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arXiv:2109.10211

On the strong superadditivity of entanglement of formation -- Conditions and examples

Asutosh Kumar

Published Sep 19, 2021• 0 citations
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Abstract

From Shor's [Comm. Math. Phys. 246(3), 453 (2004)] and Hastings's [Nature Phys. 5, 255 (2009)] studies, the strong superadditivity of entanglement of formation is, in general, not true. In this paper, we provide conditions for strong superadditivity of entanglement of formation, and consequently identify classes of states that are strongly superadditive.

DOI
10.48550/arxiv.2109.10211

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From Shor's [Comm. Math. Phys. 246(3), 453 (2004)] and Hastings's [Nature Phys. 5, 255 (2009)] studies, the strong superadditivity of entanglement of formation is, in general, not true. In this paper, we provide conditions for strong superadditivity of entanglement of formation, and consequently identify classes of states that are strongly superadditive.
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.
Employing the quantum R\'enyi $\ensuremath{\alpha}$ entropies as a measure of entanglement, we numerically find the violation of the strong superadditivity inequality for a system composed of four qubits and $\ensuremath{\alpha}>1$. This violation gets smaller as $\ensuremath{\alpha}\ensuremath{\rightarrow}1$ and vanishes for $\ensuremath{\alpha}=1$ when the measure corresponds to the entanglement of formation. We show that the R\'enyi measure aways satisfies the standard monogamy of entanglement for $\ensuremath{\alpha}=2$, and only violates a high-order monogamy inequality, in the rare cases in which the strong superadditivity is also violated. The sates numerically found where the violation occurs have special symmetries where both inequalities are equivalent. We also show that every measure satisfying monogamy for high-dimensional systems also satisfies the strong superadditivity inequality. For the case of R\'enyi measure, we provide strong numerical evidences that these two properties are equivalent.
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.