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.
Given a quantum channel \ensuremathΦ in a Hilbert space H, set H\ifmmode^\elseˆ\fi\ensuremathΦ(\ensuremathρ)=min\ensuremathρav=\ensuremathρ\ensuremathΣj=1k\ensuremathπjS(\ensuremathΦ(\ensuremathρj)), where \ensuremathρav=\ensuremathΣj=1k\ensuremathπj\ensuremathρj, the minimum is taken over all probability distributions \ensuremathπ=\ensuremathπj and states \ensuremathρj in H, and S(\ensuremathρ)=\ensuremath−Tr\ensuremathρlog\ensuremathρ is the von Neumann entropy of a state \ensuremathρ. The strong superadditivity conjecture states that H\ifmmode^\elseˆ\fi\ensuremathΦ\ensuremath⨂\ensuremathΨ(\ensuremathρ)\ensuremath⩾H\ifmmode^\elseˆ\fi\ensuremathΦ(TrK(\ensuremathρ))+H\ifmmode^\elseˆ\fi\ensuremathΨ(TrH(\ensuremathρ)) for two channels \ensuremathΦ and \ensuremathΨ in Hilbert spaces H and K, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in any dimensions.
Given a quantum channel Φ in a Hilbert space H put H^Φ(ρ)=ρav=ρminΣj=1kπjS(Φ(ρj)), where ρav=Σj=1kπjρj, the minimum is taken over all probability distributions π={πj} and states ρj in H, S(ρ)=−Trρlogρ is the von Neumann entropy of a state ρ. The strong superadditivity conjecture states that H^Φ⊗Ψ(ρ)≥H^Φ(TrK(ρ))+H^Ψ(TrH(ρ)) for two channels Φ and Ψ in Hilbert spaces H and K, 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) for the special class of Weyl channels Φ of the form Φ=Ξ∘Φdep, where Φdep is the quantum depolarizing channel and Ξ is the phase damping is given.
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.
One of the main recent achievements of quantum information theory [1] is the proof of the equivalence of several (sub-, super-) additivity conjectures for finite-dimensional quantum channels and systems [2] (see the survey in [3]). The main purpose of this paper is to give a generalization of this result to the infinite-dimensional case. Let H and H ′ be separable Hilbert spaces. A quantum channel is a linear completely positive trace-preserving map Φ: T(H ) 7→ T(H ′), where T(H ) is the ideal of all trace class operators on H . In particular, Φ generates an affine map of the convex set S(H ) of states (that is, density operators) on the space H to the set S(H ′) of states on the space H ′ [1]. Important characteristics of a quantum channel are the output entropy HΦ(ρ) = H(Φ(ρ)), a lower-semicontinuous concave function on the input state space with values in [0,+∞], and its convex closure (see [4]), denoted by b HΦ(ρ) and called the b H-function of the channel Φ. It was shown in [5] that the convex closure of the output entropy of an arbitrary quantum channel Φ is given by the expression b HΦ(ρ) = inf μ Z
We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation EF and the conjugate function E∗ of the entanglement function E(rho)=S(traceArho). We then consider the conjectured strong superadditivity of the entanglement of formation EF(rho)geEF(rhoI)+EF(rhoII), where rhoI and rhoII are the reductions of rho to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E∗. As an application, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for all non-trace-preserving quantum channels, when purity is measured by Schatten p-norms for p tending to 1.