​). Here xxx runs the set of states sigma(H)\\sigma (H)sigma(H) in the Hilbert space H,dimH=3,H, dimH=3,H,dimH=3, Wm,nW_{m,n}Wm,n​ are the Weyl operators satisfying the relation Wm,nWm′,n′=efrac2pii(m′n−mn′)3Wm+m′,n+n′,0lem,nle2,W_{m,n}W_{m',n'}=e^{\\frac {2\\pi i(m'n-mn')}{3}}W_{m+m',n+n'}, 0\\le m,n\\le 2,Wm,n​Wm′,n′​=efrac2pii(m′n−mn′)3Wm+m′,n+n′​,0lem,nle2, and 0lerm,p1,p2le1,sumlimitsm=02rm+3(p1+p2)=10\\le r_{m},p_{1},p_{2}\\le 1, \\sum \\limits _{m=0}^{2}r_{m}+3(p_{1}+p_{2})=10lerm​,p1​,p2​le1,sumlimitsm=02​rm​+3(p1​+p2​)=1. This class includes the quantum depolarizing channel as well as a number of the other channels being covariant with respect to the maximum commutative group of unitaries.
We estimate the von Neumann entropy infimum for the output of the Weyl channels being covariant with respect to the maximum commutative group. In qubit case, such the class includes the quantum depolarizing channel and the "two-Pauli" channel as well. For the dimesion d=2d=2d=2 our approach allows to prove the additivity conjecture for the Holevo-Schumacher-Westmoreland bound. Our method is based upon using the decreasing property of the relative entropy. Earlier we had shown that the method gives the proof of the additivity conjecture for the quantum depolarizing channel in any dimension.
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.
We investigate the Weyl channels being covariant with respect to the maximum commutative group of unitary operators. This class includes the quantum depolarizing channel and the “two-Pauli” channel as well. Then, we show that our estimation of the output entropy for a tensor product of the phase damping channel and the identity channel based upon the decreasing property of the relative entropy allows to prove the additivity conjecture for the minimal output entropy for the quantum depolarizing channel in any prime dimension and for the two-Pauli channel in the qubit case.
We study a natural generalization of the additivity problem in quantum information theory: given a pair of quantum channels, then what is the set of convex trace functions that attain their maximum on unentangled inputs if they are applied to the corresponding output state? We prove several results on the structure of the set of those convex functions that are ``additive'' in this more general sense. In particular, we show that all operator convex functions are additive for the Werner-Holevo channel in $3\ifmmode\times\else\texttimes\fi{}3$ dimensions, which contains the well-known additivity results for this channel as special cases.
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.