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On Strong Superadditivity of the Entanglement of Formation

Audenaert, K M R, Braunstein, S L

Published Mar 10, 2003• 58 citations
OpenAlex

Abstract

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation $E_F$ and the conjugate function $E^*$ of the entanglement function $E(\\rho)=S(\\trace_A\\rho)$. We then consider the conjectured strong superadditivity of the entanglement of formation $E_F(\\rho) \\ge E_F(\\rho_I)+E_F(\\rho_{II})$, where $\\rho_I$ and $\\rho_{II}$ 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.

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Strong superadditivity of the entanglement of formation follows from its additivity

A. A. Pomeransky

The additivity of both the entanglement of formation and the classical channel capacity is known to be a consequence of the strong superadditivity conjecture. We show that, conversely, the strong superadditivity conjecture follows from the additivity of the entanglement of formation; this means that the two conjectures are equivalent and that the additivity of the classical channel capacity is a consequence of them.

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On the additivity conjecture for channels with arbitrary constraints

Максим Евгеньевич Широков

Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the $χ$-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the $χ$-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the $χ$-function and strong superadditivity of the entanglement of formation.

arXiv:quant-ph/0308168•Aug 29, 2003•3 citations

Superadditivity of the convex closure of the output entropy of a quantum channel

Максим Евгеньевич Широков

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

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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

On strong superadditivity for a class of quantum channels

Grigori Gennadievich Amosov

Given a quantum channel $Φ$ in a Hilbert space $H$ put $\hat H_Φ(ρ)=\min \limits_{ρ_{av}=ρ}Σ_{j=1}^{k}π_{j}S(Φ(ρ_{j}))$, where $ρ_{av}=Σ_{j=1}^{k}π_{j}ρ_{j}$, the minimum is taken over all probability distributions $π=\{π_{j}\}$ and states $ρ_{j}$ in $H$, $S(ρ)=-Trρ\logρ$ is the von Neumann entropy of a state $ρ$. The strong superadditivity conjecture states that $\hat H_{Φ\otimes Ψ}(ρ)\ge \hat H_Φ(Tr_{K}(ρ))+\hat H_Ψ(Tr_{H}(ρ))$ 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 $\hat H_{Φ\otimes Ψ}(ρ)$ for the special class of Weyl channels $Φ$ of the form $Φ=Ξ\circ Φ_{dep}$, where $Φ_{dep}$ is the quantum depolarizing channel and $Ξ$ is the phase damping is given.

arXiv:quant-ph/0610098•Oct 12, 2006•0 citations

The additivity problem for constrained quantum channels

Александр Семенович Холево, Максим Евгеньевич Широков

One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:

Russian Mathematical Surveys•Apr 30, 2004•0 citations
We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation EFE_FEF​ and the conjugate function E∗E^*E∗ of the entanglement function E(rho)=S(traceArho)E(\\rho)=S(\\trace_A\\rho)E(rho)=S(traceA​rho). We then consider the conjectured strong superadditivity of the entanglement of formation EF(rho)geEF(rhoI)+EF(rhoII)E_F(\\rho) \\ge E_F(\\rho_I)+E_F(\\rho_{II})EF​(rho)geEF​(rhoI​)+EF​(rhoII​), where rhoI\\rho_IrhoI​ and rhoII\\rho_{II}rhoII​ are the reductions of rho\\rhorho to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E∗E^*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 ppp-norms for ppp tending to 1.
Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the χχχ-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the χχχ-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the χχχ-function and strong superadditivity of the entanglement of formation.
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
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.
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.
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity: